reputation in continuous-time games - yale universityef253/2010_10_20_reputation.pdf · reputation...
TRANSCRIPT
Reputation in Continuous-Time Games�
Eduardo FaingoldDepartment of Economics
Yale University
Yuliy SannikovDepartment of Economics
Princeton University
October 15, 2010
Abstract
We study reputation dynamics in continuous-time games in which a large player (e.g. government)faces a population of small players (e.g. households) and the large player’s actions are imperfectlyobservable. The major part of our analysis examines the case in which public signals about the largeplayer’s actions are distorted by a Brownian motion and the large player is either a normal type, whoplays strategically, or a behavioral type, who is committed to playing a stationary strategy. We obtain aclean characterization of sequential equilibria using ordinary differential equations and identify generalconditions for the sequential equilibrium to be unique and Markovian in the small players’ posteriorbelief. We find that a rich equilibrium dynamics arises when the small players assign positive priorprobability to the behavioral type. By contrast, when it is common knowledge that the large player isthe normal type, every public equilibrium of the continuous-time game is payoff-equivalent to one inwhich a static Nash equilibrium is played after every history. Finally, we examine variations of themodel with Poisson signals and multiple behavioral types.
Keywords: Reputation, repeated games, incomplete information, continuous time.
JEL Classification: C70, C72.
�We are grateful to David K. Levine and two anonymous referees for exceptionally helpful comments and suggestions. We alsothank Daron Acemoglu, Martin Cripps, Kyna Fong, Drew Fudenberg, George J. Mailath, Eric Maskin, Stephen Morris, BernardSalanie, Paolo Siconolfi, Andrzej Skrzypacz, Lones Smith and seminar audiences at Bocconi, Columbia, Duke, Georgetown,the Institute for Advanced Studies at Princeton, UCLA, UPenn, UNC at Chapel Hill, Washington Univ. in St. Louis, Yale, theStanford Institute of Theoretical Economics, the Meetings of the Society for Economic Dynamics in Vancouver and the 18th
International Conference in Game Theory in Stony Brook for many insightful comments.
1
1 Introduction
Reputation plays an important role in long-run relationships. Firms can benefit from reputation to fightpotential entrants (Kreps and Wilson (1982), Milgrom and Roberts (1982)), to provide high quality toconsumers (Klein and Leffler (1981)), or to generate good returns to investors (Diamond (1989)). Reputa-tion can help time-inconsistent governments commit to non-inflationary monetary policies (Barro (1986),Cukierman and Meltzer (1986)), low capital taxation (Chari and Kehoe (1993), Celentani and Pesendorfer(1996)), and repayment of sovereign debt (Cole, Dow, and English (1995)). In credence goods markets,strategic concerns to avoid a bad reputation can create perverse incentives that lead to market breakdown(Ely and Valimaki (2003)).
We study reputation dynamics in repeated games between a large player (e.g. firm, government) anda population of small players (e.g. consumers, households) in which the actions of the large player areimperfectly observable. For example, the observed quality of a firm’s product may be a noisy outcome ofthe firm’s effort to maintain quality standards; the realized inflation rate may be a noisy signal of the centralbank’s target monetary growth. Our setting is a continuous-time analogue of the Fudenberg and Levine(1992) model. Specifically, we assume that a noisy signal about the large player’s actions is publiclyobservable and that the evolution of this signal is driven by a Brownian motion. The small players areanonymous and hence behave myopically in every equilibrium, acting to maximize their instantaneousexpected payoffs. We follow the incomplete information approach to reputation pioneered by Kreps andWilson (1982) and Milgrom and Roberts (1982), which assumes that the small players are uncertain asto which type of large player they face. Specifically, the large player can be either a normal type, whois forward looking and behaves strategically, or a behavioral type, who is committed to playing a certainstrategy. As usual, we interpret the small players’ posterior probability on the behavioral type as a measureof the large player’s reputation.
Recall that in discrete time two main limit results about reputation effects are known. First, in ageneral model with multiple behavioral and non-behavioral types, Fudenberg and Levine (1992) provideupper and lower bounds on the set of equilibrium payoffs of the normal type which hold in the limit ashe gets arbitrarily patient; when the public signals satisfy an identifiability condition and the stage-gamepayoffs satisfy a non-degeneracy condition, these asymptotic bounds are tight and equal the stage-gameStackelberg payoff. Second, Cripps, Mailath, and Samuelson (2004) show that in a wide range of repeatedgames the power of reputation effects is only temporary: in any equilibrium the large player’s type must beasymptotically revealed in the long run.1 However, apart from these two limit results (and their extensionsto various important settings), not much is known about equilibrium behavior in reputation games. Inparticular, in the important case in which the signal distribution has full support, the explicit constructionof even one sequential equilibrium appears to be an elusive, if not intractable, problem.
By contrast, we obtain a clean characterization of sequential equilibria for fixed discount rates usingordinary differential equations, by setting the model in continuous time and restricting attention to a singlebehavioral type. Using the characterization, we find that a rich equilibrium dynamics arises when the
1For this result, Cripps, Mailath, and Samuelson (2004) assume that: (i) the public monitoring technology has full support andsatisfies an identifiability condition; (ii) there is a single non-behavioral type and finitely many behavioral types; (iii) the actionof each behavioral type is not part of a static Nash equilibrium of the complete information game; (iv) the small players have aunique best-reply to the action of each behavioral type.
2
small players assign positive prior probability to the behavioral type, but not otherwise. Indeed, when thesmall players are certain that they are facing the normal type, we show that the only sequential equilibriaof the continuous-time game are those which yield payoffs in the convex hull of the set of static Nashequilibria. By contrast, the possibility of behavioral types gives rise to non-trivial intertemporal incentives.In this case, we identify conditions for the sequential equilibrium to be unique and Markovian in the smallplayers’ posterior belief and examine when a reputation yields a positive value for the large player. Then,under the equilibrium uniqueness conditions, we examine the impact of the large player’s patience onthe equilibrium strategies and obtain a reputation result for equilibrium behavior, which strengthens theconclusion of the more standard reputation results concerning equilibrium payoffs. Finally, we extendsome of our results to settings with multiple equilibria, Poisson signals and multiple behavioral types.
Our characterization relies on the recursive structure of repeated games with public signals and onstochastic calculus. Both in discrete time and in continuous time, sequential equilibria in public strategies—hereafter, public sequential equilbria—can be described by the stochastic law of motion of two state vari-ables: the large player’s reputation and his continuation value. The law of motion of the large player’sreputation is determined by Bayesian updating, while the evolution of his continuation value is charac-terized by promise keeping and incentive constraints. If and only if the evolution of the state variablessatisfies these restrictions (as well as the condition that the continuation values are bounded), there is apublic sequential equilibrium corresponding to these state variables. It follows that the set of equilib-rium belief-continuation value pairs can be characterized as the greatest bounded set inside which thestate variables can be kept while respecting Bayes’ rule and the promise keeping and incentive constraints(Theorem 2).2
In continuous time, it is possible to take a significant step further and use stochastic calculus to connectthe equilibrium law of motion of the state variables with the geometry of the set of equilibrium beliefs andpayoffs. This insight—introduced in Sannikov (2007) in the context of continuous-time games without un-certainty over types—allows us to characterize public sequential equilibria using differential equations.3
We first use this insight to identify an interesting class of reputation games that have a unique sequen-tial equilibrium. Our main sufficient condition for uniqueness is stated in terms of a family of auxiliaryone-shot games in which the stage-game payoffs of the large player are adjusted by certain “reputationalweights.” When these auxiliary one-shot games have a unique Bayesian Nash equilibrium, we show thatthe reputation game must also have a unique public sequential equilibrium, which is Markovian in thelarge player’s reputation and characterized by a second-order ordinary differential equation (Theorem 4).4
We also provide sufficient conditions in terms of the primitives of the game, i.e. stage-game payoffs, driftand volatility (Proposition 4).
We use our characterization of unique Markovian equilibria to derive a number of interesting resultsabout behavior. First, we show that when the large player’s static Bayesian Nash equilibrium payoffincreases in reputation, his sequential equilibrium payoff in the continuous-time game also increases inreputation. Second, while the normal type of large player benefits from imitating the behavioral type, in
2In discrete time, this is a routine extension of the recursive methods of Abreu, Pearce, and Stacchetti (1990) to repeatedgames with uncertainty over types.
3See also Sannikov (2008) for an application of this methodology to agency problems.4Recall that in a Markov perfect equilibrium (cf. Maskin and Tirole (2001)) the equilibrium behavior is fully determined by
the payoff-relevant state variable, which, in our reputation game, is the small players’ posterior belief.
3
equilibrium this imitation is necessarily imperfect; otherwise, the public signals would be uninformativeabout the large player’s type and imitation would have no value. Third, we find a square root law ofsubstitution between the discount rate and the volatility of the noise: doubling the discount rate has thesame effect on the equilibrium as rescaling the volatility matrix of the public signal by a factor of
p2.
Finally, we derive the following reputation result for equilibrium behavior: If the small players assignpositive probability to a behavioral type committed to the Stackelberg action and the signals satisfy anidentifiability condition, then, as the large player gets patient, the equilibrium strategy of the normal typeapproaches the Stackelberg action at every reputation level (Theorem 5).
By contrast, when the small players are certain that they are facing the normal type, we find that everyequilibrium of the continuous-time game must be degenerate, i.e. a static Nash equilibrium must be playedafter every history (Theorem 3). This phenomenon has no counterpart in the discrete-time framework ofFudenberg and Levine (1992), where non-trivial equilibria of the complete information game are knownto exist (albeit with payoffs bounded away from efficiency, as shown in Fudenberg and Levine (1994)). Indiscrete time, the large player’s incentives to play a non-myopic best reply can be enforced by the threatof a punishment phase, which is triggered when the public signal about his hidden action is sufficiently“bad.” However, such intertemporal incentives may unravel as actions become more frequent, as firstdemonstrated in a classic paper of Abreu, Milgrom, and Pearce (1991) using a game with Poisson signals.Such incentives also break down under Brownian signals, as in the repeated Cournot duopoly with flexibleproduction of Sannikov and Skrzypacz (2007) and also in the repeated commitment game with long-runand short-run players of Fudenberg and Levine (2007). The basic intuition underlying these results is that,under some signal structures, when players take actions frequently the information they observe withineach time period becomes excessively noisy, and so the statistical tests that trigger the punishment regimesproduce false positives too often. However, this phenomenon arises only under some signal structures(Brownian signals and “good news” Poisson signals), as shown in contemporaneous work of Fudenbergand Levine (2007) and Sannikov and Skrzypacz (2010). We further discuss the discrete-time foundationsof our equilibrium degeneracy result in Section 5 and in our concluding remarks in Section 10.
While a rich structure of intertemporal incentives can arise in equilibrium only when the prior proba-bility on the behavioral type is positive, there is no discontinuity when the prior converges to zero: Underour sufficient conditions for uniqueness, as the reputation of the large player tends to zero, for any fixeddiscount rate the equilibrium behavior in the reputation game converges to the equilibrium behavior undercomplete information. In effect, the Markovian structure of reputational equilibria is closely related to theequilibrium degeneracy without uncertainty over types. When the stage game has a unique Nash equilib-rium, the only equilibrium of the continuous-time game without uncertainty over types is the repetitionof the static Nash equilibrium, which is trivially Markovian. In our setting with Brownian signals, con-tinuous time prevents non-Markovian incentives created by rewards and punishments from enhancing theincentives naturally created by reputation dynamics.
We go beyond games with a unique Markov perfect equilibrium and extend our characterization tomore general environments with multiple sequential equilibria. Here, the object of interest is the corre-spondence of sequential equilibrium payoffs of the large player as a function of his reputation. In The-orem 6, we show that this correspondence is convex-valued and that its upper boundary is the greatestbounded solution of a differential inclusion (see, e.g., Aubin and Cellina (1984)), with an analogous char-acterization for the lower boundary. We provide a computed example illustrating the solutions to these
4
differential inclusions.While a major part of our analysis concerns the case of a single behavioral type, most of the work on
reputation effects in discrete time examines the general case of multiple behavioral types, as in Fudenbergand Levine (1992) and Cripps, Mailath, and Samuelson (2004). To be consistent with this tradition, wefeel compelled to shed some light on continuous-time games with multiple types, and so we extend ourrecursive characterization of public sequential equilibrium to this case, characterizing the properties thatthe reputation vector and the large player’s continuation value must satisfy in equilibrium. With multipletypes, however, we do not go to the next logical level to characterize equilibrium payoffs via partialdifferential equations, thus leaving this important extension for future research. Nevertheless, we use ourrecursive characterization of sequential equilibria under multiple behavioral types to prove an analogue ofthe Cripps, Mailath, and Samuelson (2004) result that the reputation effect is a short-lived phenomenon, i.e.eventually the small players learn when they are facing a normal type. The counterpart of the Fudenberg-Levine bounds on the equilibrium payoffs of the large type as he becomes patient, for continuous-timegames with multiple types, is shown in Faingold (2008).
Finally, to provide a more complete analysis, we also address continuous-time games with Poissonsignals and extend many of our results to those games. However, as we know from Abreu, Milgrom, andPearce (1991), Fudenberg and Levine (2007), Fudenberg and Levine (2009) and Sannikov and Skrzypacz(2010), Brownian and Poisson signals have different informational properties in games with frequent ac-tions or in continuous time. As a result, our equilibrium uniqueness result extends only to games in whichPoisson signals are “good news.” Instead, when the Poisson signals are “bad news,” multiple equilibria arepossible even under the most restrictive conditions on payoffs that yield uniqueness in the Brownian andin the Poisson good news case.
The rest of the paper is organized as follows. Section 2 presents an example in which a firm caresabout her reputation concerning the quality of her product. Section 3 introduces the continuous-time modelwith Brownian signals and a single behavioral type. Section 4 provides the recursive characterization ofpublic sequential equilibria. Section 5 examines the underlying complete information game. Section 6presents the ODE characterization when the sequential equilibrium is unique, along with the reputationresult for equilibrium strategies and the sufficient conditions for uniqueness in terms of the primitives ofthe game. Section 7 extends the characterization to games with multiple sequential equilibria. Section8 deals with multiple behavioral types. Section 9 considers games with Poisson signals and proves theequilibrium uniqueness and characterization result for the case in which the signals are good news. Section10 concludes by drawing analogies between continuous and discrete time.
2 Example: product choice
Consider a firm who provides a service to a continuum of identical consumers. At each time t 2 Œ0;1/,the firm exerts a costly effort, at 2 Œ0; 1�, which affects the quality of the service provided, and eachconsumer i 2 Œ0; 1� chooses a level of service to consume, bit 2 Œ0; 3�. The firm does not observe eachconsumer individually but only the aggregate level of service in the population of consumers, denoted Nbt .Likewise, the consumers do not observe the firm’s effort level; instead, they publicly observe the qualityof the service, dXt , which is a noisy signal of the firm’s effort and follows
dXt D atdt C dZt ;
5
where .Zt /t�0 is a standard Brownian motion. The unit price for the service is exogenously fixed andnormalized to one. The discounted profit of the firm and the overall surplus of consumer i are, respectively,Z 1
0
re�rt . Nbt � at / dt andZ 10
re�rt .bit .4 � Nbt / dXt � bit dt/;
where r > 0 is the discount rate. Thus, the payoff function of the consumers features a negative exter-nality: greater usage Nbt of the service by other consumers leads each consumer to enjoy the service less.This feature is meant to capture a situation in which the quality of the service is adversely affected bycongestion, as in the case of Internet service providers.
Note that in every equilibrium of the continuous-time game the consumers must optimize myopically,i.e. they must act to maximize their expected instantaneous payoff. This is because the firm can onlyobserve the aggregate consumption in the population, so no individual consumer can have an impact onfuture play.
In the unique static Nash equilibrium, the firm exerts zero effort and the consumers choose the lowestlevel of service to consume. In Section 5, we show that the unique equilibrium of the continuous-timerepeated game is the repeated play of this static equilibrium, irrespective of the discount rate r . Thus, itis impossible for the consumers to create intertemporal incentives for the firm to exert effort, even whenthe firm is patient and despite her behavior being statistically identified—i.e. different effort levels inducedifferent drifts for the quality signal Xt . This stands in sharp contrast to the standard setting of repeatedgames in discrete time, which is known to yield a great multiplicity of non-static equilibria when the non-myopic players are sufficiently patient (Fudenberg and Levine, 1994). However, if the firm were able tocommit to any effort level a� 2 Œ0; 1�, this commitment would influence the consumers’ choices, andhence the firm could earn a higher profit. Indeed, each consumer’s choice, bi , would maximize theirexpected flow payoff, bi .a�.4 � Nb/ � 1/, and in equilibrium all consumers would choose the same levelb� D max f0; 4 � 1=a�g. The service provider would then earn a profit of max f0; 4 � 1=a�g � a�, and ata� D 1 this function achieves its maximal value of 2, the firm’s Stackelberg payoff.
Thus, following Fudenberg and Levine (1992), it is interesting to explore the repeated game withreputation effects. Assume that at time zero the consumers believe that with probability p 2 .0; 1/ thefirm is a behavioral type, who always chooses effort level a� D 1, and with probability 1� p the firm is anormal type, who chooses at to maximize her expected discounted profit. What happens in equilibrium?
The top panel of Figure 1 displays the unique sequential equilibrium payoff of the normal type as afunction of the population’s belief p, for different discount rates r . In equilibrium the consumers con-tinually update their posterior belief �t— the probability assigned to the behavioral type—using the ob-servations of the public signal Xt . The equilibrium is Markovian in �t , which uniquely determines theequilibrium actions of the normal type (bottom left panel) and the consumers (bottom right panel). Con-sistent with the bounds on equilibrium payoffs obtained in Faingold (2008), which extends the reputationbounds of Fudenberg and Levine (1992) to continuous time, the computation shows that as r ! 0 thelarge player’s payoff converges to the Stackelberg payoff of 2. We can also see from Figure 1 that theaggregate consumption in the population, Nb, increases towards the commitment level of 3 as the discountrate r decreases towards 0. While the normal type chooses action 0 for all levels of �t when r D 2, as ris closer to 0 his action increases towards the Stackelberg action a� D 1. However, the “imitation” of thebehavioral type by the normal type is never perfect, even for very low discount rates.
6
r D 0:1
r D 0:1
r D 0:1
r D 0:5
r D 0:5
r D 0:5
r D 2
r D 2
r D 2
00
0
0.2
0.2
0.4
0.4
0.6
0.6 0.8
0.50.5
1
11
1
1
2
2
3
3
Stackelberg payoff
�t�t
�t
Payoff of the normal type
Effort level of the normal type Aggregate service level
1
Figure 1: Equilibrium payoffs and actions in the product choice game.
3 The reputation game
A large player faces a continuum of small players in a continuous-time repeated game. At each timet 2 Œ0;1/, the large player chooses an action at 2 A and each small player i 2 I defD Œ0; 1� chooses anaction bit 2 B , where the action spaces A and B are compact subsets of an Euclidean space. The smallplayers are anonymous: at each time t the public information includes the aggregate distribution of thesmall players’ actions, Nbt 2 �.B/, but not the action of any individual small player.5 We assume that theactions of the large player are not directly observable by the small players. Instead, there is a noisy publicsignal .Xt /t�0, whose evolution depends on the actions of the large player, the aggregate distribution ofthe small players’ actions and noise. Specifically,
dXt D �.at ; Nbt / dt C �. Nbt / dZt ;
where .Zt /t�0 is a d -dimensional Brownian motion and the drift and volatility coefficients are determinedby Lipschitz continuous functions � W A � B ! Rd and � W B ! Rd�d , which are linearly extended to
5The aggregate distribution over the small players’ actions is the probability distribution Nbt 2 �.B/ such that Nbt .B 0/ DRfi Wbi
t2B0g di for each Borel measurable subset B 0 � B .
7
A��.B/ and �.B/, respectively.6;7 For technical reasons, we assume that there is a constant c > 0 suchthat j�.b/yj � cjyj for all y 2 Rd and b 2 B . We write .Ft /t�0 to designate the filtration generated by.Xt /t�0.
The small players have identical preferences.8 The payoff of player i 2 I depends on his own action,the aggregate distribution of the small players’ actions and the action of the large player:Z 1
0
re�rth.at ; bit ; Nbt / dt;
where r > 0 is the discount rate and h W A � B � B ! R is a continuous function, which is linearlyextended to A�B ��.B/. As is standard in the literature on imperfect public monitoring, we assume thatthe small players do not gather any information from their payoff flow beyond the information conveyedin the public signal. An important case is when h.at ; bit ; Nbt / is the expected payoff flow of player i ,whereas his ex post payoff flow is privately observable and depends only on his own action, the aggregatedistribution of the small players’ actions, and the flow, dXt , of the public signal. In this case, the ex postpayoff flow of player i takes the form
u.bit ;Nbt / dt C v.bit ;
Nbt / � dXt ;
so thath.a; bi ; Nb/ D u.bi ; Nb/C v.bi ; Nb/ � �.a; Nb/; (1)
as in the product choice game of Section 2. While this functional form is natural in many applications,none of our results hinges on it, so for the sake of generality we take the payoff function h W A�B�B ! R
to be a primitive and do not impose (1).While the small players’ payoff function is common knowledge, there is uncertainty about the type
� of the large player. At time t D 0 they believe that with probability p 2 Œ0; 1� the large player is abehavioral type (� D b) and that with probability 1 � p he is a normal type (� D n). The behavioraltype plays a fixed action a� 2 A at all times, irrespective of history. The normal type plays strategically tomaximize the expected value of his discounted payoff,Z 1
0
re�rt g.at ; Nbt / dt ;
where g W A � B ! R is a Lipschitz continuous function, which is linearly extended to A ��.B/.A public strategy of the normal type of the large player is a random process .at /t�0 with values in A
and progressively measurable with respect to .Ft /t�0. Similarly, a public strategy of small player i 2 Iis a progressively measurable process .bit /t�0 taking values in B . In the repeated game the small players
6Functions � and � are extended to distributions over B via �.a; Nb/ D RB �.a; b/ d
Nb.b/ and �. Nb/�. Nb/> DRB �.b/�.b/
>d Nb.b/.7The assumption that the volatility of .X/t�0 is independent of the large player’s actions corresponds to the full support
assumption that is standard in discrete-time repeated games. By Girsanov’s Theorem (Karatzas and Shreve, 1991, p. 191) theprobability measures over the sample paths of two diffusion processes with the same volatility coefficient but different boundeddrifts are mutually absolutely continuous, i.e. they have the same zero-probabilty events. Since the volatility of a continuous-timediffusion is effectively observable, we do not allow � to depend on at .
8All our results can be extended to a setting where the small players observe the same public signal, but have heterogeneouspreferences.
8
formulate a belief about the large player’s type following their observations of .Xt /t�0. A belief processis a progressively measurable process .�t /t�0 taking values in Œ0; 1�, where �t designates the probabilitythat the small players assign at time t to the large player being the behavioral type.
Definition 1. A public sequential equilibrium consists of a public strategy .at /t�0 of the normal type ofthe large player, a public strategy .bit /t�0 for each small player i 2 I and a belief process .�t /t�0 suchthat at all times t � 0 and after all public histories,
(a) the strategy of the normal type of the large player maximizes his expected payoff
Et
�Z 10
re�rs g.as; Nbs/ dsˇ̌̌� D n
�;
(b) the strategy of each small player i maximizes his expected payoff
.1 � �t /Et�Z 10
re�rs h.as; bis; Nbs/ dsˇ̌̌� D n
�C �t Et
�Z 10
re�rs h.a�; bis; Nbs/ dsˇ̌̌� D b
�(c) beliefs .�t /t�0 are determined by Bayes rule given the common prior �0 D p.
A strategy profile satisfying conditions (a) and (b) is called sequentially rational. A belief process .�t /t�0satisfying condition (c) is called consistent.
This definition can be simplified in two ways. First, because the small players have identical prefer-ences, any strategy profile obtained from a public sequential equilibrium by a permutation of the smallplayers’ labels remains a public sequential equilibrium. Given this immaterial indeterminacy, we shallwork directly with the aggregate behavior strategy . Nbt /t�0 rather than with the individual strategies.bit /t�0. Second, in any public sequential equilibrium the small players’ strategies must be myopicallyoptimal, for their individual behavior is not observed by any other player in the game and it cannot in-fluence the evolution of the public signal. Thus, with slight abuse of notation, we will say that a tuple.at ; Nbt ; �t /t�0 is a public sequential equilibrium when, for all t � 0 and after all public histories, condi-tions (a) and (c) are satisfied as well as the myopic incentive constraint
b 2 arg maxb02B
.1 � �t / h.at ; b0; Nbt / C �t h.a�; b0; Nbt / 8b 2 supp Nbt :
Finally, we make the following remarks concerning the definition of strategies and the solution con-cept:
Remark 1. Since the aggregate distribution over the small players’ actions is publicly observable, theabove definition of public strategies is somewhat non-standard, in that it requires the behavior to dependonly on the sample path of .Xt /t�0. Notice, however, that in our game this restricted definition of publicstrategies incurs no loss of generality. For a given strategy profile, the public histories along which there areobservations of Nbt that differ from those on the path of play correspond to deviations by a positive measureof small players. Since the play that follows such joint deviations is irrelevant for equilibrium incentivesour restricted definition of public strategies does not alter the set of public sequential equilibrium outcomes.
9
Remark 2. The framework can be extended to accommodate public sequential equilibria in mixed strate-gies. A mixed public strategy of the large player is a random process . Nat /t�0 progressively measurablewith respect to .Ft /t�0 with values in �.A/. To accommodate mixed strategies, the payoff functionsg.�; Nb/ and h.�; bi ; Nb/ and the drift �.�; Nb/ are linearly extended to �.A/. Because there is a continuumof anonymous small players, the assumption that each of them plays a pure strategy is without loss ofgenerality.
Remark 3. For both pure- and mixed-strategy equilibria, the restriction to public strategies is without lossof generality. For pure strategies, it is redundant to condition a player’s current action on his private history,as every private strategy is outcome-equivalent to a public strategy. For mixed strategies, the restriction topublic strategies is without loss of generality in repeated games with signals that have a product structure,as in the repeated games that we consider.9 To form a belief about his opponent’s private histories, in agame with product structure a player can ignore his own past actions because they do not influence thesignal about his opponent’s actions. Formally, a mixed private strategy of the large player in our gamewould be a random process .at /t�0 with values in A and progressively measurable with respect to afiltration .Gt /t�0, which is generated by the public signals and the large player’s private randomization.For any private strategy of the large player, an equivalent mixed public strategy can then be defined byletting Nat be the conditional distribution of at given Ft . Strategies .at /t�0 and . Nat /t�0 induce the sameprobability distributions over public signals and hence give the large player the same expected payoffconditional on Ft .
3.1 Relation to the literature
At this point the reader may be wondering how exactly our continuous-time formulation relates to thecanonical model of Fudenberg and Levine (1992). Besides continuous vs. discrete time, a fundamentaldifference is that we assume the existence of a single behavioral type, while in the Fudenberg and Levine(1992) model the type space of the large player is significantly more general, including multiple behavioraltypes and, possibly, arbitrary non-behavioral types. Another distinction is that in our model the uninformedplayers are infinitely-lived small anonymous players, whereas in each period of the canonical model thereis a single uninformed player who lives only in that period. But, in this dimension our formulation nests thecanonical model, since when the payoff h.a; bi ; Nb/ is independent of Nb our model is formally equivalentto one in which there is a continuous flow of uninformed players, each of whom lives only for an instantof time. In this case, bi D Nb and therefore each individual uninformed player can influence the evolutionof the public signal—both drift and volatility—, just as in the canonical model.
Moreover, the goal of our analysis is conceptually different from that of Fudenberg and Levine (1992).While their main result determines upper and lower bounds on the equilibrium payoffs of the normaltype which hold in the limit as he gets arbitrarily patient, we provide a characterization of sequentialequilibrium—both payoffs and behavior—under a fixed rate of discounting. For this reason, it is notsurprising that in some dimensions our assumptions are more restrictive than the assumptions in Fudenbergand Levine (1992).
9A public monitoring technology has a product structure if each public signal is controlled by exactly one large player and thepublic signals corresponding to different large players are conditionally independent given the action profile—cf. Fudenberg andLevine (1994, Section 5). Since our reputation game has only one large player, this condition holds trivially.
10
Although our focus is on equilibrium characterization for fixed discount rates, it is worth noting that theFudenberg-Levine limit bounds on equilibrium payoffs have a counterpart in our continuous-time setting,as shown in Faingold (2008). For each a 2 A, let B.a/ designate the set of Nash equilibria of the partialgame between the small players given some action that is observationally equivalent to a, i.e.
B.a/defD8<: Nb 2 �.B/ W 9 Qa 2 A such that �. Qa; Nb/ D �.a; Nb/ and
b 2 arg maxb02B
h. Qa; b0; Nb/ 8b 2 supp Nb
9=; :The following theorem is similar to Faingold (2008, Theorem 3.1):10
Theorem 1 (Reputation Effect). For every " 2 .0; 1/ and ı > 0 there exists Nr > 0 such that in every publicsequential equilibrium of the reputation game with prior p 2 Œ"; 1 � "� and discount rate r 2 .0; Nr�, theexpected payoff of the normal type is bounded below by
minNb2B.a�/g.a�; Nb/ � ı;
and bounded above bymaxa2A
maxNb2B.a/
g.a; Nb/ C ı:
The upper bound, Ngs defD maxa2A max Nb2B.a/ g.a;Nb/, is the generalized Stackelberg payoff. It is
the greatest payoff that a large player with commitment power can get, taking into account the limitedobservability of the large players’ actions and the incentives that arise in the partial game among the smallplayers. Perhaps more interesting is the lower bound, min Nb2B.a�/ g.a
�; Nb/, which can be quite high insome games. Of particular interest are games in which it approximates
gs defD supa2A
minNb2B.a/g.a; Nb/;
the greatest possible lower bound. While the supremum above may not be attained, given " > 0 we canfind some a� 2 A such that min Nb2B.a�/ g.a
�; Nb/ lies within " of gs. Thus, if the behavioral type playssuch a�, there is a lower bound on equilibrium payoffs converging to gs � " as r ! 0.11
It is therefore natural to examine when the reputation bounds, gs and Ngs, coincide. A wedge may arisefor two reasons: (i) there may be multiple actions that are observationally equivalent to some action a,under some Nb; and (ii) the partial game among the small players may have multiple static Nash equilibria.Recall that also in Fudenberg and Levine (1992) the upper and lower bounds may be different. However,in their setting the indeterminacy is less severe, for in their model there is a single myopic player in eachperiod, and hence (ii) is just the issue of multiplicity of optima in a single-agent programming problem,
10Faingold (2008) examines reputation effects in continuous-time games with a general type space for the large player, as inFudenberg and Levine (1992), and public signals that follow a controlled Lévy process. Although Faingold (2008) does notconsider cross-section populations of small players, a straightforward extension of the proof of Faingold (2008, Theorem 3.1) canbe used to prove Theorem 1 above.
11When the support of the prior contains all possible behavioral types, as in Fudenberg and Levine (1992) and Faingold (2008),there is a lower bound on the equilibrium payoffs of the normal type which converges exactly to gs. In contrast, when there isonly a finite set of behavioral types—e.g., as in the current paper and in Cripps, Mailath, and Samuelson (2004)—, the lowerbound maybe strictly less than gs.
11
a less severe problem than the issue of multiplicity of Nash equilibria in a game. Indeed, Fudenberg andLevine (1992) show that when the action sets are finite, the upper and lower bounds must be equal underthe following assumptions:
� the actions of the large player are identified, i.e. there do not exist Na, Na0 2 �.A/, with Na ¤ Na0, andNb 2 �.B/ such that . Na; Nb/ and . Na0; Nb/ generate the same distribution over public signals;
� the payoff matrix of the small players is non-degenerate, i.e. there do not exist b 2 B and Nb 2 �.B/such that h.�; Nb/ D h.�; b/ and b ¤ Nb.12
However, in our setting these assumptions do not generally imply Ngs D gs, due to the externality acrossthe small players that we allow.
4 The structure of public sequential equilibrium
This section develops a recursive characterization of public sequential equilibria which is used through therest of the paper. Recall that in a public sequential equilibrium, beliefs must be consistent with the publicstrategies and the strategies must be sequentially rational given beliefs. For the consistency of beliefs,Proposition 1 presents equation (2), which describes how the small players’ posterior evolves with thepublic signal .Xt /t�0. The sequential rationality of the normal type can be verified by examining theevolution of his continuation value, i.e. his expected future discounted payoff given the history of publicsignals up to time t . First, Proposition 2 presents a necessary and sufficient condition for a random process.Wt /t�0 to be the process of continuation values of the normal type. Then, Proposition 3 characterizessequential rationality using a condition that is connected to the law of motion of .Wt /t�0. Propositions 2and 3 are analogous to Propositions 1 and 2 of Sannikov (2007).
We begin with Proposition 1, which characterizes the stochastic evolution of the small players’ poste-rior beliefs.13
Proposition 1 (Belief Consistency). Fix the prior p 2 Œ0; 1� on the behavioral type. A belief process.�t /t�0 is consistent with a public strategy profile .at ; Nbt /t�0 if and only if �0 D p and
d�t D .at ; Nbt ; �t / � �. Nbt /�1.dXt � ��t .at ; Nbt / dt/; (2)
where for each .a; Nb; �/ 2 A ��.B/ � Œ0; 1�, .a; Nb; �/ defD �.1 � �/�. Nb/�1 ��.a�; Nb/ � �.a; Nb/� ;
��.a; Nb/ defD ��.a�; Nb/C .1 � �/�.a; Nb/:
12To be precise, in Fudenberg and Levine (1992) the upper and lower bounds are slightly different from Ngs and gs. Since theyassume that the support of the prior contains all possible behavioral types, the myopic players never play a weakly dominatedaction. Accordingly, they define B0.a/ to be the set of all undominated actions Nb which are a best-reply to some action that isobservationally equivalent to a under Nb. Then, the definitions of the upper and lower bounds are similar to those of Ngs and gs,but with B0.a/ replacing B.a/. This turns out to be crucial for their proof that the upper and lower bounds coincide under theidentifiability and non-degeneracy conditions above.
13Similar versions of the filtering equation (2) have been used in the literature on strategic experimentation in continuous time(cf. Bolton and Harris (1999), Keller and Rady (1999) and Moscarini and Smith (2001)). For a general treatment of filtering incontinuous time, see Liptser and Shiryaev (1977).
12
Proof. The strategy of each type of the large player induces a probability measure over the paths of thepublic signal .Xt /t�0. From Girsanov’s Theorem we can find the ratio �t between the likelihood that apath .XsI s 2 Œ0; t �/ arises for type b and the likelihood that it arises for type n. This ratio is characterizedby
d�t D �t �t � dZns ; �0 D 1 ; (3)
where �t D �. Nbt /�1��.a�; Nbt / � �.at ; Nbt /
�and Zn
t DR t0 �.Nbs/�1.dXs � �.as; Nbs/ ds/ is a Brownian
motion under the probability measure generated by the strategy of type n.Suppose that .�t /t�0 is consistent with .at ; Nbt /t�0. Then, by Bayes’ rule, the posterior after observing
a path .XsI s 2 Œ0; t �/ is
�t D p�t
p�t C .1 � p/: (4)
By Itô’s formula,
d�t D p.1 � p/.p�t C .1 � p//2
d�t � 2p2.1 � p/.p�t C .1 � p//3
�2t �t � �t2
dt
D �t .1 � �t /�t � dZnt � �2t .1 � �t /�t � �t dt (5)
D �t .1 � �t /�t � �. Nbt /�1.dXt � ��t .at ; Nbt / dt/;
which is equation (2).Conversely, suppose that .�t /t�0 solves equation (2) with initial condition �0 D p. Define �t using
expression (4), i.e.
�t D 1 � pp
�t
1 � �t :
Then, applying Itô’s formula to the expression above gives equation (3), hence �t must equal the ratiobetween the likelihood that a path .XsI s 2 Œ0; t �/ arises for type b and the likelihood that it arises for typen. Thus, �t is determined by Bayes’ rule and the belief process is consistent with .at ; Nbt /t�0. �
Note that in the statement of Proposition 1, .at /t�0 is the strategy that the small players believe that thenormal type is following. Thus, when the normal type deviates from his equilibrium strategy, the deviationaffects only the drift of .Xt /t�0, but not the other terms in equation (2).
Coefficient of equation (2) is the volatility of beliefs: it reflects the speed with which the small playerslearn about the type of the large player. The definition of plays an important role in the characterizationof public sequential equilibrium presented in Sections 6 and 7. The intuition behind equation (2) is asfollows. If the small players are convinced about the type of the large player, then �t .1 � �t / D 0; sothey never change their beliefs. When �t 2 .0; 1/ then .at ; Nbt ; �t / is larger, and learning is faster, whenthe noise �. Nbt / is smaller or the drifts produced by the two types differ more. From the small players’perspective, the noise driving equation (2), �. Nbt /�1.dXt���t .at ; Nbt / dt/, is a standard Brownian motionand their belief .�t /t�0 is a martingale. From equation (5) we see that, conditional on the large playerbeing the normal type, the drift of �t is non-positive: in the long run, either the small players learn thatthey are facing the normal type, or the normal type plays like the behavioral type.
We turn to the analysis of the second important state variable in the strategic interaction between thelarge player and the small players: the continuation value of the normal type, i.e. his future expected dis-counted payoff following a public history, under a given strategy profile. More precisely, given a strategy
13
profile S D .as; Nbs/s�0, the continuation value of the normal type at time t is
Wt .S/defDEt
�Z 1t
re�r.s�t/g.as; Nbs/ dsˇ̌̌� D n
�: (6)
The following proposition characterizes the law of motion ofWt .S/. Throughout the paper, we write L
to designate the space of Rd -valued progressively measurable processes .ˇt /t�0 with E� R T0 jˇt j2 dt
�<
1 for all 0 < T <1.
Proposition 2 (Continuation Values). A bounded process .Wt /t�0 is the process of continuation values ofthe normal type under a public strategy profile S D .at ; Nbt /t�0 if and only if for some .ˇt /t�0 in L,
dWt D r.Wt � g.at ; Nbt // dt C rˇt � .dXt � �.at ; Nbt / dt/: (7)
Proof. First, note that Wt .S/ is a bounded process by (6). Let us show that Wt D Wt .S/ satisfies (7) forsome .ˇt /t�0 in L. Denote by Vt .S/ the expected discounted payoff of the normal type conditional onthe public information at time t , i.e.
Vt .S/defDEt
�Z 10
re�rsg.as; Nbs/ dsˇ̌̌� D n
�DZ t
0
re�rsg.as; Nbs/ ds C e�rtWt .S/ (8)
Thus, .Vt /t�0 is a martingale when the large player is the normal type. By the Martingale RepresentationTheorem, there exists .ˇt /t�0 in L such that
dVt .S/ D re�rtˇ>t �. Nbt / dZnt ; (9)
where dZnt D �. Nbt /�1.dXt � �.at ; Nbt / dt/ is a Brownian motion when the large player is the normal
type. Differentiating (8) with respect to t yields
dVt .S/ D re�rtg.at ; Nbt / dt � re�rtWt .S/ dt C e�rtdWt .S/: (10)
Combining equations (9) and (10) yields (7), which is the desired result.Conversely, let us show that if .Wt /t�0 is a bounded process satisfying (7) thenWt D Wt .S/. Indeed,
when the large player is the normal type, the process
Vt DZ t
0
re�rsg.as; Nbs/ ds C e�rtWt
is a martingale under the strategy profile S D .at ; Nbt /, because dVt D re�rtˇ>t �. Nbt / dZnt by (7). More-
over, as t ! 1 the martingales Vt and Vt .S/ converge because both e�rtWt and e�rtWt .S/ tend to 0.Therefore,
Vt D Et ŒV1� D Et ŒV1.S/� D Vt .S/ ) Wt D Wt .S/for all t; as required. �
Thus, equation (7) describes how Wt .S/ evolves with the public history. Note that this equation musthold regardless of the large player’s actions before time t . This fact is used in the proof of Proposition 3below, which deals with incentives.
14
We finally turn to conditions for sequential rationality. The condition for the small players is straight-forward: they maximize their static payoff because a deviation of an individual small player cannot influ-ence the future course of equilibrium play. The characterization of the large player’s incentive constraintsis more complicated: the normal type acts optimally if he maximizes the sum of his current flow payoffand the expected change in his continuation value.
Proposition 3 (Sequential Rationality). A public strategy profile .at ; Nbt /t�0 is sequentially rational withrespect to a belief process .�t /t�0 if and only if there exists .ˇt /t�0 in L and a bounded process .Wt /t�0satisfying (7), such that for all t � 0 and after all public histories,
at 2 arg maxa02A
g.a0; Nbt /C ˇt � �.a0; Nbt /; (11)
b 2 arg maxb02B
�th.a�; b0; Nbt /C .1 � �t /h.at ; b0; Nbt /; 8b 2 supp Nbt : (12)
Proof. Consider a strategy profile .as; Nbs/s�0 and an alternative strategy . Qas/s�0 for the normal type.Denote by Wt the continuation payoff of the normal type at time t when he follows strategy .as/s�0 aftertime t , while the population follows . Nbs/s�0. If the normal type plays . Qas/s�0 up to time t and thenswitches to .as/s�0, his expected payoff conditional on the public information at time t is given by
QVt DZ t
0
re�rsg. Qas; Nbs/ ds C e�rtWt :
By Proposition 2, the continuation values .Wt /t�0 follow equation (7) for some .ˇt /t�0 in L. Thus, theabove expression for QVt implies
d QVt D re�rt .g. Qat ; Nbt / �Wt / dt C e�rtdWtD re�rt ..g. Qat ; Nbt / � g.at ; Nbt // dt C ˇt � .dXt � �.at ; Nbt / dt//;
and hence the profile . Qat ; Nbt /t�0 yields the normal type an expected payoff of
QW0 D EŒ QV1� D E�QV0 C
Z 10
d QVt�D W0 C E
�Z 10
d QVt�
D W0 C E�Z 10
re�rt .g. Qat ; Nbt / � g.at ; Nbt / C ˇt � .�. Qat ; Nbt / � �.at ; Nbt // dt�;
where the expectation is taken under the probability measure induced by . Qat ; Nbt /t�0, so that .Xt /�0 hasdrift �. Qat ; Nbt /.
Suppose that .at ; Nbt ; �t /t�0 satisfies the incentive constraints (11) and (12). Then, for every . Qat /t�0we have W0 � QW0, and therefore the normal type must be sequentially rational at time t D 0. A similarargument can be used to show that the normal type is sequentially rational at all times t > 0, after allpublic histories. Finally, the small players must also be sequentially rational, since they are anonymousand hence myopic. Conversely, suppose that the incentive constraint (11) fails. Choose a strategy . Qat /t�0satisfying (11) for all t � 0 and all public histories. Then QW0 > W0, and hence the large player is notsequentially rational at t D 0. Likewise, if (12) fails, then a positive measure of small players would notbe maximizing their instantaneous expected payoffs. Since the small players are anonymous, and hencemyopic, their strategies would not be sequentially rational. �
15
We can summarize our characterization in the following theorem.
Theorem 2 (Sequential Equilibrium). Fix a prior p 2 Œ0; 1� on the behavioral type. A strategy profile.at ; Nbt /t�0 and a belief process .�t /t�0 form a public sequential equilibrium with continuation values.Wt /t�0 for the normal type if and only if for some process .ˇt /t�0 in L,
(a) .�t /t�0 satisfies (2) with initial condition �0 D p,
(b) .Wt /t�0 is a bounded process satisfying (7), given .ˇt /t�0,
(c) .at ; Nbt /t�0 satisfies the incentive constraints (11) and (12), given .ˇt /t�0 and .�t /t�0.
Thus, Theorem 2 provides a recursive characterization of public sequential equilibrium. Let E WŒ0; 1� � R denote the correspondence that maps a prior probability on the behavioral type into the cor-responding set of public sequential equilibrium payoffs of the normal type. An equivalent statement ofTheorem 2 is that E is the greatest bounded correspondence such that a controlled process .�t ; Wt /t�0,defined by (2) and (7), can be kept in Graph.E/ by controls .at ; Nbt ; ˇt /t�0 which are require to satisfy(11) and (12).14 In Section 5 we apply Theorem 2 to the repeated game with prior p D 0, the completeinformation game. In Sections 6 and 7 we characterize E.p/ for p 2 .0; 1/.
5 Equilibrium degeneracy under complete information
In this section we examine the structure of public equilibrium in the underlying complete informationgame, i.e. the continuous-time repeated game in which it is common knowledge that the large player isthe normal type. We show that non-trivial intertemporal incentives cannot be sustained in equilibrium,regardless of the level of patience of the large player.
Theorem 3 (Equilibrium Degeneracy). Suppose the small players are certain that they are facing thenormal type, i.e. p D 0. Then, irrespective of the discount rate r > 0, in every public equilibrium of thecontinuous-time game the large player cannot achieve a payoff outside the convex hull of his static Nashequilibrium payoffs, i.e.
E.0/ D co
8̂<̂:g.a; Nb/ W
a 2 arg maxa02A
g.a0; Nb/b 2 arg max
b02Bh.a; b0; Nb/; 8b 2 supp Nb
9>=>; :Here is the idea behind this result. In order to give incentives to the large player to play an action that
results in a greater payoff than in static Nash equilibrium, his continuation value must respond to the publicsignal .Xt /t�0. But, when the continuation value reaches the greatest payoff among all public equilibriaof the repeated game, such incentives cannot be provided. In effect, if the large player’s continuation valuewere sensitive to the public signal when his continuation value equals the greatest public equilibrium pay-off, then with positive probability the continuation value would escape above this upper bound, and this isnot possible. Therefore, at the upper bound the continuation value cannot be sensitive to the public signal,hence the large player must be playing a myopic best-reply there. This implies that at the upper bound the
14This means that the graph of any bounded correspondence with this property must be contained in the graph of E .
16
flow payoff of the large is no greater than the greatest static Nash equilibrium payoff. Moreover, anothernecessary condition to prevent the continuation value from escaping is that the drift of the continuationvalue process must be less than or equal to zero at the upper bound. But since this drift is proportionalto the difference between the continuation value and the flow payoff, it follows that the greatest publicequilibrium payoff must be no greater than the flow payoff at the upper bound, which, as we have arguedabove, must be no greater than the greatest static equilibrium payoff.
Proof of Theorem 3. Let Nv be the greatest Nash equilibrium payoff of the large player in the static completeinformation game. We will show that it is impossible to achieve a payoff greater than Nv in any publicequilibrium of the continuous-time game. (The proof for the least Nash equilibrium payoff is similar andtherefore omitted.) Suppose there was a public equilibrium .at ; Nbt /t�0 with continuation values .Wt /t�0for the normal type in which the large player’s expected discounted payoff, W0, was greater than Nv. ByPropositions 2 and 3, for some .ˇt /t�0 in L the large player’s continuation value must satisfy
dWt D r.Wt � g.at ; Nbt // dt C rˇt � .dXt � �.at ; Nbt / dt/;
where at maximizes g.a0; Nbt / C ˇt � �.a0; Nbt / over all a0 2 A and Nbt maximizes h.at ; b0; Nbt / over allb0 2 �.B/. Let ND defDW0 � Nv > 0.
Claim. There exists c > 0 such that, so long as Wt � Nv C ND=2, either the drift of Wt is greater thanr ND=4 or the norm of the volatility of Wt is greater than c.
This claim is an implication of the following lemma, whose proof is relegated to Appendix A.
Lemma 1. For any " > 0 there exists ı > 0 such that for all t � 0 and after all public histories, jˇt j � ıwhenever g.at ; Nbt / � Nv C ".Indeed, letting " D ND=4 in this lemma yields a ı > 0 such that the norm of the volatility of Wt , whichequals r jˇt j, must be greater than or equal to c D rı whenever g.at ; Nbt / � Nv C ND=4. Moreover, wheng.at ; Nbt / < Nv C ND=4 and Wt � Nv C ND=2, then the drift of Wt , which equals r.Wt � g.at ; Nbt //, must begreater than r ND=4, and this concludes the proof of the claim.
Since we have assumed that ND D W0 � Nv > 0, the claim above readily implies that Wt must growarbitrarily large with positive probability, and this is a contradiction since .Wt /t�0 is bounded. �
While Theorem 3 has no counterpart in discrete time, it is not a result of continuous-time technicali-ties.15 The large player’s incentives to depart from a static best response become fragile when he is flexibleto react to new information quickly. The foundations of this result are similar to the collapse of intertem-poral incentives in discrete-time games with frequent actions, as in Abreu, Milgrom, and Pearce (1991)in a prisoners’ dilemma with Poisson signals, and in Sannikov and Skrzypacz (2007) and Fudenberg andLevine (2007) in games with Brownian signals. Borrowing intuition from these papers, suppose that thelarge player must hold his action fixed for an interval of time of length � > 0. Suppose that the largeplayer’s equilibrium incentives to take the Stackelberg action are created through a statistical test that trig-gers an equilibrium punishment when the signal is sufficiently “bad.” A profitable deviation has a gain
15Fudenberg and Levine (1994) show that in discrete-time repeated games with large and small players, often there are publicperfect equilibria with payoffs above static Nash, albeit bounded away from efficiency.
17
O.�/
O.p�/
deviation
(std.dev)�O.p�/
incentives
false positives
punishment
1
Figure 2: A statistical test to prevent a given deviation.
on the order of �, therefore such deviation can be prevented only if it increases the probability of trig-gering punishment by at least O.�/. Sannikov and Skrzypacz (2007) and Fudenberg and Levine (2007)show that, under Brownian signals, the log-likelihood ratio for a test against any particular deviation isnormally distributed, and that a deviation shifts the mean of this distribution by O.
p�/. Thus, a success-
ful test against a deviation would generate a false positive with probability of O.p�/. This probability,
which reflects the value destroyed in each period by the punishment, is disproportionately large for small� compared to the value created during a period of length �. This intuition implies that in equilibriumthe large player cannot sustain a payoff above static Nash as � ! 0. Figure 2 illustrates the densities ofthe log-likelihood ratio under the “recommended” action of the large player and a deviation, and the areasresponsible for the large player’s incentives and for the false positives.
The above result relies crucially on the assumption that the volatility of the public signal is independentof the large player’s actions. If the large player could influence the volatility, his actions would becomepublicly observable, since in continuous time the volatility of a diffusion process is effectively observable.Motivated by this observation, Fudenberg and Levine (2007) consider discrete-time approximations ofcontinuous-time games with Brownian signals, allowing the large player to control the volatility of thepublic signal. In a variation of the product choice game, when the volatility is decreasing in the largeplayer’s effort level, they show the existence of nontrivial equilibrium payoffs in the limit as the periodlength tends to zero.16
Finally, Fudenberg and Levine (2009) examine discrete-time repeated games with public signals drawnfrom multinomial distributions that depend on the length of the period. They assume that for each profileof stationary strategies, the distribution over signals—properly normalized and embedded in continuoustime—satisfies standard conditions for weak convergence to a Brownian motion with drift as the periodlength tends to zero.17 When the Brownian motion is approximated by a binomial process, they showthat the set of equilibrium payoffs of the discrete-time game approaches the degenerate equilibrium of thecontinuous-time game. By contrast, when the Brownian motion is approximated by a trinomial processand the discount rate is low enough, they show that the greatest equilibrium payoff of the discrete-time
16They also find the surprising result that when the volatility of the Brownian signal is increasing in the large player’s effort,the set of equilibrium payoffs of the large player collapses to the set of static Nash equilibrium payoffs, despite the fact that in thecontinuous-time limit the actions of the large player are effectively observable.
17Namely, they satisfy the conditions of Donker’s Invariance Principle (Billingsley, 1999, Theorem 8.2).
18
0 1
Wt D U.�t /
�t
.at ; Nbt /
Belief
Large player’s payoff
1
Figure 3: The large player’s payoff in a Markov perfect equilibrium.
model converges to a payoff strictly above the static Nash equilibrium payoff.
6 Reputation games with a unique sequential equilibrium
In many interesting applications, including the product choice game of Section 2, the public sequentialequilibrium is unique and Markovian in the small players’ posterior belief. That is, at each time t , the smallplayers’ posterior belief, �t , uniquely determines the equilibrium actions at D a.�t / and Nbt D b.�t /, aswell as the continuation value of the normal type, Wt D U.�t /, as depicted in Figure 3. This sectionpresents general conditions for the sequential equilibrium to be unique and Markovian, and characterizesthe equilibrium using an ordinary differential equation.
First, we derive our characterization heuristically. By Theorem 2, in a sequential equilibrium .at ; Nbt ; �t /t�0the small players’ posterior beliefs, .�t /t�0, and the continuation values of the normal type, .Wt /t�0,evolve according to
d�t D �j .at ; Nbt ; �t /j2=.1 � �t / dt C .at ; Nbt ; �t / � dZnt (13)
anddWt D r.Wt � g.at ; Nbt // dt C rˇ>t �. Nbt / dZn
t ; (14)
for some random process .ˇt /t�0 in L, where dZnt D �. Nbt /�1.dXt ��.at ; Nbt / dt/ is a Brownian motion
under the normal type.18 The negative drift in (13) reflects the fact that, conditional on the large playerbeing the normal type, the posterior on the behavioral type must be a supermartingale.
If we assume that the equilibrium is Markovian, then by Itô’s formula the continuation value Wt DU.�t / of the normal type must follow
dU.�t / D 1
2j .at ; Nbt ; �t /j2
�U 00.�t / � 2U 0.�t /=.1 � �t /
�dt C U 0.�t / .at ; Nbt ; �t / � dZn
t ; (15)
18Equation (13) is just equation (2) re-written from the point of view of the normal type, rather than the point of view of thesmall players. This explain the change of drift.
19
assuming that the value function U W .0; 1/! R is twice continuously differentiable. Thus, matching thedrifts in (14) and (15) yields the optimality equation:
U 00.�/ D 2U 0.�/1 � � C
2r.U.�/ � g.a.�/; b.�///j .a.�/; b.�/; �/j2 ; � 2 .0; 1/: (16)
Then, to determine the Markovian strategy profile .at ; Nbt / D .a.�t /; b.�t //, we can match the volatilitycoefficients in (14) and (15):
rˇ>t D U 0.�t / .at ; Nbt ; �t />�. Nbt /�1:Plugging this expression in the incentive constraint (11) and applying Theorem 2 yields
.at ; Nbt / 2 N .�t ; �t .1 � �t /U 0.�t /=r/; (17)
where N W Œ0; 1� �R� A ��.B/ is the correspondence defined by19
N .�; z/defD
8̂<̂:.a; Nb/ W
a 2 arg maxa02A
g.a0; Nb/C z.�.a�; Nb/ � �.a; Nb//>.�. Nb/�. Nb/>/�1�.a0; Nb/b 2 arg max
b02B�h.a�; b0; Nb/C .1 � �/h.a; b0; Nb/ 8b 2 supp Nb
9>=>; : (18)
Effectively, for each .�; z/ 2 Œ0; 1��R, correspondence N returns the set of Bayesian Nash equilibriaof an auxiliary one-shot game in which the large player is a behavioral type with probability � and thepayoff of the normal type is perturbed by a “reputational term” weighted by z. In particular, N .�; 0/ isthe set of static Bayesian Nash equilibria when the prior on the behavioral type is �.
Our characterization of unique sequential equilibria, stated below as Theorem 4, is valid under Condi-tions 1 and 2 below. Condition 1 guarantees that the volatility of beliefs, which appears in the optimalityequation in the denominator of a fraction, is bounded away from zero across all reputation levels. Con-dition 2 ensures that correspondence N is nonempty and single-valued, so that equilibrium behavior isdetermined by condition (17).
Condition 1. For each � 2 Œ0; 1� and each Bayesian Nash equilbrium .a; Nb/ of the static game with prior�, we have �.a; Nb/ ¤ �.a�; Nb/.
Thus, under Condition 1, in every static Bayesian Nash equilibrium the behavior of the normal type isstatistically distinguishable from the behavioral type. Therefore, in order for the normal type to play eithera� or some observationally equivalent action, he ought to be given intertemporal incentives. This rules outsequential equilibria in which the posterior beliefs settle in finite time.
Note that when the flow payoff of the small players depends on the actions of the large player onlythrough the public signal (cf. equation (1) and the discussion therein), Condition 1 becomes equivalentto the following simpler condition: for every static Nash equilibrium .a; Nb/ of the complete informationgame, �.a; Nb/ ¤ �.a�; Nb/. This condition is similar to the non-credible commitment assumption ofCripps, Mailath, and Samuelson (2004), which we discuss in greater detail in Section 8.
Condition 2. Either of the following conditions holds:
19Note that N .�; z/ also depends on the strategy a� of the behavioral type, although our notation does not make this depen-dence explicit.
20
a. N is a non-empty single-valued correspondence that returns a mass-point distribution of smallplayers’ actions for each .�; z/ 2 Œ0; 1��R. Moreover, N is Lipschitz continuous on every boundedsubset of Œ0; 1� �R.
b. the restriction of N to Œ0; 1�� Œ0;1/ is non-empty, single-valued and returns a mass-point distribu-tion of small players’ actions for each .�; z/ 2 Œ0; 1�� Œ0;1/. Moreover, N is Lipschitz continuouson every bounded subset of Œ0; 1� � Œ0;1/, and the static Bayesian Nash equilibrium payoff of thenormal type, g.N .�; 0//, is increasing in the prior �.
Condition 2.b has practical importance, for it holds in many games in which Condition 2.a fails. In-deed, the payoff of the normal type in (18), when adjusted by a negative reputational weight z, may failto be concave even when g.�; Nb/ and �.�; Nb/ are strictly concave for all Nb. In Section 6.2 we provide con-ditions on the primitives of the game—stage-game payoffs, drift and volatility—which are sufficient forConditions 2.b.
Note that Condition 2 depends on the action a� of the behavioral type: it may hold for some behavioraltypes but not others. However, to keep the notation concise we have chosen not to index correspondenceN by a�.
Finally, Condition 1 is not necessary for equilibrium uniqueness. When Condition 2 holds but Condi-tion 1 fails, the reputation game still has a unique public sequential equilibrium, which is Markovian andcharacterized by (13), (16) and (17) up until the stopping time when the posterior first hits a value � forwhich there is a static Bayesian equilibrium .a� ; Nb�/ with �.a� ; Nb�/ D �.a�; Nb�/; from this time on, theposterior no longer updates and the behavior is .a� ; Nb�/ statically.20 In particular, when the payoff flow ofthe small players depends on the large player’s action only through the public signal (cf. equation (1) andthe discussion therein), then, when Condition 1 fails and Condition 2 holds, for every prior p the uniquepublic sequential equilibrium of the reputation game is the repeated play of the static Nash equilibrium ofthe complete information game, and the posterior is �t D p identically.
Theorem 4. Assume Conditions 1 and 2. Under Condition 2.a (resp. 2.b), the correspondence of publicsequential equilibrium payoffs of the normal type, E W Œ0; 1� � R, is single-valued and coincides, onthe interval .0; 1/, with the unique bounded solution (resp. unique bounded increasing solution) of theoptimality equation:
U 00.�/ D 2U 0.�/1 � � C
2r.U.�/ � g.N .�; �.1 � �/U 0.�/=r//j .N .�; �.1 � �/U 0.�/=r/; �/j2 ; � 2 .0; 1/: (19)
Moreover, at p D 0 and 1, E.p/ satisfies the boundary conditions
lim�!pU.�/ D E.p/ D g.N .p; 0// and lim
�!p �.1 � �/U0.�/ D 0: (20)
Finally, for each prior p 2 Œ0; 1� there exists a unique public sequential equilibrium, which is Markovianin the small players’ posterior belief: at each time t and after each public history, the equilibrium actionsare determined by (17), the posterior evolves according to (13) with initial condition �0 D p and thecontinuation value of the normal type is Wt D U.�t /.
20Indeed, an argument similar to the proof of Theorem 3 can be used to show that when the prior is � the large player cannotachieve any payoff other than g.a� ; Nb�/.
21
The intuition behind Theorem 4 is similar to the idea behind the equilibrium degeneracy result undercomplete information (Theorem 3). With Brownian signals it is impossible to create incentives to sustaina greater payoff than in a Markov perfect equilibrium, for otherwise, in a public sequential equilibriumthat achieves the greatest difference W0 � U.�0/ > 0 across all priors �0 2 Œ0; 1�, at time t D 0 the jointvolatility of .�t ; Wt /t�0 must be parallel to the slope of U.�t /, since Wt � U.�t / cannot increase forany realization of dXt . It follows that rˇ>0 �. Nb0/ D U 0.�0/ .a0; Nb0; �0/> and hence, when N is single-valued, the equilibrium action profile played at time zero must equal that played in a Markov perfectequilibrium at reputation �0. The optimality equation (19) then implies that Wt � U.�t / has a positivedrift at time zero, which implies that with positive probabilityWt �U.�t / > W0 �U.�0/ for some t > 0,and this is a contradiction.
Proof of Theorem 4. The proofs of existence and uniqueness of a bounded solution of the optimality equa-tion (19) are presented in Appendix C, along with a number of intermediate lemmas. First, Proposi-tion C.3 shows that under Conditions 1 and 2.a the optimality equation has a unique bounded solutionU W .0; 1/ ! R. Then, Proposition C.2 shows that U must satisfy the boundary conditions (39), whichinclude (20). Finally, Proposition C.4 implies that under Conditions 1 and 2.b the optimality equation hasa unique increasing bounded solution U , which also satisfies the boundary conditions.
We now show that for each prior p 2 .0; 1/ there is no public sequential equilibrium in which thenormal type receives a payoff different from U.p/. Towards a contradiction, suppose that for some p 2.0; 1/ there is a public sequential equilibrium .at ; Nbt ; �t /t�0 in which the normal type receives payoffW0 > U.p/. By Theorem 2, the small players’ belief process .�t /t�0 follows (13), the continuation valueof the normal type .Wt /t�0 follows (14) for some .ˇt /t�0 in L, and the equilibrium actions .at ; Nbt /t�0satisfy the incentive constraints (11) and (12). Moreover, by Itô’s formula, the process .U.�t //t�0 follows(15). Then, using (14) and (15), the process Dt
defDWt � U.�t /, which starts at D0 > 0, has drift
rDt C rU.�t /„ ƒ‚ …rWt
�rg.at ; Nbt /C j .at ; Nbt ; �t /j2�U 0.�t /1 � �t
� U00.�t /2
�;
and volatilityrˇ>t �. Nbt / � .at ; Nbt ; �t /> U 0.�t /:
Claim. There exists ı > 0 such that, so long as Dt � D0=2, either the drift of Dt is greater than rD0=4or the norm of the volatility of Dt is greater than ı.
This claim is an implication of Lemma C.8 from Appendix C, which shows that for each " > 0 we canfind ı > 0 such that for all t � 0 and after all public histories, either the drift ofDt is greater than rDt �",or the norm of the volatility of Dt is greater than ı. Thus, letting " D rD0=4 in Lemma C.8 proves theclaim.
The proof of Lemma C.8 is relegated to Appendix C, but here we explain the main idea behind it.When the norm of the volatility of Dt is exactly zero we have rˇ>t �. Nbt / D .at ; Nbt ; �t /> U 0.�t /, so by(11) and (12) we have
at 2 arg maxa02A rg.a0; Nbt /C U 0.�t / .at ; Nbt ; �t />��1. Nbt /„ ƒ‚ …rˇ>t
�.a0; Nbt /;
b 2 arg maxb02B �th.a�; b0; Nbt /C .1 � �t /h.at ; b0; Nbt / 8b 2 supp Nbt ;
22
and hence .at ; Nbt / D N .�t ; �t .1 � �t /U 0.�t /=r/. Then, by (19) the drift of Dt must equal rDt . Theproof of Lemma C.8 uses a continuity argument to show that in order for the drift of Dt to be belowrDt � ", the volatility of Dt must be uniformly bounded away from 0.
Since we have assumed that D0 > 0 the claim above readily implies that Dt must grow arbitrarilylarge with positive probability, which is a contradiction since both Wt and U.�t / are bounded processes.This contradiction shows that a public sequential equilibrium that yields the normal type a payoff greaterthan U.p/ cannot exist. Similarly, it can be shown that no equilibrium can yield a payoff less than U.p/.
We turn to the construction of a sequential equilibrium that yields a payoff of U.p/ to the normal type.Consider the stochastic differential equation (13) with .at ; Nbt /t�0 defined by (17). Since the function� 7! .N .�; �.1 � �/U 0.�/=r/; �/ is Lipschitz continuous, this equation has a unique solution .�t /t�0with initial condition �0 D p. We now show that .at ; Nbt ; �t /t�0 is a public sequential equilibrium inwhich Wt D U.�t / is the process of continuation values of the normal type. By Proposition 1, the beliefprocess .�t /t�0 is consistent with .at ; Nbt /t�0. Moreover, since Wt D U.�t / is a bounded process withdrift r.Wt � g.at ; Nbt // by (15) and (19), Proposition 2 implies that .Wt /t�0 is the process of continuationvalues of the normal type under .at ; Nbt /t�0. Thus, the process .ˇt /t�0, given by the representation of Wtin Proposition 2, must satisfy rˇ>t �.bt / D U 0.�t / .at ; bt ; �t />. Finally, to see that the strategy profile.at ; Nbt /t�0 is sequentially rational under .�t /t�0, recall that .at ; Nbt / D N .�t ; �t .1 � �t /U 0.�t /=r/,hence
at D arg maxa02A rg.a0; Nbt /C U 0.�t / .at ; Nbt ; �t />�. Nbt /�1„ ƒ‚ …rˇ>t
�.a0; Nbt / ;
Nbt D arg maxb02B �th.a�; b0; Nbt /C .1 � �t /h.at ; b0; Nbt /;(21)
and therefore sequential rationality follows from Proposition 3. We conclude that .at ; Nbt ; �t /t�0 is apublic sequential equilibrium that yields a payoff of U.p/ to the normal type.
Finally, we show that in any public sequential equilibrium .at ; Nbt ; �t /t�0 the equilibrium actions areuniquely determined by the small players’ belief by (17). Indeed, if .Wt /t�0 is the process of continuationvalues of the normal type, then the pair .�t ; Wt / must stay on the graph of U , because there is no publicsequential equilibrium with continuation value different from U.�t /, as we have shown above. Therefore,the volatility of Dt D Wt � U.�t / must be identically zero, i.e. rˇ>t �. Nbt / D U 0.�t / .at ; Nbt ; �t />. ThusProposition 3 implies condition (21), and therefore .at ; Nbt / D N .�t ; �t .1 � �t /U 0.�t /=r/. �
Theorem 4 is a striking result because it highlights equilibrium properties that hold in discrete timeonly in approximation, and also because it summarizes reputational incentives through a single variablez D �.1��/U 0.�/=r . By contrast, in the standard discrete-time model equilibrium behavior is not pinneddown by the small players’ posterior belief and Markov perfect equilibria may fail to exist.21 However,as we explain in our conclusions in Section 10, we expect the characterization of Theorem 4 to capturethe limit equilibrium behavior in games in which players can respond to new information quickly andinformation arrives in a continuous fashion.22 The uniqueness and characterization result shows that our
21See Mailath and Samuelson (2006, pp. 560–566) for a discrete-time finite-horizon variation of the product-choice game inwhich no Markov perfect equilibrium exists.
22We expect our methods to apply broadly to other continuous-time games with payoff-relevant state variables, such as theCournot competition with mean-reverting prices of Sannikov and Skrzypacz (2007). In that model the market price is the payoff-relevant state variable.
23
continuous-time formulation provides a tractable model of reputation phenomena that can be valuable forapplications, as we demonstrate by several examples below. The fact that the reputational incentives aresummarized by a single variable is particularly useful for applications with multi-dimensional actions.
The characterization of the unique sequential equilibrium by a differential equation, in addition tobeing tractable for numerical computation, is useful to derive qualitative conclusions about equilibriumbehavior, both at the general level and for specific classes of games. First, in addition to the existence,uniqueness and characterization of sequential equilibrium, Theorem 4 provides a simple sufficient condi-tion for the equilibrium value of the normal type to be monotonically increasing in reputation; namely,the condition that the stage-game Bayesian Nash equilibrium payoff of the normal type is increasing inthe prior on the behavioral type. By contrast, in the canonical discrete-time model, it is not known underwhich conditions reputation has a positive value.
Second, Theorem 4 implies that, under Conditions 1 and 2, in equilibrium the normal type neverperfectly imitates the behavioral type. Since the equilibrium actions are determined by (17), if the normaltype imitated the behavioral type perfectly at time t , then .a�; Nbt / would be a Bayesian Nash equilibriumof the static game with prior �t , and this would violate Condition 1.
Third, the characterization implies the following square root law of substitution between the discountrate and the volatility of the signals:
Corollary 1. Under Conditions 1 and 2, multiplying the discount rate by a factor of ˛ > 0 has the sameeffect on the equilibrium value of the normal type as rescaling the volatility matrix � by a factor of
p˛.
Proof. This follows directly from observing that the discount rate r and the volatility matrix � enter theoptimality equation only through the product r��>. �
Fourth, the differential equation characterization of Theorem 4 is also useful to study reputationalincentives in the limit as the large player gets patient. In Section 6.1 below, we use the optimality equationto prove a new general result about reputation effects at the level of equilibrium behavior (as opposed toequilibrium payoffs, as in Fudenberg and Levine (1992) and Faingold (2008)). Under Conditions 1 and 2,if the behavioral type plays the Stackelberg action and the public signals satisfy an identifiability condition,as the large player gets patient the equilibrium action of the normal type approaches the Stackelberg actionat every reputation level (Theorem 5).
Finally, to illustrate how our characterization can be useful to study specific classes of games, wediscuss a few examples:
(a) Signal manipulation. An agent chooses a costly effort a1 2 Œ0; 1�, which affects two public signals,X1 and X2. The agent also chooses a level of signal manipulation a2 2 Œ0; 1�, which affects onlysignal X1. Thus, as in Holmstrom and Migrom’s (1991) multitasking agency model, the agent’shidden action is multi-dimensional and the signals are distorted by a Brownian motion. Specifically,
dX1;t D .a1;t C a2;t / dt C dZ1;t ;
dX2;t D a1;t dt C � dZ2;t ;
where � � 1, i.e. the signal that cannot be manipulated is less informative. There is a competitivemarket of small identical principals, each of whom is willing to hire the agent at a spot wage of
24
Nbt , which equals the market expectation of the agent’s effort given the public observations of thesignals. Thus, the principals do not care directly about the agent’s signal manipulation activity, butin equilibrium manipulation can affect the principals’ behavior through their statistical inference ofthe agent’s effort. The agent’s cost of effort and manipulation is quadratic, so that his flow payoff is
Nbt � 12.a21;t C a22;t /:
Finally, the behavioral type is the Stackelberg type, i.e. he is committed to a� D .1; 0/ (full effortand no signal manipulation).
The signal manipulation game satisfies Conditions 1 and 2 directly (both 2.a and 2.b), and henceits sequential equilibrium is unique and Markovian in reputation, and the value function of thenormal type is increasing and characterized by the optimality equation. We can examine the agent’sintertemporal incentives using the characterization. In equilibrium, when the reputational weight isz � 0, the agent will choose .a1; a2/ to maximize
�12.a01
2 C a022/C zh1 � a1 � a2 1 � a1
i " 1 0
0 1=�2
#"a01a02
#over all .a01; a02/ 2 Œ0; 1� � Œ0; 1�. The first-order conditions are
�a1 C z.1 � a1 � a2/C z��2.1 � a1/ D 0;
�a2 C z.1 � a1 � a2/ D 0;
which yield
a1 D .1C 1=�2/z C z2=�21C .2C 1=�2/z C z2=�2 ;
a2 D z
1C .2C 1=�2/z C z2=�2 :
As shown in Figure 4, when the reputational weight is close to zero (as in equilibrium when � isnear 0 or 1), the agent exerts low effort and engages in nearly zero manipulation. As the reputationalweight z grows, the agent’s effort increases monotonically and converges to maximal effort a�1 D 1,while the manipulation action is single-peaked and approaches zero manipulation. In Figure 4 wecan also see the effect of the informativeness of the non-manipulable signal: as � increases, so thatthe non-manipulable signal becomes noisier, the agent’s equilibrium effort decreases and the amountof signal manipulation increases at every level of the reputational weight z > 0. The intuition isthat when the volatility of the non-manipulable signal increases, it becomes cheaper to maintain areputation by engaging in signal manipulation relative to exerting true effort. Indeed, a greater �implies that the small players update their beliefs by less when they observe unexpected changes inthe non-manipulable signalX2; on the other hand, irrespective of � the actions a1 and a2 are perfectsubstitutes in terms of their impact in the informativeness of X1.
Finally, we emphasize that characterizations like this one are impossible to achieve in discrete time,because equilibria typically fail to be Markovian and incentives cannot be characterized by a singlereputational weight parameter.
25
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 10 20 30 40 50 60 70 80
Effort(σ=1)
Manipula<on(σ=5)
Reputa<onalweightz
Effort(σ=5)
Manipula<on(σ=1)
Figure 4: Effort and manipulation in the signal manipulation game.
(b) Monetary policy. As in Cukierman and Meltzer (1986), at each time t � 0 a central bank chooses acostly policy variable at 2 Œ0; 1�, which is the target rate of monetary growth. This policy variableis unobservable to the “population” and affects the stochastic evolution of the aggregate price level,Pt , as follows:
Pt D expXt ; where dXt D at dt C � dZt :At each time t , the population formulates a rational expectation aet of the current rate of moneygrowth, given their past observations of X . In reduced form, the behavior of the population givesrise to a law of motion of the aggregate level of employment nt (in logarithmic scale), as follows:
dnt D �. Nn � nt / dt C &.dXt � aet dt/; (22)
where Nn is the long-run level of employment. Thus, the residual change in employment is propor-tional to the unanticipated inflation, as in a Phillips curve.
The central bank cares about stimulating the economy but inflation is costly to society; specifically,the average discounted payoff of the central bank isZ 1
0
re�rt�˛nt � a2t
2
�dt; (23)
where ˛ > 0 is the marginal rate of substitution between stimulus and inflation. To calculate theexpected payoff of the central bank, first note that (22) implies
nt D n0e��t C Nn.1 � e��t /C &Z t
0
e�.s�t/ .dXs � aes ds/:
Plugging into (23) yields the following expression for the central bank’s discounted payoff:
n0r
r C � C Nn�1 � r
r C ��C Z 1
0
re�rt�˛&
Z t
0
e�.s�t/ .dXs � aes ds/ �a2t2
�dt;
26
which, by integration-by-parts, equals
n0r
r C � C Nn�1 � r
r C ��C Z 1
0
re�rt� ˛&
r C � .dXt � aet dt/ �
a2t2dt�:
Thus, modulo a constant, the expected flow payoff of the central bank is
˛&
r C � .at � aet / �
a2t2:
In a static world, the unique Nash equilibrium would have the central bank targeting an inflationrate of ˛&
rC� > 0. By contrast, if the central bank had the credibility to commit to a rule, it wouldfind it optimal to commit to zero inflation target. Indeed, under commitment, the population wouldanticipate the target, and hence the term at � aet would be identically zero and the central bankwould be left with minimizing the cost of inflation.23
Consider now the reputation game with a behavioral type committed to a� D 0. To study the centralbank’s incentives, consider his maximization problem in the definition of correspondence N . Whenthe reputational weight is z, the central bank chooses a 2 Œ0; 1� to maximize
˛&a0
r C � �a02
2� zaa
0
�2over all a0 2 Œ0; 1�;
and therefore setsa D ˛&
.r C �/.1C z=�2/ :
Thus, when the reputational weight z is near zero (which happens in equilibrium when � is closeto 0 or 1), the central bank succumbs to his myopic incentive to inflate prices and sets the targetrate around ˛&
rC� ; as the reputational weight grows, the target inflation decreases monotonicallyand approaches zero asymptotically. Moreover, in equilibrium the central bank sets a higher targetinflation rate at each reputational weight when: (i) it values stimulus more (i.e. ˛ is greater), (ii)the effect of unanticipated inflation on employment takes longer to disappear (i.e. � is smaller),(iii) changes in employment are more sensitive to unexpected inflation (i.e. & is greater), or (iv) thefluctuations in the aggregate price level are more volatile (i.e. � is greater)
(c) Product choice. The repeated product choice game of Section 2 satisfies Conditions 1 and 2 (both2.a and 2.b), and hence its sequential equilibrium is unique, Markovian and the equilibrium valuefunction of the normal type, U , is increasing in reputation. For each .�; z/ 2 Œ0; 1� � R the uniqueprofile .a.�; z/; b.�; z// 2 N .�; z/ is
a.�; z/ D(0 W z � 11 � 1=z W z > 1
;
b.�; z/ D(0 W � C .1 � �/a.�; z/ � 1=44 � .� C .1 � �/a.�; z//�1 W � C .1 � �/a.�; z/ > 1=4 :
23This discrepancy between the Nash and the Stackelberg outcomes is an instance of the familiar time consistency problem inmacroeconomic policy studied by Kydland and Prescott (1977).
27
The particular functional form we have chosen for the product choice game is not important tosatisfy Conditions 1 and 2. More generally, consider the case in which the quality of the productstill follows24
dXt D atdt C dZt ;but the flow profit of the firm and the flow surplus of the consumers take the more general form
Nbt � c.at / and v.bit ;Nbt / dXt � bit dt;
respectively, where c.at / is the cost of effort and v.bit ; Nbt / is the consumer’s valuation of the qualityof the product. Assume that c and v are twice continuously differentiable and satisfy the standardconditions c0 > 0, c00 � 0, v1 > 0 and v11 � 0, where the subscripts denote partial derivatives. Inaddition, assume that
v12 < �v11; (24)
so that either there are negative externalities among the consumers, i.e. v12 < 0 as in the exampleabove, or there are positive, but weak, externalities among them. Then, Conditions 1 and 2.b aresatisfied and the conclusion of Theorem 4 still applies. An example in which condition (24) fails isanalyzed in Section 7.
Finally, note that condition (24) arises naturally in the important case in which there is a continuousflow of short-lived consumers (as in Fudenberg and Levine (1992), Cripps, Mailath, and Samuelson(2004) and Faingold (2008)) instead of a cross-section population of small long-lived consumers(cf. the discussion in Section 3.1). In fact, when there is a single short-lived consumer at each timet , v is independent of Nb and therefore (24) reduces to the condition v11 < 0.
6.1 Reputation effects at the behavioral level (as r ! 0)
With the equilibrium uniqueness result in place, we can examine the impact of the large player’s patienceon the equilibrium strategies of the large player and the population of small players. In Theorem 5 be-low, we present a reputation effects result that is similar in spirit to Theorem 1 but concerns equilibriumbehavior rather than equilibrium payoffs.
DefineN 0 defDf.a; b; �/ W 9z 2 R such that .a; b/ 2 N .�; z/g:
We need the folowing assumption:
Condition 3. The following two conditions are satisfied:
a. 9K > 0 such that j�.a�; b/ � �.a; b/j � K ja� � aj, 8.a; b/ in the range of N ,
b. 9R > 0 such that jb� � bj � R .1 � �/ja� � aj, 8.a; b; �/ 2 N 0, 8b� 2 B.a�/.
24When the drift is independent of the aggregate strategy of the small players and is monotonic in the large player’s action,imposing a linear drift is just a normalization.
28
Condition 3.a is an identifiability condition: it implies that at every action profile in the range of N ,the action of the normal type can be statistically distinguished from a� by estimating the drift of thepublic signal. Condition 3.b is a mild technical condition. To wit, the upper hemi-continuity of N impliesthat for any convergent sequence .an; bn; �n/n2N such that .an; bn/ 2 N .�n; zn/ for some zn, if eitherlim an D a� or lim�n D 1 then lim bn D b� 2 B.a�/; Condition 3.b strengthens this continuity propertyby requiring the modulus of continuity to be proportional to .1 � �n/ja� � anj. In Section 6.2 below, weprovide sufficient conditions for Condition 3 (as well as for Conditions 1 and 2) in terms of the primitivesof the game.
Recall that the Stackelberg payoff is
Ngs D maxa2A
maxb2B.a/
g.a; b/:
Say that a� is a Stackelberg action if it attains the Stackelberg payoff, i.e. Ngs D maxb2B.a�/ g.a�; b/.
Finally, under Conditions 1 and 2, we write .ar.�/; br.�// to designate the unique Markov perfect equi-librium when the discount rate is r and, likewise, we write Ur.�/ for the equilibrium value of the normaltype.
Theorem 5. Assume Conditions 1, 2 and 3 and that a� is the Stackelberg action. Let fb�g D B.a�/. Then,for each � 2 .0; 1/,
limr!0Ur.�/ D Ng
s D g.a�; b�/ and limr!0.ar.�/; br.�// D .a
�; b�/:
The rate of convergence above is not uniform in the reputation level: as � approaches 0 or 1, the con-vergence gets slower and slower. The proof of Theorem 5, presented in Appendix C.5, uses the differentialequation characterization of Theorem 4, as well as the payoff bounds of Theorem 1.
6.2 Primitive sufficient conditions
To illustrate the applicability of Theorems 4 and 5 we provide sufficient conditions for Conditions 1, 2 and3 in terms of the basic data of the game—stage-game payoff functions, drift and volatility. The conditionsare stronger than necessary, but are easy to check and have a transparent interpretation.
To ease the exposition, we focus on the case in which the actions and the public signal are one-dimensional.25 We use subscripts to denote partial derivatives.
Proposition 4. Suppose A, B � R are compact intervals, a� D maxA and the public signal is one-dimensional. If g W A � B ! R, h W A � B � B ! R and � W A � B ! R are twice continuouslydifferentiable, � W B ! R n f0g is continuous and for each .a; b; Nb/ 2 A � B � B ,
(a) g11.a; Nb/ < 0, h22.a; b; Nb/ < 0, .h22Ch23/.a; Nb; Nb/ < 0,�g11.h22Ch23/�g12h12
�.a; Nb; Nb/ > 0,
(b) g2.a; Nb/.h2.a�; Nb; Nb/ � h2.a; Nb; Nb// � 0,
25In models with multi-dimensional actions / signals, the appropriate versions of conditions (a)–(d) below are significantlymore cumbersome. This is reminiscent of the analysis of equilibrium uniqueness in one-shot games and has nothing to do withreputation dynamics.
29
(c) �1.a; Nb/ > 0, �11.a; Nb/ � 0,
then Conditions 2.b and 3 are satisfied. If, in addition,
(d) g1.a�; b�/ < 0, where fb�g D B.a�/,
then Condition 1 is also satisfied.
Evidently, any conditions implying equilibrium uniqueness in the reputation game ought to be strongconditions, as they must, at least, imply equilibrium uniqueness in the stage game. Interestingly, Proposi-tion 4 shows that in our continuous-time framework such conditions are not much stronger than standardconditions for equilibrium uniqueness in static games. Indeed, (a) and (b) are standard sufficient con-ditions for the static-game analogue of Condition 2.b, so the only extra condition that comes from thereputation dynamics is (c), a monotonicity and concavity assumption on the drift which is natural in manysettings. We view this as the main reason why our continuous-time formulation is attractive from theapplied perspective. Many interesting applications, such as the product choice game of Section 2 andthe signal manipulation and monetary policy games presented after Theorem 4 satisfy the conditions ofProposition 4.
For Condition 2.b, first we ensure that the best-reply correspondences are single-valued by assumingthat the payoffs are strictly concave in own action, i.e. g11 < 0 and h22 < 0, and that .h22Ch23/.a; b; b/ <0 for all .a; b/. The latter condition guarantees that the payoff complementarity between the actions ofthe small players is not too strong, ruling out coordination effects which can give rise to multiple equilib-ria. Next, the assumption that g11.h22 C h23/ � g12h12 > 0 (along the diagonal of the small players’actions) implies that the graphs of the best-reply functions intersect only once. This is also a familiarcondition: together with g11 < 0 and h22 < 0 it implies the negative definiteness of the matrix ofsecond-order derivatives of the payoff functions of the complete information one-shot game. In addi-tion, condition (c) guarantees that the large player’s best-reply remains unique when the reputational termz�.b/�2.�.a�; Nb/ � �.a; Nb//�.a0; Nb/, with z � 0, is added to his static payoff. The monotonicity of thestage-game Bayesian Nash equilibrium payoff of the normal type follows from condition (b), which re-quires that whenever the small players can hurt the normal type by decreasing their action (so that g2 > 0),their marginal utility cannot be smaller when they are facing the behavioral type than when they are fac-ing the normal type (otherwise they would have an incentive to decrease their action when their beliefsincrease, causing a decrease in the payoff of the normal type).
Condition 3.a follows from �1 > 0 and Condition 3.b follows from h22 < 0 and h22Ch23 � 0 (alongthe diagonal of the small players’ actions). Finally, for Condition 1, observe that under the assumptionthat �1 > 0 there is no a ¤ a� with �.a; b�/ D �.a�; b�/. Thus Condition 1 follows directly fromassumption (d), which states that a� is not part of a static Nash equilibrium of the complete informationgame. The proof of Proposition 4 is presented in Appendix C.6.
Under a set of assumptions similar to those of Proposition 4, Liu and Skrzypacz (2010) prove unique-ness of sequential equilibrium in discrete-time reputation games in which the short-run players observeonly the outcomes of a fixed, finite number of past periods. They also find that in the complete infor-mation version of their repeated game, the only equilibrium is the repeated play of the stage-game Nashequilibrium, a result similar to our Theorem 3.
30
Remark 4. The assumption that a� is the greatest action in A is only for convenience. The conclusion ofProposition 4 remains valid under any a� 2 A, provided condition (c) is replaced by
�1.a; Nb/ > 0; .�.a�; Nb/ � �.a; Nb//�11.a; Nb/ � 0 8.a; Nb/ 2 A � B;
and condition (d) by
g1.a�; b�/
8̂<̂:< 0 W a� D maxA;¤ 0 W a� 2 .minA;maxA/;> 0 W a� D minA:
Remark 5. Among conditions (a)–(d), perhaps the ones that are most difficult to check are�g11.h22 C h23/ � g12h12
�.a; Nb; Nb/ > 0 and g2.a; Nb/.h2.a�; Nb; Nb/ � h2.a; Nb; Nb// � 0:
But, assuming the other conditions in (a), a simple sufficient condition for the above is that
g12.a; Nb/ � 0; h12.a; Nb; Nb/ � 0 and g2.a; Nb/ � 0;
as is the case in the product choice game of Section 2.
7 General characterization
This section extends the analysis of Section 6 to environments with multiple sequential equilibria. Whenthe correspondence N is not single-valued (so that Condition 2 is violated), the correspondence of se-quential equilibrium payoffs, E , may also fail to be single-valued. Theorem 6 below characterizes E inthis general case.
To prove the general characterization, we maintain Condition 1 but replace Condition 2 by:
Condition 4. N .�; z/ ¤ ; 8.�; z/ 2 Œ0; 1� �R.
This assumption is automatically satisfied in the mixed-strategy extension of the game (cf. remark 2).However, to simplify the exposition, we have chosen not to deal with mixed strategies explicitly andimpose Condition 4 instead.
Consider the optimality equation of Section 6. When N is a multi-valued correspondence, there canbe multiple bounded functions which solve
U 00.�/ D 2U 0.�/1 � � C
2r.U.�/ � g.a.�/; Nb.�///j .a.�/; Nb.�/; �/j2 ; � 2 .0; 1/ (25)
for different selections � 7! .a.�/; Nb.�// 2 N .�; �.1 � �/U 0.�/=r/. An argument similar to the proofof Theorem 4 can be used to show that for each such solution U and each prior p, there exists a sequen-tial equilibrium that achieves the payoff U.p/ for the normal type. Therefore, a natural conjecture is thatthe correspondence of sequential equilibrium payoffs, E , contains all values between its upper bound-ary, the greatest solution of (25), and its lower boundary, the least solution of (25). Accordingly, thepair .a.�/; Nb.�// 2 N .�; �.1 � �/U 0.�/=r/ should minimize the right-hand side of (25) for the upperboundary, and maximize it for the lower boundary.
31
However, the differential equation
U 00.�/ D H.�;U.�/; U 0.�//; � 2 .0; 1/; (26)
where H W Œ0; 1� �R2 ! R is given by
H.�; u; u0/ defD 2u0
1 � � Cmin
(2r.u � g.a; Nb//j .a; Nb; �/j2 W .a; Nb/ 2 N .�; �.1 � �/u0=r/
); (27)
may fail to have a solution in the classical sense. In general, N is upper hemi-continuous but not neces-sarily lower hemi-continuous, so the right-hand side of (26) can be discontinuous. Due to this technicaldifficulty, our analysis relies on a generalized notion of solution, called viscosity solution (cf. Definition 2below), which applies to discontinuous equations.26 We show that the upper boundary U.�/ defD sup E.�/
is the greatest viscosity solution of the upper optimality equation (26), and that the lower boundaryL.�/
defD inf E.�/ is the least solution of the lower optimality equation, which is defined analogously, byreplacing minimum by maximum in the definition of H .
While in general viscosity solutions can fail to be differentiable, we show that the upper boundaryU is continuously differentiable and has an absolutely continuous derivative. In particular, when N iscontinuous in a neighborhood of .�; �.1 � �/U 0.�// for some � 2 .0; 1/, so that H is also continuousin that neighborhood, any viscosity solution of (26) is a classical solution around �. Otherwise, we showthat U 00.�/; which exists almost everywhere since U 0 is absolutely continuous, must take values in theinterval between H.�;U.�/; U 0.�// and H�.�; U.�/; U 0.�//, where H� is the upper semi-continuousenvelope of H , i.e. the least u.s.c. function which is greater than or equal to H everywhere. (Note that His necessarily lower semi-continuous, i.e. H D H�.)
Definition 2. A bounded function U W .0; 1/ ! R is a viscosity super-solution of the upper optimalityequation if for every � 2 .0; 1/ and every twice continuously differentiable test function V W .0; 1/! R,
U�.�/ D V.�/ and U� � V H) V 00.�/ � H�.�; V .�/; V 0.�// :
A bounded function U W .0; 1/ ! R is a viscosity sub-solution if for every � 2 .0; 1/ and every twicecontinuously differentiable test function V W .0; 1/! R,
U �.�/ D V.�/ and U � � V H) V 00.�/ � H�.�; V .�/; V 0.�//:
A bounded function U is a viscosity solution if it is both a super-solution and a sub-solution.27
Appendix D presents the complete analysis, which we summarize here. Propositions D.1 and D.2show that U , the upper boundary of E , is a bounded viscosity solution of the upper optimality equation.Lemma D.3 then shows that U must be a continuously differentiable function with absolutely continuousderivative (so its second derivative exists almost everywhere), and hence U must solve the differentialinclusion
U 00.�/ 2 ŒH.�; U.�/; U 0.�// ; H�.�; U.�/; U 0.�//� a.e. (28)
26For an introduction to viscosity solutions we refer the reader to Crandall, Ishii, and Lions (1992).27This is equivalent to Definition 2.2 in Crandall, Ishii, and Lions (1992).
32
In particular, when H is continuous in a neighborhood of .�; U.�/; U 0.�// then U satisfies equation (26)in the classical sense in a neighborhood of �. Finally, Proposition D.3 shows that U must be the greatestbounded solution of (28).
We summarize our characterization in the following theorem.
Theorem 6. Assume Conditions 1 and 4 and that E is nonempty-valued. Then, E is a compact- andconvex-valued continuous correspondence whose upper and lower boundaries are continuously differen-tiable functions with absolutely continuous derivatives. Moreover, the upper boundary of E is the greatestbounded solution of the differential inclusion (28), and the lower boundary is the least bounded solutionof the analogous differential inclusion, where maximum is replaced by minimum in the definition of H .
To illustrate the case of multiple sequential equilibria, we provide two examples.
(a) Product choice with positive externalities. This is a simple variation of the product choice gameof Section 2. Suppose the firm chooses effort level at 2 Œ0; 1�, where a� D 1 is the action of thebehavioral type, and that each consumer chooses a level of service bit 2 Œ0; 2�. The public signalabout the firm’s effort follows
dXt D at dt C dZt :The payoff flow of the normal type is . Nbt �at / dt and each consumer i 2 I receives the payoff flowbitNbt dXt �bit dt . Thus, the consumers’ payoff function features a positive externality: greater usageNbt of the service by other consumers allows each individual consumer to enjoy the service more.
The unique Nash equilibrium of the static game is .0; 0/. The correspondence N .�; z/ determinesthe action of the normal type uniquely by
a D(0 if z � 11 � 1=z otherwise:
(29)
The consumers’ actions are uniquely Nb D 0 only when .1 � �/a C �a� < 1=2. When .1 � �/a C�a� � 1=2 the partial game amongst the consumers, who face a coordination problem, has two pureNash equilibria with Nb D 0 and Nb D 2 (and one mixed equilibrium when .1 � �/a C �a� > 1=2).Thus, the correspondence N .�; z/ is single-valued only on a subset of its domain.
How is this multiplicity reflected in the equilibrium correspondence E.p/? Figure 5 displays theupper boundary of E.p/ computed for discount rates r D 0:1, 0:2 and 0:5. The lower boundary forthis example is identically zero, because the static game among the consumers has an equilibriumwith Nb D 0. For each discount rate, the upper boundary U is divided into three regions. In theregion near � D 0, where the upper boundary is displayed as a solid line, the correspondence� 7! N .�; �.1 � �/U 0.�/=r/ is single-valued and U satisfies the upper optimality equation in theclassical sense. In the region near � D 1, where the upper boundary is displayed as a dashed line,the correspondence N is continuous and takes multiple values (two pure and one mixed). There,U also satisfies the upper optimality equation in the classical sense, with the small players’ actiongiven by Nb D 2. In the middle region, where the upper boundary is shows as a dotted line, we have
U 00.�/ 2�2U 0.�/1 � � C
2r.U.�/ � 2C a/j .a; 2; �/j2 ;
2U 0.�/1 � � C
2r.U.�/ � 0C a/j .a; 0; �/j2
�;
33
r D 0:1
r D 0:2
r D 0:5
0
0.5
0.5
1
1
2
1.5
p
Payoff of the normal type
1
Figure 5: The upper boundary of E.p/.
where a is given by (29) with z D �.1 � �/U 0.�/=r , and 0 and 2 are the two pure values of Nbthat the correspondence N returns. In that range, the correspondence N .�; �.1 � �/U 0.�/=r/ isdiscontinuous in its arguments: if we lower U 0.�/ slightly the equilibrium among the consumerswith Nb D 2 breaks down. These properties of the upper boundary follow from its characterizationas the greatest solution of the upper optimality equation.
(b) Bad reputation. This is a continuous-time version of the reputation game of Ely and Valimaki (2003)with noisy public signals (see also Ely, Fudenberg, and Levine (2008)). The large player is a carmechanic, who chooses the probability a1 2 Œ0; 1� with which he replaces the engine on cars thatneed an engine replacement, and the probability a2 2 Œ0; 1� with which he replaces the engineon cars that need a mere tune-up. Thus, the stage-game action of the mechanic, a D .a1; a2/, istwo-dimensional.
Each small player (car owner), without the knowledge of whether his car needs an engine replace-ment or a tune-up, decides on the probability b 2 Œ0; 1�with which he brings the car to the mechanic.The behavioral type of mechanic—a bad behavioral type—replaces the engines of all cars, irrespec-tive of which repair is more suitable, i.e. a� D .1; 1/. Car owners observe noisy information aboutthe number of engines replaced by the mechanic in the past:
dXt D Nbt .a1;t C a2;t / dt C dZt :The payoffs are given by
g.a; Nb/ D Nb.a1 � a2/ and h.a; b; Nb/ D b.a1 � a2 � 1=2/:Thus the normal type of mechanic is a good type, in that he prefers to replace the engine only whenit is needed. As for the car owners, they prefer to take their car to the mechanic only when they
34
believe the mechanic is sufficiently honest, i.e. a1�a2 � 1=2; otherwise, they prefer not to take thecar to the mechanic at all.
While this game violates Condition 1, we are able to characterize the set of public sequential equi-librium payoffs of the normal type via a natural extension of Theorem 6 (see Remark 6 below).The correspondence E is illustrated in Figure 6. The lower boundary of this correspondence, L, isidentically 0. The upper boundary, U , displayed as a solid line, is characterized by three regions: (i)for � � 1=2, U.�/ D 0, (ii) for � 2 Œ��; 1=2�, U.�/ D r log 1��
�, and (iii) for � � ��, U.�/ solves
the upper optimality equation in the classical sense:
U 00.�/ D 2U 0.�/1 � � C
2r.U.�/ � 1/�2.1 � �/2 ;
with boundary conditions
U.0/ D 1; U.��/ D r log1 � ����
and U 0.��/ D � r
��.1 � ��/ ;
which is attained by the action profile .a; Nb/ D ..1; 0/; 1/.28 In particular, the upper boundary U isalways below r log 1��
�for � < 1=2, as illustrated in Figure 6.
For � � 1=2 the set N .�; �.1 � �/U 0.�/=r/ includes the profile ..1; 0/; 1/ as well as the profiles..a1; a2/; 0/ with a1 � a2 � 1=.2.1 � �//, and the former profile must be played on the upperboundary of the equilibrium set in this belief range.29 Moreover, for each prior � < 1=2, there is a(non-Markovian) sequential equilibrium attaining the upper boundary in which ..1; 0/; 1/ is playedon the equilibrium path so long as the car owners’ posterior remains below 1=2 and the continuationvalue of the normal type is strictly positive.30 However, since r log 1��
�! 0 as r ! 0 for all � > 0,
for all priors the greatest equilibrium payoff of the normal type converges to 0 as r ! 0. This resultis analogous to Ely and Valimaki (2003, Theorem 1).
Remark 6. The characterization of Theorem 6 can be extended to games in which Condition 1 fails asfollows. Denote by
V.�/ D co˚g.a; Nb/ W .a; Nb/ 2 N .�; 0/ and �.a; Nb/ D �.a�; Nb/ :
28This differential equation turns out to have a closed-form solution:
U.�/ D 1 ��1 � r log
1 � ����
�h .1 � ��/���.1 � �/
i 12 .1C
p1C8r/
; � 2 Œ0; ���:
The cutoff �� is pin down by the condition U 0.��/ D �r=.��.1 � ��//, which yields �� D 1=�1C e 5�p
1C8r4r
�.
29For � 2 Œ��; 1=2�, the slope U 0.�/ D �r=.�.1 � �// is such that, at the profile ..1; 0/; 1/ 2 N .�; �.1 � �/U 0.�/=r/, themechanic is indifferent among all actions .a1; 0/, a1 2 Œ0; 1�. Moreover, the correspondence N is discontinuous at .�; �.1 ��/U 0.�/=r/ for all � 2 Œ��; 1=2� and U satisfies the differential inclusion (28) with strict inequality.
30In this equilibrium, the posterior on the behavioral type, �t , and the continuation values of the normal type, Wt , followd�t D �t .1 � �t / .dXt � .�t C 1/ dt/ and dWt D r.Wt � 1/ dt C �t .1 � �t /U 0.�t / .dXt � dt/ up until the first time when.�t ; Wt / hits the lower boundary; from then on, the equilibrium play follows the static equilibrium ..0; 0/; 0/ and the posteriorno longer updates. When �t is below ��, the optimality equation implies that the pair .�t ; Wt / remains at the upper boundary;but, for �t 2 Œ��; 1=2�, the differential inclusion, which is satisfied with strict inequality, implies that Wt eventually falls strictlybelow the upper boundary.
35
0
0.5
1
1.5
0.0 0.5 1.0
U(φ)
rlog((1‐φ)/φ)
φ*
φ
Figure 6: The correspondence E in the bad reputation game (for r D 0:4).
The set V.�/ can be attained by equilibria in which the normal type always “looks” like the behavioraltype, and his reputation stays fixed. For games in which Condition 1 holds, the correspondence V.�/ isempty, and for the bad reputation example above, V.�/ D f0g for all �: Then, for any belief � 2 .0; 1/,the upper boundary, U.�/, is the maximum between max V.�/ and the greatest solution of the differentialinclusion (28) that remains bounded both to the left and to the right of � until the end of the interval .0; 1/or until it reaches the correspondence V . The lower boundary is characterized analogously. For our badreputation example, the upper boundary of E solves the differential inclusion on .0; 1=2� and reaches thecorrespondence V at the belief level 1=2.
Remark 7. A caveat of Theorem 6 is that it assumes the existence of a public sequential equilibrium, al-though existence is not guaranteed in general unless we assume the availability of a public randomizationdevice. Standard existence arguments based on fixed-point methods do not apply to our continuous-timegames (even assuming finite action sets and allowing mixed strategies), because continuous time rendersthe set of partial histories at any time t uncountable.31 This is similar to the existence problem that arisesin the context of subgame perfect equilibrium in extensive-form games with continuum action sets, as inHarris, Reny, and Robson (1995). Moreover, while it can be shown that the differential inclusion (28) isguaranteed to have a bounded solution U W .0; 1/! R under Conditions 1 and 4, this does not imply theexistence of a sequential equilibrium, because a selection � 7! .a.�/; Nb.�// satisfying (25) may not existwhen N is not connected-valued. However, if the model is suitably enlarged to allow for public random-ization, then existence of sequential equilibrium is restored. In our supplementary appendix, Faingold andSannikov (2010), we explain the formalism of public randomization in continuous time and demonstratethat a sequential equilibrium in publicly randomized strategies is guaranteed to exist under Conditions 1
31At a technical level, the problem is the lack of a topology on strategy sets under which expected discounted payoffs arecontinuous and strategy sets are compact.
36
and 4. It is interesting to note that also in Harris, Reny, and Robson (1995) public randomization is the keyto obtain existence of subgame perfect equilibrium.
8 Multiple behavioral types
In this section we consider reputation games with multiple behavior types. We extend the recursive char-acterization of sequential equilibrium of Section 4 and prove an analogue of Cripps, Mailath, and Samuel-son’s (2004) result that reputation effects are a temporary phenomenon. These results lay the groundworkfor future analysis of these games, and we leave the extension of the characterizations of Sections 6 and 7for future research.
Suppose there are K < 1 behavioral types, where each type k 2 f1; : : : ; Kg plays a fixed actiona�k2 A at all times, after all histories. Initially the small players believe that the large player is behavioral
type k with probability pk 2 Œ0; 1�, so that p0 D 1 �PKkD1 pk > 0 is the prior probability on the normal
type. All else is exactly as described in Section 3.To derive the recursive characterization of sequential equilibrium, the main difference from Section 4
is that now the small players’ belief is a vector in the K-dimensional simplex, �K . Accordingly, for eachk D 0; : : : ; K we write �k;t to designate the small players’ belief that the large player is of type k, andwrite �t D .�0;t ; : : : ; �K;t /.Proposition 5 (Belief Consistency). Fix a prior p 2 �K . A belief process .�t /t�0 is consistent with astrategy profile .at ; Nbt /t�0 if and only if �0 D p and for each k D 0; : : : ; K,
d�k;t D k.at ; Nbt ; �t / � �. Nbt /�1.dXt � ��t .at ; Nbt / dt/; (30)
where for each .a; Nb; �/ 2 A ��.B/ ��K ,
0.a; Nb; �/ defD�0�. Nb/�1.�.a; Nb/ � ��.a; Nb//; k.a; Nb; �/ defD�k�. Nb/�1.�.a�k ; Nb/ � ��.a; Nb//; k D 1; : : : ; K;
��.a; Nb/ defD�0�.a; Nb/CKXkD1
�k�.a�k ;Nb/:
Proof. As in the proof of Proposition 1, the relative likelihood �k;t that a signal path arises .XsI s 2 Œ0; t �/from the behavior of type k instead of the normal type is characterized by
d�k;t D �k;t �k;t � dZ0s ; �k;0 D 1;
where �k;tdefD �. Nbt /�1.�.a�k ; Nbt / � �.at ; Nbt // and dZ0t
defD �. Nbt /�1.dXt � �.at ; Nbt / dt/ is a Brownianmotion under the normal type. By Bayes’ rule,
�k;t Dpk�k;t
p0 CPKkD1 pk�k;t
:
Applying Itô’s formula to this expression yields
d�k;t Dpk
p0 CPKkD1 pk�k;t
d�k;t �KXk0D1
pk�k;t
.p0 CPKkD1 pk�k;t /2
pk0d�k0;t C .: : :/ dt;
37
where we do not need to derive the .: : :/ dt term because we know that .�k;t /t�0 is a martingale from thepoint of view of the small players. Since dZ�t D �. Nbt /�1.dXt � ��t .at ; Nbt / dt/ is a Brownian motionfrom the viewpoint of the small players,
d�k;t Dpk
p0 CPKkD1 pk�k;t
�k;t�k;t � dZ�t �� KXk0D1
pk�k;t
.p0 CPKkD1 pk�k;t /2
pk0�k0;t�k0;t� � dZ�t
D �k;t .�k;t �KXk0D1
�k0;t�k0;t / � dZ�t
D �k;t�. Nbt /�1.�.a�k ; Nbt / � ��t .at ; Nbt // � dZ�t ;which is the desired result. �
Proceeding towards the recursive characterization of sequential equilibrium, it can be readily verifiedthat the characterization of the continuation value of the normal type, viz. Proposition 2, remains validunder multiple behavior types. Also the characterization of sequential rationality given in Proposition 3continues to hold, provided the small players’ incentive constraint (12) is replaced by
Nbt 2 arg maxb2B
�0;th.at ; b; Nbt /CKXkD1
�k;th.a�k ; b;
Nbt / 8b 2 supp Nbt : (31)
Thus we have the following analogue of Theorem 2:
Theorem 7 (Sequential Equilibrium). Fix the prior p 2 �K . A public strategy profile .at ; Nb/t�0 anda belief process .�t /t�0 form a sequential equilibrium with continuation values .Wt /t�0 for the normaltype if and only if there exists a random process .ˇt /t�0 in L such that
(a) .�t /t�0 satisfies equation (30) with initial condition �0 D p,
(b) .Wt /t�0 is a bounded process satisfying equation (7), given .ˇt /t�0,
(c) .at ; Nbt /t�0 satisfy the incentive constraints (11) and (31), given .ˇt /t�0 and .�t /t�0.
It is beyond the scope of this paper to apply Theorem 7 to obtain characterizations similar to Theo-rems 4 and 6 under multiple behavioral types. However, to illustrate the applicability of Theorem 7, weuse it to prove an analogue of Cripps, Mailath, and Samuelson’s (2004) result that in every equilibrium thereputation of the large player disappears in the long run, when the large player is the normal type.
The following condition extends Condition 1 to the setting with multiple behavioral types:
Condition 10. For each � 2 �K and each static Bayesian Nash equilbrium .a; Nb/ of the game with prior�, we have �.a; Nb/ … co f�.a�
k; Nb/ W k D 1; : : : ; Kg.
Note that when the flow payoff of the small players depends on the actions of the large player onlythrough the public signal (cf. equation (1) and the discussion therein), then Condition 10 becomes equiv-alent to a simpler condition: for each static Nash equilbrium .a; Nb/ of the complete information game,�.a; Nb/ … co f�.a�
k; Nb/ W k D 1; : : : ; Kg.
The following result, whose proof is presented in Appendix E, is similar to Cripps, Mailath, andSamuelson (2004, Theorem 4), albeit under a different assumption.
38
Theorem 8. Under Condition 10, in every public sequential equilibrium limt!1 �0;t D 1 almost surelyunder the normal type.
When there is a single behavioral type, Condition 10 cannot be dispensed with. If it fails, then forsome prior there is a static Bayesian Nash equilibrium (BNE) in which the behavior of the normal type isindistinguishable from the behavioral type. Thus, the repeated play of this BNE is a sequential equilibriumof the reputation game in which the posterior belief never changes.
While the conclusion of Theorem 8 is similar to the conclusion of Cripps, Mailath, and Samuelson(2004, Theorem 4), our assumptions are different. In general, the discrete-time analogue of Condition 10—viz. the requirement that in every static Bayesian Nash equilibrium the distribution over signals inducedby the normal type is not a convex combination of the distributions induced by the behavioral types—isneither stronger nor weaker than the assumptions of Cripps, Mailath, and Samuelson (2004, Theorem 4),which are:
(i) the stage-game actions of the large player are identifiable;
(ii) no behavioral type plays an action which is part of a static Nash equilibrium of the complete infor-mation game;
(iii) the small players’ best-reply to each behavioral type is unique.
However, when there is a single behavioral type, Condition 10 is implied by conditions (i) and (ii) above.It is therefore surprising that our result does not require condition (iii), which is known to be a necessaryassumption in the discrete-time setting of Cripps, Mailath, and Samuelson (2004). The reason why we candispense with condition (iii) in our continuous-time framework is related to our equilibrium degeneracyresult under complete information (Theorem 3). In discrete time, when condition (i) and (ii) hold butcondition (iii) fails, it is generally possible to construct sequential equilibria where the normal type playsthe action of the behavioral type after every history, in which case the large player’s type is not revealed inthe long run. In such equilibrium, the incentives of the normal type arise from the threat of a punishmentphase in which the small players play the best-reply to the behavioral type that the normal type dislikes themost. However, we know from the analysis of Section 5 that intertemporal incentives of this sort cannotbe provided in our continuous-time setting.
9 Poisson signals
This section examines a variation of the model in which the public signals are driven by a Poisson process,instead of a Brownian motion. First, we derive a recursive characterization of public sequential equilibriumakin to Theorem 2. Then, for a class of games in which the signals are “good news,” we show that thesequential equilibrium is unique, Markovian and characterized by a functional differential equation.
As discussed in Section 5, our result that the equilibria of the underlying complete information gameis degenerate under Brownian signals (Theorem 3) is reminiscent of Abreu, Milgrom, and Pearce’s (1991)analysis of a repeated prisoners’ dilemma with Poisson signals. This seminal paper shows that when thearrival of a Poisson signal is “good news”—evidence that the players are behaving non-opportunistically—the equilibria of the repeated game must collapse to the static Nash equilibrium in the limit as the period
39
length tends to zero, irrespective of the discount rate. On the other hand, when Poisson arrivals are “badnews”—evidence of non-cooperative behavior—Abreu, Milgrom, and Pearce (1991) show that the greatestsymmetric public equilibrium payoff is greater than, and bounded away from, the static Nash equilibriumpayoff as the length of the period shrinks, provided the discount rate is low enough and the signal issufficiently informative. Thus, in complete information games, the structure of equilibrium is qualitativelydifferent under Poisson and Brownian signals.32 A similar distinction exists in the context of reputationgames, and we exploit it below to derive a characterization of sequential equilibrium in the good newscase.33
Throughout the section we assume that the public signal, now denoted .Nt /t�0, is a counting processwith Poisson intensity �.at ; Nbt / > 0, where � W A � B ! RC is a continuous function. This means that.Nt /t�0 is increasing and right-continuous, has left limits everywhere, takes values in the non-negativeintegers and has the property that Nt �
R t0 �.as;
Nbs/ ds is a martingale. A public strategy profile is nowa random process .at ; Nbt /t�0 with values in A � �.B/ which is predictable with respect to the filtrationgenerated by .Nt /t�0.34 Otherwise, the structure of the reputation game is as described in Section 3.
The next proposition, which is analogous to Proposition 1, characterizes the evolution of the smallplayers’ posterior beliefs under Poisson signals.
Proposition 6 (Belief Consistency). Fix a prior probability p 2 Œ0; 1� on the behavioral type. A beliefprocess .�t /t�0 is consistent with a public strategy profile .at ; Nbt /t�0 if and only if �0 D p and
d�t D ��t�.at ; Nbt / .dNt � ��t�.at ; Nbt / dt/; (32)
where for each .a; Nb; �/ 2 A ��B � Œ0; 1�,
��.a; Nb/ defD�.1 � �/.�.a�; Nb/ � �.a; Nb//��.a; Nb/ ;
��.a; Nb/ defD��.a�; Nb/C .1 � �/�.a; Nb/:
Proof. Similar to the proof of Proposition 1, but replaces Girsanov’s Theorem and Itô’s formula by theirappropriate counterparts in the Poisson setting (cf. Brémaud (1981, pp. 165–168, 337–339)). �
Equation (32) above means that the posterior jumps by �t � �t� D ��t�.at ; Nbt / when a Poissonevent arrives, and that between two consecutive arrivals the posterior follows the differential equation
d�t=dt D ��t .1 � �t /.�.a�; Nbt / � �.at ; Nbt //;
32Sannikov and Skrzypacz (2010) examine this difference in detail, showing that long-run players must use the Brownianand Poisson signals in distinct ways to create incentives. Specifically, Brownian signals can be used effectively only throughpayoff transfers along tangent hyperplanes, as in Fudenberg, Levine, and Maskin (1994), whereas Poisson jumps can also createincentives by “burning value,” i.e. moving orthogonally to the tangent hyperplane.
33In a recent paper, Board and Meyer-ter-Vehn (2010) also compare the structure of equilibria under Brownian, Poisson goodnews and Poisson bad news signals in a product choice game in continuous-time. In the class of games they examine, the firmhas two strategic types.
34For the definition of predictability see (Brémaud, 1981, p. 8). Any real-valued process .Yt /t�0 which is predictable with
respect to the filtration .Ft /t�0 must have the property that each Yt is measurable w.r.t. the � -field Ft� defDWs<t Fs .
40
which is pathwise deterministic. Thus, coefficient ��.a; Nb/ plays here a role similar to that played by .a; Nb; �/ in the Brownian setting, measuring the sensitivity of the belief updating to the fluctuations inthe public signal. From the small players’ viewpoint the process Nt �
R t0 �
�s�.as; Nbs/ ds is a martingaleand so is the belief process .�t /t�0.
The following proposition, which is analogous to Proposition 2, characterizes the evolution of thecontinuation value of the normal type.
Proposition 7 (Continuation Values). A bounded process .Wt /t�0 is the process of continuation values ofthe normal type under a strategy profile .at ; Nbt /t�0 if and only if there exists a predictable process .�t /t�0such that
dWt D r.Wt� � g.at ; Nbt // dt C r�t .dNt � �.at ; Nbt / dt/: (33)
Proof. Similar to the proof of Proposition 2, but replaces the Martingale Representation Theorem with theanalogue result for Poisson-driven martingales (cf. Brémaud (1981, p. 68)). �
Equation (33) means that the continuation value of the normal type jumps by Wt �Wt� D r�t whena Poisson event arrives, and has drift given by r.Wt � g.at ; Nbt / � �t�.at ; Nbt // between two consecutivePoisson arrivals.
Turning to sequential rationality, for each .�; �/ 2 Œ0; 1� � R consider the set M.�; �/ of all actionprofiles .a; Nb/ 2 A ��.B/ satisfying
a 2 arg maxa02A
g.a0; Nb/C ��.a0; Nb/;
b 2 arg maxb02B
�h.a�; b0; Nb/C .1 � �/h.a; b0; Nb/ 8b 2 supp Nb:
The next result, which is analogous to Proposition 3 and has a similar proof, characterizes sequentialrationality using these local incentive constraints.
Proposition 8 (Sequential Rationality). Let .at ; Nbt /t�0 be a public strategy profile, .�t /t�0 a belief pro-cess and .�t /t�0 the predictable process from Proposition 7. Then, .at ; Nbt /t�0 is sequentially rationalwith respect to .�t /t�0 if and only if
.at ; Nbt / 2M.�t�; �t / almost everywhere: (34)
We can summarize the recursive characterization of sequential equilibria of Poisson reputation gamesin the following theorem.
Theorem 9 (Sequential Equilibrium). Fix the prior probability p 2 Œ0; 1� on the behavioral type. Apublic strategy profile .at ; Nbt /t�0 and a belief process process .�t /t�0 form a sequential equilibrium withcontinuation payoffs .Wt /t�0 for the normal type if and only if there exists a predictable process .�t /t�0such that
(a) .�t /t�0 solves equation (32) with initial condition �0 D p;
(b) .Wt /t�0 is bounded and satisfies equation (33), given .�t /t�0;
(c) .at ; Nbt /t�0 satisfies the incentive constraint (34), given .�t /t�0 and .�t /t�0.
41
9.1 Good news and unique sequential equilibrium
In this section we identify a class of Poisson reputation games with a unique sequential equilibrium. Weassume that the payoff functions and the signal structure satisfy conditions similar to those of Proposi-tion 4, and, in addition, assume that the Poisson signals are good news. First, we show that under completeinformation, the unique equilibrium of the continuous-time game is the repeated play of the static equilib-rium (Theorem 10), as in Abreu, Milgrom, and Pearce (1991). Second, for reputation games, we show thatthe sequential equilibrium is unique, Markovian and characterized by a functional differential equation(Theorem 11). Finally, we discuss briefly how the analysis would change when the Poisson signals are badnews and, in particular, why we expect the uniqueness and characterization results to break down in thiscase.
We impose the following assumptions: A, B � R are compact intervals, a� D maxA, the Poissonintensity depends only on the large player’s action—so we write �.a/ for each a 2 A—, the functions g,h and � are twice continuously differentiable, and for each .a; b; Nb/ 2 A � B � B ,
(a) g11.a; Nb/ < 0, h22.a; b; Nb/ < 0, .h22 C h23/.a; Nb; Nb/ < 0, g12.a; Nb/ � 0, h12.a; Nb; Nb/ � 0,
(b) g2.a; Nb/ � 0,
(c) �0.a/ > 0, �00.a/ � 0,
(d) g1.a�; b�/ < 0, where fb�g D B.a�/.
For future reference, we call this set of assumptions the good news model.Note that conditions (a), (b) and (d) above are similar to conditions (a), (b) and (d) of Proposition 4,
albeit slightly stronger (cf. Remark 5). Condition (c) means that the signals are “good news” for the largeplayer. Indeed, under �0 > 0 the arrival of a Poisson signal is perceived by the small players as evidencein favor of the behavioral type; this is beneficial for the normal type, since under conditions (a) and (b)the flow payoff of the normal type is increasing in reputation. This is in contrast to the case in which thesignals are “bad news,” which arises when conditions (a), (b) and (d) hold but condition (c) is replaced by:
(c0) �0.a/ < 0, �00.a/ � 0.
We discuss the bad news case briefly at the end of the section.We begin the analysis with the complete information game. In the good news model, the equilibrium of
the continuous-time game collapses to the static Nash equilibrium, similar to what happens in the Browniancase (Theorem 3) and in the repeated prisoners’ dilemma of Abreu, Milgrom, and Pearce (1991).
Theorem 10. In the good news model, if the small players are certain that they are facing the normaltype, i.e. p D 0, then the unique public equilibrium of the continuous-time game is the repeated play ofthe unique static Nash equilibrium, irrespective of the discount rate.
Proof. As shown in the first part of the proof of Proposition 4, the static game has a unique Nash equi-librium .aN ; bN /, where bN is a mass-point distribution. Let .at ; Nbt /t�0 be an arbitrary public equi-librium with continuation values .Wt /t�0 for the normal type. Suppose, towards a contradiction, that
42
W0 > g.aN ; bN /. (The proof for the reciprocal case, W0 < g.aN ; bN /, is similar and therefore omitted.)
By Theorem 9, for some predictable process .�t /t�0 the large player’s continuation value must follow
dWt D r.Wt � g.at ; Nbt / � �t�.at //„ ƒ‚ …drift
dt C r�t„ƒ‚…jump size
dNt ;
where at maximizes g.a0; Nbt /C�t�.a0/ over all a0 2 A, and Nbt maximizes h.at ; b0; Nbt / over all b0 2 �.B/.Let ND defDW0 � g.aN ; bN / > 0.
Claim. There exists c > 0 such that, so long as Wt � g.aN ; bN /C ND=2, either the drift of Wt is greaterthan r ND=4 or, upon the arrival of a Poisson event, the size of the jump in Wt is greater than c.
The claim follows from the following lemma, whose proof is in Appendix F.
Lemma 2. For any " > 0 there exists ı > 0 such that for all t � 0 and after all public histories,g.at ; Nbt /C �t�.at / � g.aN ; bN /C " implies �t � ı.Indeed, letting " D ND=4 in this lemma gives a ı > 0with the property that whenever g.at ; Nbt /C�t�.at / �rg.aN ; bN / C " the size of the jump in Wt , which equals r�t , must be greater than or equal to c defD rı.Moreover, when g.at ; Nbt /C �t�.at / < rg.aN ; bN /C ND=4 and Wt � g.aN ; bN /C ND=2, then the driftofWt , which equals r.Wt �g.at ; Nbt /� �t�.at //, must be greater than r ND=4, and this concludes the proofof the claim.
Since we have assumed that ND D W0 � g.aN ; bN / > 0, the claim above readily implies that Wt mustgrow arbitrarily large with positive probability, and this is a contradiction since .Wt /t�0 is bounded. �
We now turn to the incomplete information case, i.e. p 2 .0; 1/. Theorem 11 below shows that inthe good news model the reputation game has a unique sequential equilibrium, which is Markovian andcharacterized by a functional differential equation. As in the Brownian case, the equilibrium actions aredetermined by the posterior on the behavioral type and the equilibrium value function of the normal type.Thus, in order to state our characterization, we first need to examine how a candidate value function forthe normal type affects the incentives of the players. In effect, Proposition 9 below—which is the Poissoncounterpart of Proposition 4—establishes the existence, uniqueness and Lipschitz continuity of actionprofiles satisfying the incentive constraint (34), where r�t is the jump in the continuation value Wt whenthe pair .�t ; Wt / is constrained to lie in the graph of a candidate value function V W .0; 1/! R. The proofof Proposition 9 is presented in Appendix F.
We write C inc.Œ0; 1�/ to denote the complete metric space of real-valued continuous increasing func-tions over the interval Œ0; 1�, equipped with the supremum distance.
Proposition 9. In the good news model, for each .�; V / 2 Œ0; 1� � C inc.Œ0; 1�/ there is a unique actionprofile .a.�; V /; b.�; V // 2 A � B satisfying the incentive constraint
.a.�; V /; b.�; V // 2M��;�V.�C��.a.�; V /// � V.�/�=r�: (35)
Moreover, .a; b/ W Œ0; 1� � C inc.Œ0; 1�/ ! A � B is a continuous function, and for each � 2 Œ0; 1�, V 7!.a.�; V /; b.�; V // is a Lipschitz continuous function on the metric space C inc.Œ0; 1�/, with a Lipschitzconstant that is uniform in �.
43
The characterization of sequential equilibrium, presented in Theorem 11 below, uses the followingdifferential equation, called the optimality equation:
U 0.�/ D rg.a.�; U /; b.�; U //C �.a.�; U //�U.�/ � rU.�/��.a.�; U //��.a.�; U //
; � 2 .0; 1/; (36)
where�U.�/
defDU.� C��.a.�; U /// � U.�/and the function .a.�/; b.�// is defined implicitly by (35). This is a functional retarded differential equation,because of the delayed term U.� C ��/ on the right-hand side and the fact that the size of the lag,��.a.�; U //, is endogenously determined by the global behavior of the solution U over Œ�; 1/, ratherthan by the value of U at a single point.
The main result of this section is:
Theorem 11. In the good news model, the correspondence of sequential equilibrium payoffs of the normaltype, E W Œ0; 1� ! R, is single-valued and coincides, on the interval .0; 1/, with the unique boundedincreasing solution U W .0; 1/ ! R of the optimality equation (36). Moreover, at p 2 f0; 1g, E.p/
satisfies the boundary conditions
lim�!pU.�/ D E.p/ D g.M.p; 0// and lim
�!p �.1 � �/U0.�/ D 0:
Finally, for each prior p 2 Œ0; 1� there is a unique public sequential equilibrium, which is Markovian in thesmall players’ posterior belief: at each time t and after each public history, the equilibrium action profileis .a.�t�; U /; b.�t�; U //, where .a.�/; b.�// is the continuous function defined implicitly by condition(35); the small players’ posterior .�t /t�1 follows equation (32) with initial condition �0 D p; and thecontinuation value of the normal type is Wt D U.�t /.
However, equilibrium uniqueness generally breaks down when signals are bad news. Under assump-tions (a), (b), (c0) and (d), multiple non-Markovian sequential equilibria may exist, despite the fact thatM.�; �/ remains a singleton for each .�; �/ 2 Œ0; 1� � .�1; 0�. Indeed, this multiplicity already arises inthe underlying complete information game, as in the repeated prisoner’s dilemma of Abreu, Milgrom, andPearce (1991). Intuitively, the large player can have incentives to play an action different from the staticbest-reply by a threat that if a bad signal arrives, he will be punished by perpetual reversion to the staticNash equilibrium.35 In the reputation game, multiple non-Markovian equilibria may exist for a similar rea-son. Recall that in the Markov perfect equilibrium of a game in which Poisson signals are good news, thereaction of the large player’s continuation payoff to the arrival of a signal is completely determined by theupdated beliefs and the equilibrium value function. Payoffs above those in the Markov perfect equilibriumcannot be sustained by a threat of reversion to the Markov equilibrium, because those punishments wouldhave to be applied after good news, and therefore they cannot create incentives to play actions closer toa�. By contrast, when signals are bad news, such punishments can create incentives to play actions closerto a� effectively.
While we do not characterize sequential equilibria for the case in which the signals are bad news, weconjecture that the upper and lower boundaries of the graph of the correspondence of sequential equi-librium payoffs of the large player solve a pair of (coupled) functional differential equations. Incentive
35Naturally, for such profile to be an equilibrium the signals must be sufficiently informative.
44
provision must satisfy the feasibility condition that after bad news, the belief-continuation value pair tran-sitions to a point between the upper and lower boundaries. Subject to this feasibility condition, payoffmaximization implies that when the initial belief-continuation pair is on the upper boundary of the graphof the correspondence of sequential equilibrium payoffs, conditional on the absence of bad news it muststay on the upper boundary. A similar statement must hold for the lower boundary. This observation givesrise to a coupled pair of differential equations that characterize the upper and lower boundaries. We leavethe formalization of this conjecture for future research.
10 Conclusion
Our result that many continuous-time reputation games have a unique public sequential equilibrium, whichis Markovian in the population’s belief, does not have an analogue in discrete time. One may wonder whathappens to the equilibria of discrete-time games in the limit as the time between actions � shrinks to 0,as in Abreu, Milgrom, and Pearce (1991), Faingold (2008), Fudenberg and Levine (2007), Fudenberg andLevine (2009) , Sannikov and Skrzypacz (2007) and Sannikov and Skrzypacz (2010). While it is beyondthe scope of this paper to answer this question rigorously, we will make some guided conjectures and leavethe formal analysis to future research.
To be specific, consider what happens as the length of the period of fixed actions, �, shrinks to zeroin a game with Brownian signals satisfying Conditions 1 and 2. For a fixed value of � > 0, the upperand lower boundaries of the set of equilibrium payoffs will generally be different. For a given actionprofile, it is uniquely determined how the small players’ belief responds to the public signal, but there issome room to choose continuation values within the bounds of the equilibrium payoff set. Consider thetask of maximizing the expected payoff of the large player by the choice of a current equilibrium actionand feasible continuation values. In many instances the solution to this optimization problem would takethe form of a tail test: the large player’s continuation values are chosen on the upper boundary of theequilibrium payoff set, unless the public signal falls below a cut-off, in which case continuation values aretaken from the lower boundary. This way of providing incentives is reminiscent of the use of the cutofftests to trigger punishments in Sannikov and Skrzypacz (2007). As in the complete-information game ofSection 5, it becomes less and less efficient to use such tests to provide incentives as � ! 0. Therefore,it is natural to conjecture that as � ! 0, the distance between the upper and lower boundaries of theequilibrium value set converges to 0.
45
Appendix
A Proof of Lemma 1
Fix an arbitrary constant M > 0. Consider the set ˆ0 of all tuples .a; Nb; ˇ/ 2 A ��.B/ �Rd satisfying
a 2 arg maxa02A
g.a0; Nb/C ˇ � �.a0; Nb/; b 2 arg maxb02B
h.a; b0; Nb/; 8b 2 supp Nb; g.a; Nb/ � Nv C "; (37)
and jˇj �M . Note thatˆ0 is a compact set, as it is a closed subset of the compact space A��.B/�fˇ 2Rd W jˇj � M g, where �.B/ is equipped with the weak* topology. Therefore, the continuous function.a; Nb; ˇ/ 7! jˇj achieves its minimum, � , on ˆ0, and we must have � > 0 because of the conditiong.a; Nb/ � Nv C ". It follows that jˇj � ı defDmin fM;�g for any .a; b; ˇ/ satisfying conditions (37). �
B Bounds on coefficient .a; Nb; �/This technical appendix proves a bound on .a; Nb; �/ which is used in the subsequent analysis.
Lemma B.1. Assume Condition 1. There exists a constant C > 0 such that for all .a; Nb; �; z/ 2 A ��.B/ � .0; 1/ �R,
.a; Nb/ 2 N .�; z/ H) .1C jzj/ j .a;Nb; �/j
�.1 � �/ � C:
Proof. If the thesis of the lemma were false, there would be a sequence .an; Nbn; �n; zn/n2N with �n 2.0; 1/ and .an; Nbn/ 2 N .�n; zn/ for all n 2 N, such that both znj .an; Nbn; �n/j=.�n.1 � �n// andj .an; Nbn; �n/j=.�n.1 � �n// converged to 0. Passing to a sub-sequence if necessary, we can assume that.an; Nbn; �n/ converges to some .a; Nb; �/ 2 A��.B/�Œ0; 1�. Then, by the definition of N and the continu-ity of g, �, � and h, the profile .a; Nb/must be a Bayesian Nash equilibrium of the static game with prior �,since zn.�.a�; Nbn/ � �.an; Nbn//>.�. Nbn/�. Nbn/>/�1 D zn .an; Nbn; �n/>�. Nbn/�1=.�n.1 � �n// ! 0.Hence, �.a; Nb/ ¤ �.a�; Nb/ by Condition 1, and therefore lim infn j .an; Nbn; �n/j=.�n.1 � �n// �j�. Nb/�1.�.a�; Nb/ � �.a; Nb//j > 0, which is a contradiction. �
The following corollary shows that the right-hand side of the optimality equation satisfies a quadraticgrowth condition whenever the beliefs are bounded away from 0 and 1. This technical result is used in theexistence proofs of Appendices C and D.
Corollary B.1 (Quadratic growth). Assume Condition 1. For all M > 0 and " > 0 there exists K > 0
such that for all � 2 Œ"; 1 � "�, .a; Nb/ 2 A � B , u 2 Œ�M;M� and u0 2 R,
.a; Nb/ 2 N .�; �.1 � �/u0=r/ H)ˇ̌̌ 2u01 � � C
2r.u � g.a; Nb//j .a; Nb; �/j2
ˇ̌̌� K.1C ju0j2/:
Proof. Follows directly from Lemma B.1 and the bounds u 2 Œ�M;M� and � 2 Œ"; 1 � "�. �
46
C Appendix for Section 6
Throughout this section we maintain Conditions 1 and 2.a. For the case when Condition 2.b holds, all thearguments in this section remain valid provided we change the definition of N and set N .�; z/ equal toN .�; 0/ for z < 0. Proposition C.4 shows that under Conditions 1 and 2.b the resulting solution U mustbe increasing, so the values of N .�; z/ for z < 0 are irrelevant.
C.1 Existence of a bounded solution of the optimality equation
In this subsection we prove the following proposition.
Proposition C.1. The optimality equation (19) has at least one C2-solution that takes values in the intervalŒg; Ng� of feasible payoffs of the large player.
The proof relies on standard results from the theory of boundary-value problems for second-orderequations. We now review the part of that theory that is relevant for our existence result. Given a contin-uous function H W Œa; b� � R2 ! R and real numbers c and d , consider the following boundary valueproblem:
U 00.x/ D H.x;U.x/; U 0.x//; x 2 Œa; b�U.a/ D c; U.b/ D d: (38)
Given real numbers ˛ and ˇ, we are interested in sufficient conditions for (38) to admit a C2-solutionU W Œa; b� ! R with ˛ � U.x/ � ˇ for all x 2 Œa; b�. One such sufficient condition is called Nagumocondition, which posits the existence of a positive continuous function W Œ0;1/! R satisfyingZ 1
0
v dv
.v/D1
andjH.x; u; u0/j � .ju0j/ 8.x; u; u0/ 2 Œa; b� � Œ˛; ˇ� �R:
In the proof of Proposition C.1 below we use the following standard result, which follows from Theo-rems II.3.1 and I.4.4 in de Coster and Habets (2006):
Lemma C.1. Suppose that ˛ � c � ˇ, ˛ � d � ˇ and that H W Œa; b� � R2 ! R satisfies the Nagumocondition relative to ˛ and ˇ. Then:
(a) the boundary value problem (38) admits a solution satisfying ˛ � U.x/ � ˇ for all x 2 Œa; b�;
(b) there is a constant R > 0 such that every C2-function U W Œa; b�! R that satisfies ˛ � U.x/ � ˇfor all x 2 Œa; b� and solves
U 00.x/ D H.x;U.x/; U 0.x//; x 2 Œa; b�;
satisfies jU 0.x/j � R for all x 2 Œa; b�.
We are now ready to prove Proposition C.1.
47
Proof of Proposition C.1. Since the right-hand side of the optimality equation blows up at � D 0 and� D 1, our strategy of proof is to construct the solution as the limit of a sequence of solutions on expandingclosed subintervals of .0; 1/. Indeed, letH W .0; 1/�R2 ! R denote the right-hand side of the optimalityequation, i.e.
H.�; u; u0/ D 2u0
1 � � C2r.u � g.N .�; �.1 � �/u0=r///j .N .�; �.1 � �/u0=r/; �/j2 ;
and for each n 2 N consider the boundary value problem:
U 00.�/ D H.�;U.�/; U 0.�//; � 2 Œ1=n; 1 � 1=n�;U.1=n/ D g;U.1 � 1=n/ D Ng:
By Corollary B.1, there exists a constant Kn > 0 such that
jH.�; u; u0/j � Kn.1C ju0j2/ 8.�; u; u0/ 2 Œ1=n; 1 � 1=n� � Œg; Ng� �R:
SinceR10 K�1n .1 C v2/�1v dv D 1, for each n 2 N the boundary value problem above satisfies the
hypothesis of Lemma C.1 relative to ˛ D g and ˇ D Ng. Therefore, for each n 2 N there exists a C2-function Un W Œ1=n; 1 � 1=n� ! R which solves the optimality equation on Œ1=n; 1 � 1=n� and satisfiesg � Un � Ng. Since for m � n the restriction of Um to Œ1=n; 1 � 1=n� also solves the optimality equationon Œ1=n; 1�1=n�, by Lemma C.1 and the quadratic growth condition above the first and second derivativesof Um are uniformly bounded for m � n, and hence the sequence .Um; U 0m/m�n is bounded and equicon-tinuous over the domain Œ1=n; 1 � 1=n�. By the Arzelà-Ascoli Theorem, for every n 2 N there exists asubsequence of .Um; U 0m/m�n which converges uniformly on Œ1=n; 1 � 1=n�. Then, using a diagonaliza-tion argument, we can find a subsequence of .Un/n2N , denoted .Unk
/k2N , which converges pointwise toa continuously differentiable function U W .0; 1/ ! Œg; Ng� such that on every closed subinterval of .0; 1/the convergence takes place in C1.
Finally, U must solve the optimality equation on .0; 1/, since U 00nk.�/ D H.�;Unk
.�/; U 0nk.�// con-
verges to H.�;U.�/; U 0.�// uniformly on every closed subinterval of .0; 1/, by the continuity of H andthe uniform convergence .Unk
; U 0nk/! .U; U 0/ on closed subintervals of .0; 1/. �
C.2 Boundary conditions
Proposition C.2. If U is a bounded solution of the optimality equation (19) on .0; 1/, then it satisfies thefollowing boundary conditions at p D 0 and 1:
lim�!pU.�/ D g.N .p; 0// ; lim
�!p �.1 � �/U0.�/ D 0 ; lim
�!p �2.1 � �/2U 00.�/ D 0 : (39)
Proof. Directly from Lemmas C.4, C.5 and C.6 below. Lemmas C.2 and C.3 are intermediate steps. �
Lemma C.2. If U W .0; 1/ ! R is a bounded solution of the optimality equation, then U has boundedvariation.
Proof. Suppose there exists a bounded solution U of the optimality equation with unbounded variationnear p D 0 (the case p D 1 is similar). Then let .�n/n2N be a decreasing sequence of consecutive localmaxima and minima of U; such that �n is a local maximum for n odd and a local minimum for n even.
48
Thus for n odd we have U 0.�n/ D 0 and U 00.�n/ � 0. From the optimality equation it follows thatg.N .�n; 0// � U.�n/. Likewise, for n even we have g.N .�n; 0// � U.�n/. Thus, the total variation ofg.N .�; 0// on .0; �1� is no smaller than the total variation of U and therefore g.N .�; 0// has unboundedvariation near zero. However, this is a contradiction, since g.N .�; 0// is Lipschitz continuous underCondition 2. �
Lemma C.3. Let U W .0; 1/! R be any bounded continuously differentiable function. Then
lim inf�!0 �U 0.�/ � 0 � lim sup
�!0�U 0.�/ ; and
lim inf�!1 .1 � �/U
0.�/ � 0 � lim sup�!1
.1 � �/U 0.�/ :
Proof. Suppose, towards a contradiction, that lim inf�!0 �U 0.�/ > 0 (the case lim sup�!0 �U 0.�/ < 0
is analogous). Then, for some c > 0 and N� > 0, we must have �U 0.�/ � c for all � 2 .0; N��, whichimplies U 0.�/ � c=� for all � 2 .0; N��. But then U cannot be bounded, since the anti-derivative of 1=�,which is log�, tends to1 as � ! 0, a contradiction. The proof for the case � ! 1 is analogous. �
Lemma C.4. If U is a bounded solution of the optimality equation, then lim�!p �.1 � �/U 0.�/ D 0 forp 2 f0; 1g.
Proof. Suppose, towards a contradiction, that �U 0.�/¹ 0 as � ! 0. Then, by Lemma C.3,
lim inf�!0 �U 0.�/ � 0 � lim sup
�!0�U 0.�/ ;
with at least one strict inequality. Without loss of generality, assume lim sup�!0 �U 0.�/ > 0. Hencethere exist constants 0 < k < K; such that �U 0.�/ crosses levels k and K infinitely many times in aneighborhood of 0. Thus, by Lemma B.1, there exists C > 0 such that j .a; Nb; �/j � C � whenever�U 0.�/ 2 .k;K/ and � 2 .0; 1
2/. On the other hand, by the optimality equation, for some constant L > 0
we have jU 00.�/j � L�2 . This bound implies that for all � 2 .0; 1
2/ such that �U 0.�/ 2 .k;K/,
j.�U 0.�//0j � j�U 00.�/j C jU 0.�/j D .1C j�U00.�/j
jU 0.�/j / jU0.�/j � .1C L
k/ jU 0.�/j ;
which implies
jU 0.�/j � j.�U0.�//0j
1C L=k :
It follows that on every interval where �U 0.�/ crosses k and stays in .k;K/ until crossing K, the totalvariation of U is at least .K � k/=.1CL=k/. Since this happens infinitely many times in a neighborhoodof � D 0; function U must have unbounded variation in that neighborhood, and this is a contradiction (byvirtue of Lemma C.2.) The proof that lim�!1.1 � �/U 0.�/ D 0 is analogous. �
Lemma C.5. If U W .0; 1/! R is a bounded solution of the optimality equation, then for p 2 f0; 1g,
lim�!pU.�/ D g.N .p; 0// :
49
Proof. First, by Lemma C.2, U must have bounded variation and so the lim�!p U.�/ exists. Considerp D 0 and assume, towards a contradiction, that lim�!0 U.�/ D U0 < g.aN ; bN /; where .aN ; bN / DN .0; 0/ is the Nash equilibrium of the stage game (the proof for the reciprocal case is similar). ByLemma C.4, lim�!0 �U 0.�/ D 0, which implies that N .�; �.1� �/U 0.�/=r/ converges to .aN ; bN / as� ! 0. Recall the optimality equation:
U 00.�/ D 2U 0.�/1 � � C
2r.U.�/ � g.N .�; �.1 � �/U 0.�/=r//j .N .�; �.1 � �/U 0.�/=r/; �/j2 D 2U 0.�/
1 � � Ch.�/
�2;
where h.�/ is a continuous function that converges to
2r.U0 � g.aN ; bN //j�.bN /�1.�.a�; bN / � �.aN ; bN //j2 < 0:
as � ! 0. Since U 0.�/ D o.1=�/ by Lemma C.3, it follows that for some N� > 0 there exists a constantK > 0 such that U 00.�/ < �K=�2 for all � 2 .0; N�/. But then U cannot be bounded, since the second-order anti-derivative of �1=�2, which is log�, tends to �1 as � ! 0. The proof for the case p D 1 issimilar. �
Lemma C.6. If U W .0; 1/! R is a bounded solution of the optimality equation, then
lim�!p �
2.1 � �/2U 00.�/ D 0 ; for p 2 f0; 1g :
Proof. Consider p D 1. Fix an arbitrary M > 0 and choose � 2 .0; 1/ so that .1 � �/jU 0.�/j < M forall � 2 .�; 1/. By Lemma B.1 there exists C > 0 such that j .N .�; �.1 � �/U 0.�/=r/; �/j � C.1 � �/for all � 2 .�; 1/. Hence, by the optimality equation, for all � 2 .�; 1/ we have
.1 � �/2jU 00.�/j � 2.1 � �/jU 0.�/j C .1 � �/2 2r jU.�/ � g.N .�; �.1 � �/U 0.�/=r//jj .N .�; �.1 � �/U 0.�/=r/; �/j2
� 2.1 � �/jU 0.�/j C 2rC�2jU.�/ � g.N .�; �.1 � �/U 0.�/=r//j �! 0 as � ! 1;
by Lemmas C.4 and C.5. The case p D 0 is analogous. �
C.3 Uniqueness
Lemma C.7. If two bounded solutions of the optimality equation, U and V , satisfy U.�0/ � V.�0/ andU 0.�0/ � V 0.�0/ with at least one strict inequality, then U.�/ < V.�/ and U 0.�/ < V 0.�/ for all� > �0. Similarly, if U.�0/ � V.�0/ and U 0.�0/ � V 0.�0/ with at least one strict inequality, thenU.�/ < V.�/ and U 0.�/ > V 0.�/ for all � < �0.
Proof. Suppose that U.�0/ � V.�0/ and U 0.�0/ < V 0.�0/. If U 0.�/ < V 0.�/ for all � > �0 then wemust also have U.�/ < V.�/ on that range. Otherwise, let
�1defD inf f� 2 Œ�0; 1/ W U 0.�/ � V 0.�/g:
Then, U 0.�1/ D V 0.�1/ by continuity, and U.�1/ < V.�1/ since U.�0/ � V.�0/ and U 0.�/ < V 0.�/ onŒ�0; �1/. By the optimality equation, it follows thatU 00.�1/ < V 00.�1/, and henceU 0.�1�"/ > V 0.�1�"/for sufficiently small " > 0, and this contradicts the definition of �1.
50
For the case when U.�0/ < V.�0/ and U 0.�0/ D V 0.�0/ the optimality equation implies thatU 00.�0/ < V 00.�0/. Therefore, U 0.�/ < V 0.�/ on .�0; �0 C "/; and the argument proceeds as above.
Finally, the argument for � < �0 when U.�0/ � V.�0/ and U 0.�0/ � V 0.�0/ with at least one strictinequality is similar. �
Proposition C.3. The optimality equation has a unique bounded solution.
Proof. By Proposition C.1 a bounded solution of the optimality equation exists. Suppose U and V are twosuch bounded solutions. Assuming that V.�/ > U.�/ for some � 2 .0; 1/, let �0 2 .0; 1/ be the pointwhere the difference V � U is maximized, which is well-defined because lim�!p U.�/ D lim�!p V.�/for p 2 f0; 1g by Proposition C.2. Thus we have V.�0/�U.�0/ > 0 and V 0.�0/�U 0.�0/ D 0. But then,by Lemma C.7, the difference V.�/ � U.�/ must be strictly increasing for � > �0, a contradiction. �
Finally, the following proposition shows that under Conditions 1 and 2.b, the unique solution U W.0; 1/ ! R of the modified optimality equation in which N .�; z/ is set equal to N .�; 0/ when z < 0
must be an increasing function. In particular, U must be the unique bounded increasing solution of theoptimality equation.
Proposition C.4. Under Conditions 1 and 2.b, U W .0; 1/! R is an increasing function, and is thereforethe unique bounded increasing solution of the optimality equation.
Proof. By Proposition C.3, the modified optimality equation—with N .�; z/ set equal to N .�; 0/ whenz < 0—has a unique bounded solution U W .0; 1/ ! R, which satisfies the boundary conditionslim�!0 U.�/ D g.N .0; 0// and lim�!1 U.�/ D g.N .1; 0// by Lemma C.5.
Towards a contradiction, suppose that U is not increasing, so that U 0.�/ < 0 for some � 2 .0; 1/. Takea maximal subinterval .�0; �1/ � .0; 1/ on which U is strictly decreasing. Since g.N .�; 0// is increasingin �, we have lim�!0 U.�/ D g.N .0; 0// � g.N .1; 0// D lim�!1 U.�/, hence .�0; �1/ ¤ .0; 1/.Without loss of generality, assume �1 < 1. Then, �1 must be an interior local minimum, so U 0.�1/ D 0
and U 00.�1/ � 0. Also, we must have U.�1/ � g.N .�1; 0//, for otherwise
U 00.�1/ D 2r.U.�1/ � g.N .�1; 0///
j .N .�1; 0/; �1/j2< 0:
But then, since
lim�!�0
U.�/ > U.�1/ � g.N .�1; 0// � g.N .0; 0// D lim�!0U.�/;
it follows that �0 > 0. Therefore, U 0.�0/ D 0 and
U 00.�0/ D 2r.U.�0/ � g.N .�0; 0///
j .N .�0; 0/; �0/j2� 2r.U.�0/ � g.N .�1; 0///
j .N .�0; 0/; �0/j2> 0;
so �0 is a strict local minimum, a contradiction. �
51
C.4 The continuity lemma used in the proof of Theorem 4
Lemma C.8. Let U W .0; 1/ ! R be the unique bounded solution of the optimality equation and letd W A ��.B/ � Œ0; 1�! R and f W A ��.B/ � Œ0; 1� �R! R be the continuous functions defined by
d.a; Nb; �/ defD(r.U.�/ � g.a; Nb// � j .a;b;�/j2
1�� U 0.�/ � 12j .a; b; �/j2U 00.�/ W � 2 .0; 1/
r.g.N .�; 0// � g.a; Nb// W � D 0 or 1(40)
andf .a; Nb; �; ˇ/ defD rˇ>�. Nb/ � �.1 � �/.�.a�; Nb/ � �.a; Nb//>.�. Nb/>/�1„ ƒ‚ …
.a;b;�/>
U 0.�/: (41)
For every " > 0 there exists ı > 0 such that for all .a; Nb; �; ˇ/ satisfying
a 2 arg maxa02A
g.a0; Nb/C ˇ � �.a0; Nb/Nb 2 arg max
b02B�h.a�; b0; Nb/C .1 � �/h.a; b0; Nb/ 8b 2 supp Nb; (42)
either d.a; Nb; �/ > �" or f .a; Nb; �; ˇ/ � ı.
Proof. Since �.1��/U 0.�/ is bounded (by Proposition C.2) and there exists c > 0 such that j�. Nb/ � yj �cjyj for all y 2 Rd and Nb 2 �.B/, there exist constants M > 0 and m > 0 such that jf .a; Nb; �; ˇ/j > mfor all ˇ 2 Rd with jˇj > M .
Consider the set ˆ of all tuples .a; b; �; ˇ/ 2 A ��.B/ � Œ0; 1� �Rd with jˇj � M that satisfy (42)and d.a; Nb; �/ � �". Since U satisfies the boundary conditions (39) by Proposition C.2, d is a continuousfunction and hence ˆ is a closed subset of the compact set f.a; b; �; ˇ/ 2 A � �.B/ � Œ0; 1� � Rd Wjˇj � M g, and therefore ˆ is a compact set.36 The boundary conditions (39) also imply that the functionjf j is also continuous, so it achieves its minimum, denoted �, on ˆ. We must have � > 0, since for all.a; Nb; �; ˇ/ 2 ˆ we have d.a; Nb; �/ D 0 whenever f .a; Nb; �; ˇ/ D 0 by the optimality equation, as weargued in the proof of Theorem 4. It follows that for all .a; Nb; �; ˇ/ satisfying (42), either d.a; Nb; �/ > �"or jf .a; Nb; �; ˇ/j � ı defDmin fm; �g, as required. �
C.5 Proof of Theorem 5
First, we need the following lemma:
Lemma C.9. Under Conditions 1, 2 and 3, for any �0 2 .0; 1/ and k > 0 the initial value problem
v0.�/ D 2v.�/
1 � � C2.g.a�; b�/ � g.eN .�; �.1 � �/v.�///j .eN .�; �.1 � �/v.�//; �/j2 ; v.�0/ D k; (43)
where eN .�; z/defD(
N .�; z/; if Condition 2.a holds,N .�;maxfz; 0g/; if Condition 2.b holds,
has a unique solution on the interval .0; 1/. Moreover, the solution satisfies lim inf�!1 v.�/ < 0.
36Recall that �.B/ is compact in the topology of weak convergence of probability measures.
52
Proof. Assume Conditions 1, 2.a and 3. (The proof for the case when Condition 2.b holds is very similarand thus ommitted.) Fix �0 2 .0; 1/ and k > 0. First, note that on every compact subset of .0; 1/ � R,the right-hand side of (43) is Lipschitz continuous in .�; v/ by Conditions 1 and 2.a and Lemma B.1. Thisimplies that a unique solution exists on a neighborhood of �0. Let J�0
denote the maximal interval ofexistence / uniqueness of the solution of (43), and let us show that J�0
D .0; 1/. Indeed, the Lipschitzcontinuity of g, Condition 3 and the bound from Lemma B.1 imply that the solution of (43) satisfies:
8" > 0 9K" > 0 such that jv0.�/j � K".1C jv.�/j/; 8� 2 J�0\ Œ"; 1 � "�:
Given this linear growth condition, a standard argument shows that jvj and jv0j cannot blow-up in anyclosed sub-interval of .0; 1/. Thus, we must have J�0
D .0; 1/, i.e. the solution v is well-defined on thewhole interval .0; 1/.
It remains to show that lim inf�!1 v.�/ < 0, but first we will prove the intermediate result thatlim sup�!1.1 � �/v.�/ � 0. By the large player’s incentive constraint in the definition of N , for each.a; b; �; z/ 2 A � B � .0; 1/ �R with .a; b/ 2 N .�; z/,
g.a; b/ � g.a�; b/ � z.�.a�; b/ � �.a; b//>.�.b/�.b/>/�1.�.a�; b/ � �.a; b// DD z j�.b/�1.�.a�; b/ � �.a; b//j2 D zj .a; b; �/j2=.�.1 � �//2: (44)
This implies that, for each � 2 .0; 1/ and .a; b/ 2 N .�; �.1 � �/v.�//,2.g.a�; b�/ � g.a; b//
j .a; b; �/j2 D 2.g.a�; b�/ � g.a�; b//j .a; b; �/j2 C 2.g.a�; b/ � g.a; b//
j .a; b; �/j2
� 2jg.a�; b�/ � g.a�; b/jj .a; b; �/j2 � 2v.�/
�.1 � �/: (45)
Moreover, by the Lipschitz continuity of g and Condition 3, there is a constant K1 > 0 such that for each� 2 .0; 1/ and .a; b/ 2 N .�; �.1 � �/v.�//,
jg.a�; b�/ � g.a�; b/j � K1��1j .a; b; �/j:
Plugging this inequality in (45) yields, for each � 2 .0; 1/ and .a; b/ 2 N .�; �.1 � �/v.�//,2.g.a�; b�/ � g.a; b//
j .a; b; �/j2 � � 2v.�/
�.1 � �/ C2K1
�j .a; b; �/j :
But since by Lemma B.1 there is a constant C > 0 such that for each � 2 .0; 1/ and .a; b/ 2 N .�; �.1 ��/v.�//,
j .a; b; �/j � C�.1 � �/1C �.1 � �/jv.�/j ;
it follows that for each � 2 .0; 1/ and .a; b/ 2 N .�; �.1 � �/v.�//,2.g.a�; b�/ � g.a; b//
j .a; b; �/j2 � � 2v.�/
�.1 � �/ C2K1.1C �.1 � �/jv.�/j/
C�2.1 � �/ :
Plugging this inequality in (43) and simplifying yields
v0.�/ � C1jv.�/j C C2
1 � � ; 8� 2 Œ�0; 1/;
53
where C1 D 2.K1=C � 1/=�0 and C2 D 2K1=.C�20/. This differential inequality implies
v.�/ � keC1.���0/ C C2eC1.���0/
Z �
�0
e�C1.x��0/
1 � x dx; 8� 2 Œ�0; 1/;
and hence,v.�/ � C3 � C4 log.1 � �/; 8� 2 Œ�0; 1/;
where C3 and C4 are positive constants. But since limx!0 x log x D 0, it follows that lim sup�!1 .1 ��/v.�/ � 0, as was to be proved.
Finally, let us show that lim inf�!1 v.�/ < 0. Suppose not, i.e. suppose lim inf�!1 v.�/ � 0. Thenwe must have lim�!1.1 � �/v.�/ D 0, since we have already shown that lim sup�!1 .1 � �/v.�/ � 0above. Hence, lim�!1N .�; �.1 � �/v.�// D N .1; 0/ D . Qa; b�/, where Qa 2 arg maxa2A g.a; b�/. Butsince g.a�; b�/ � g. Qa; b�/ < 0 by Condition 1, and �.a�; b�/ ¤ �. Qa; b�/ by Condition 3.a, there is aconstant K2 > 0 such that
2.g.a�; b�/ � g.N .�; �.1 � �/v.�///j .N .�; �.1 � �/v.�//; �/j2 � � 2K2
.1 � �/2 ; 8� � 1:
Also, since lim�!1.1 � �/v.�/ D 0, we have
2v.�/
1 � � �K2
.1 � �/2 ; 8� � 1:
Plugging these two inequalities in (43) and simplifying yields
v0.�/ � � K2
.1 � �/2 ; 8� � 1;
which implies lim�!1 v.�/ D �1. But this is a contradiction, since we have assumed lim inf�!1 v.�/ �0. �
We are now ready to prove Theorem 5.
Proof of Theorem 5. Fix an arbitrary �0 2 .0; 1/. If we show that limr!0 U 0r.�0/=r D 1 it will followthat limr!0 ar.�0/ D a�, since there is a constant K0 > 0 such that for each r > 0,
Ng � g � g.ar.�0/; br.�0// � g.a�; br.�0// � K0ja� � ar.�0/j2 U 0r.�0/=r;
by the large player’s incentive constraint (44) and Condition 3.a. Since limr!0 ar.�0/ D a� implieslimr!0 br.�0/ D b�, to conclude the proof we need only show that limr!0 U 0r.�0/=r D1.
En route to a contradiction, suppose there is some k > 0 such that lim infr!0 U 0r.�0/=r � k.
Claim. 8" > 0, 8�1 2 .�0; 1/ 9Nr > 0 such that 8r 2 .0; Nr�,
U 0r.�0/ � kr ) U 0r.�/ � r.v.�/C "/ 8� 2 Œ�0; �1�;
where v is the unique solution of the initial value problem (43).
54
To prove this claim, fix " > 0 and �1 2 .�0; 1/ and recall that, by Theorem 1, for each ı > 0 thereexists rı > 0 such that for each 0 < r < rı and � 2 Œ�0; �1� we have Ur.�/ < g.a�; b�/C ı, and hence
U 00r .�/ <2U 0r.�/1 � � C
2r.ı C g.a�; b�/ � g.N .�; �.1 � �/U 0r.�/=r///j .N .�; �.1 � �/U 0r.�/=r/; �/j2
:
Thus, for each ı > 0 and 0 < r < rı ,
U 0r.�0/ � kr ) U 0r.�/ � Vr;ı.�/ 8� 2 Œ�0; �1�; (46)
where Vr;ı solves the initial value problem37
V 0r;ı.�/ D2Vr;ı.�/
1 � � C 2r.ı C g.a�; b�/ � g.eN .�; �.1 � �/Vr;ı.�/=r///j .eN .�; �.1 � �/Vr;ı.�/=r/; �/j2
; Vr;ı.�0/ D kr:
where eN .�; z/ D(
N .�; z/; if Condition 2.a holdsN .�;maxfz; 0g/; if Condition 2.b holds.
Clearly, Vr;ı must be homegeneous of degree one in r , so it must be of the form Vr;ı.�/ D rvı.�/, wherevı is independent of r and solves the initial value problem
v0ı.�/ D2vı.�/
1 � � C2.ı C g.a�; b�/ � g.eN .�; �.1 � �/vı.�///
j .eN .�; �.1 � �/vı.�//; �/j2; vı.�0/ D k; (47)
which coincides with (43) when ı D 0. Thus, by (46), it suffices to show that for some ı > 0 we havevı.�/ � v.�/C " for all � 2 Œ�0; �1�, where v is the unique solution of (43). In effect, over the domainŒ�0; �1�, the right-hand side of (47) is jointly continuous in .ı; �; vı/ and Lipschitz continuous in vıuniformly in .�; ı/. Therefore, by standard results on existence, uniqueness and continuity of solutions toordinary differential equations, for every ı > 0 small enough, a unique solution, vı , exists on the intervalŒ�0; �1� and the mapping ı 7! vı is continuous in the sup-norm. Hence, for some ı0 > 0 small enough,vı0.�/ � v.�/C " for all � 2 Œ�0; �1�. Letting Nr D rı0
thus concludes the proof of the claim.The claim above implies
lim inf�!1 lim inf
r!0 U 0r.�/=r < 0; (48)
since lim inf�!1 v.�/ < 0 by Lemma C.9, and lim infr!0 U 0r.�0/=r � k by assumption. Thus, underCondition 2.b we readily get a contradiction, since in this case Ur.�/ must be increasing in � for eachr > 0, by Theorem 4.
Now suppose Condition 2.a holds. Since g.N .1; 0// > g.a�; b�/ by Condition 1, there is some � > 0such that
g.N .�; 0// > g.a�; b�/C �; 8� � 1:This fact, combined with (48) and the upper bound from Theorem 1, implies that there is some �1 2 .�0; 1/and r > 0 such that
U 0r.�1/ < 0 and Ur.�1/ < g.a�; b�/C � < g.N .�; 0//; 8� 2 Œ�1; 1/: (49)
37Implication (46) follows from the fact that U 00r .�/ < V 0r;ı .�/ whenever U 0r .�/ D Vr;ı .�/.
55
We claim that Ur.�/ < g.a�; b�/C � for all � 2 Œ�1; 1/. Otherwise, Ur must have a local minimum atsome point �2 2 .�1; 1/ where
Ur.�2/ < g.a�; b�/C �; (50)
since Ur.�1/ < g.a�; b�/C� and U 0r.�1/ < 0. Since at the local minimum �2 we must have U 0r.�2/ D 0and U 00r .�2/ � 0, the optimality equation implies
0 � U 00.�2/ D 2r.Ur.�2/ � g.N .�2; 0///
j .N .�2; 0/; �/j2;
and hence Ur.�2/ � g.N .�2; 0// � 0, which is impossible by (49) and (50). We have thus proved thatUr.�/ < g.a
�; b�/C � for all � 2 Œ�1; 1/. But this is a contradiction, since Ur must satisfy the boundarycondition lim�!1 Ur.�/ D g.N .1; 0// by Theorem 4. �
C.6 Proof of Proposition 4
The proof relies on two lemmas, presented below. Throughout this section we maintain all the assumptionsof Proposition 4.
Lemma C.10. N .�; z/ ¤ ;; 8.�; z/ 2 Œ0; 1� � Œ0;1/.
Proof. Fix .�; z/ 2 Œ0; 1� � Œ0;1/ and consider the correspondence � W A � B � A � B ,
�.a; b/ D
8̂<̂:. Oa; Ob/ W
Oa 2 arg maxa0
g.a0; b/C z�.b/�2.�.a�; b/ � �.a; b//�.a0; b/Ob 2 arg max
b0�h.a�; b0; b/C .1 � �/h.a; b0; b/
9>=>; :Thus, an action profile .a; b/ 2 A � B belongs to N .�; z/ if and only if it is a fixed point of � . ByBrouwer’s fixed point theorem it is enough to show that � is single-valued and continuous. Indeed, sinceg, h, � and � are continuous, � is non-empty-valued and upper hemi-continuous. To see that � is actuallysingle-valued (and hence continuous), fix .a; b/ 2 A�B and note that the assumptions g11 < 0, h22 < 0,�1 > 0 and �11 � 0 imply that g.�; b/ C z�.b/�2.�.a�; b/ � �.a; b//�.�; b/ and �h.a�; �; b/ C .1 ��/h.a; �; b/ are strictly concave and hence that �.a; b/ is a singleton. �
The proof of Proposition 4 below uses a first-order condition to characterize the action profile .a; b/ 2N .�; z/. To express this condition, define a function F W A � B � Œ0; 1� � Œ0;1/! R2 as follows:
F.a; b; �; z/defD�g1.a; b/Cz.�.a
�; b/ � �.a; b/�.b/2
/�1.a; b/; � h2.a�; b; b/C .1��/ h2.a; b; b/
�: (51)
Thus, the first-order condition for .a; b/ 2 N .�; z/ can be written as:
F.a; b; �; z/ � . Oa � a; Ob � b/ � 0 8. Oa; Ob/ 2 A � B; (52)
where “�” designates inner product.
Lemma C.11. The function F W A � B � Œ0; 1� � Œ0;1/ ! R2, defined by (51), satisfies the followingconditions:
56
(A) 9L > 0 such that 8.a; b/ 2 A � B and 8.�; z/, . O�; Oz/ 2 Œ0; 1� � Œ0;1/,
jF.a; b; O�; Oz/ � F.a; b; �; z/j � L j. O�; Oz/ � .�; z/jI
(B) 8K > 0 9M > 0 such that 8.�; z/ 2 Œ0; 1� � Œ0;K� and 8.a; b/, . Oa; Ob/ 2 A � B ,�F. Oa; Ob; �; z/ � F.a; b; �; z/� � . Oa � a; Ob � b/ � �M .j Oa � aj2 C j Ob � bj2/:
Proof. For each .a; b/ 2 A � B and .�; z/, . O�; Oz/ 2 Œ0; 1� � Œ0;1/,
F.a; b; O�; Oz/ � F.a; b; �; z/ D"�.b/�2.�.a�; b/ � �.a; b//�1.a; b/.Oz � z/
.h2.a�; b; b/ � h2.a; b; b//. O� � �/
#;
which yields condition (A) with L D 2max.a;b/ �.b/�2j�.a; b/jj�1.a; b/j C jh2.a; b; b/j.Turning to condition (B), fix K > 0. By the mean value theorem, for each .a; b/, . Oa; Ob/ 2 A � B and
.�; z/ 2 Œ0; 1� � Œ0;K� there exists some . Na; Nb/ such that
�F. Oa; Ob; �; z/ � F.a; b; �; z/� � . Oa � a; Ob � b/ D h Oa � a Ob � b
iD.a;b/F. Na; Nb; �; z/
"Oa � aOb � b
#;
where D.a;b/F. Na; Nb; �; z/ is the derivative of F.�; �; �; z/ at . Na; Nb/. Thus,
.F. Oa; Ob; �/ � F.a; b; �; z// � . Oa � a; Ob � b/ � �M.j Oa � aj2 C j Ob � bj2/;
where
MdefD� sup
nx>D.a;b/F.a; b; �; z/ x W .a; b; �; z; x/ 2 A � B � Œ0; 1� � Œ0;K� �R2 with jxj D 1
o:
It remains to show that M > 0. Since g, h and � are twice continuously differentiable and � iscontinuous, the supremum above is attained at some point in A�B � Œ0; 1�� Œ0;K�� .R2 n f0g/. Thus, itsuffices to prove thatD.a;b/F.a; b; �; z/ is negative definite for each .a; b; �; z/ 2 A�B � Œ0; 1�� Œ0;K�.Fix an arbitrary .a; b; �; z/ 2 A � B � Œ0; 1� � Œ0;K�. We have
D.a;b/F.a; b; �; z/ D
D"g11 C z��2..�� � �/�11 � �21/ g12 C z��2..�� � �/�12 C .��2 � �2/�1/
.1 � �/h12 �.h�22 C h�23/C .1 � �/.h22 C h23/
#;
where, to save on notation, we omit the arguments of the functions on the right-hand side and use super-script � to indicate when a function is evaluated at .a�; b; b/ or .a�; b/ rather than at .a; b; b/ or .a; b/.Thus, for each " � 0,
D.a;b/F.a; b; �; z/ D ‰."/Cƒ."/;where
‰."/defD"
g11 g12
.1 � �/h12 �.h�22 C h�23/C .1 � �/.h22 C h23/C "
#
57
and
ƒ."/defD"z��2..�� � �/�11 � �21/ z��2..�� � �/ � �12 C .��2 � �2/�1/
0 �"
#:
The matrix ‰.0/ is negative definite by condition (a), hence there is "0 > 0 small enough such that ‰."0/remains negative definite. Moreover, ƒ."0/ is negative semi-definite by condition (c), since "0 > 0 andz � 0. It follows that D.a;b/F.a; b; �; z/ D ‰."0/Cƒ."0/ is negative definite, and hence M > 0. �
We are now ready to prove Proposition 4.
Proof of Proposition 4. (a), (b) & (c)) Condition 2.b. First, N .�; z/ is nonempty for each .�; z/ 2Œ0; 1� � Œ0;1/ by Lemma C.10. Moreover, .a; b/ 2 N .�; z/ if and only if .a; b/ satisfies the first-ordercondition (52), because the functions g.�; b/ C z.�.a�; b/ � �.a; b//�.�; b/ and �h.a�; �; b/ C .1 ��/h.a; �; b/ are differentiable and concave for each fixed .a; b/. We will now show that N .�; z/ is asingleton for each .�; z/ 2 Œ0; 1� � Œ0;1/. Let .�; z/ 2 Œ0; 1� � Œ0;1/ and pick .a; b/, . Oa; Ob/ 2 N .�; z/.Thus,
F.a; b; �; z/ � . Oa � a; Ob � b/ � 0 and F. Oa; Ob; �; z/ � .a � Oa; b � Ob/ � 0;by the first-order conditions. Subtracting the former inequality from the latter yields
.F. Oa; Ob; �; z/ � F.a; b; �; z// � . Oa � a; Ob � b/ � 0;
which is possible only if .a; b/ D . Oa; Ob/, by part .B/ of Lemma C.11. We have thus shown that N .�; z/
contains a unique action profile for each .�; z/ 2 Œ0; 1� � Œ0;1/.Turning to Lipschitz continuity, fix K > 0 and let L > 0 and M > 0 designate the constants from
Lemma C.11. Fix �, O� 2 Œ0; 1� and z, Oz 2 Œ0;K� and let .a; b/ 2 N .�; z/ and . Oa; Ob/ 2 N . O�; Oz/. By thefirst-order conditions,
0 � .F. Oa; Ob; O�; Oz/ � F.a; b; �; z// � . Oa � a; Ob � b/D .F. Oa; Ob; O�; Oz/ � F.a; b; O�; Oz// � . Oa � a; Ob � b/
C .F.a; b; O�; Oz/ � F.a; b; �; z// � . Oa � a; Ob � b/
� �M.j Oa � aj2 C j Ob � bj2/ C L
qj O� � �j2 C jOz � zj2
qj Oa � aj2 C j Ob � bj2;
where the last inequality follows from Lemma C.11 and the Cauchy-Schwarz inequality. Therefore,qj Oa � aj2 C j Ob � bj2 � L
M
qj O� � �j2 C jOz � zj2
and we have thus shown that N is Lipschitz continuous over Œ0; 1� � Œ0;K�.It remains to show that g.N .�; 0// is increasing in �. Let .a.�/; b.�// designate the unique static
Bayesian Nash equilibrium when the prior is �. Since a and b are Lipschitz continuous, and hence ab-solutely continuous, it suffices to show that 0 � d
d�g.a.�/; b.�// D g1
dad�C g2 db
d�almost everywhere,
which is equivalent to showing that g2 dbd�� 0 a.e. by the first-order condition of the large player.38
Then, by condition (b), it is enough to show that db.�/=d� has the same sign as h2.a�; b.�/; b.�// �
38Indeed, if g1.a.�/; b.�// ¤ 0 then a must be constant in a neighborhood of �, in which case .da=d�/ .�/ D 0.
58
h2.a.�/; b.�/; b.�// almost everywhere. First, consider the case in which a.�/ and b.�/ are interiorsolutions, so the first-order conditions are satisfied with equality. By the implicit function theorem,"
da=d�db=d�
#D"
g11 g12
.1 � �/h12 �.h�22 C h�23/C .1 � �/.h22 C h23/
#�1�"
0
��.h�2 � h2/
#;
which implies
db=d� D
ˇ̌̌̌ˇ g11 0
.1 � �/h12 ��.h�2 � h2/
ˇ̌̌̌ˇˇ̌̌̌
ˇ g11 g12
.1 � �/h12 �.h�22 C h�23/C .1 � �/.h22 C h23/
ˇ̌̌̌ˇ:
By condition (a) the denominator is positive and the numerator has the same sign as h�2 � h2. Thus,dd�g.a.�/; b.�// � 0 for almost every � such that .a.�/; b.�// is in the interior of A � B . Also, for
almost every �, if either a.�/ or b.�/ is a corner solution, then either (i) b is constant in a neighborhoodof �, and hence g2db=d� D 0 trivially, or (ii) the first-order condition for b holds with equality and a isconstant in a neighborhood of �, in which case differentiating the first-order condition of the small players(with da=d� D 0) yields the desired result.
(a) & (c)) Condition 3. Condition 3.a follows directly from �1 > 0. Turning to Condition 3.b, foreach .a; �/ 2 A � Œ0; 1� let bBR.a; �/ 2 B designate the joint best-reply of the small players to .a; �/, i.e.
bBR.a; �/ 2 arg maxb
�h.a�; b; bBR.a; �//C .1 � �/h.a; b; bBR.a; �//: (53)
To see why such bBR.a; �/ must be unique, note that a necessary (and sufficient) condition for b to be abest-reply of the small players to .a; �/ is the first-order condition
G.a; b; �/defD�h2.a�; b; b/C .1 � �/h2.a; b; b/
8̂<̂:� 0 W b D minBD 0 W b 2 .minB;maxB/� 0 W b D maxB:
Since h22Ch23 < 0 along the diagonal of the small players’ actions, we haveG2 < 0 and henceG.a; b; �/is strictly decreasing in b for each .a; �/. This implies that bBR.a; �/ is unique for each .a; �/. Next, notethat for each .a; b; �/ 2 A � B � Œ0; 1�,
.G.a; b�; �/ �G.a; b; �//.b� � b/ � �K1jb� � bj2; (54)
where K1 D min.a0;b0;�/ jG2.a0; b0; �0/j > 0. Moreover, G is C1, and hence for each .a; �/ 2 A � Œ0; 1�there is some Na 2 A such that
G.a�; b; �/ �G.a; b; �/ D G1. Na; b; �/.a� � a/ D .1 � �/h12. Na; b; b/.a� � a/;
and so there is a constant K2 > 0 such that for each .a; b; �/ 2 A � B � Œ0; 1�,
jG.a�; b; �/ �G.a; b; �/j � K2.1 � �/ja� � aj: (55)
Noting that b� D bBR.a�; �/ and letting b D bBR.a; �/, the first-order conditions yield
G.a�; b�; �/.b � b�/ � 0 and G.a; b; �/.b� � b/ � 0;
59
and hence.G.a�; b�; �/ �G.a; b; �//.b� � b/ � 0:
This inequality, together with (54) and (55), implies
0 � .G.a�; b�; �/ �G.a; b; �//.b� � b/D .G.a�; b�; �/ �G.a; b�; �//.b� � b/C .G.a; b�; �/ �G.a; b; �//.b� � b/
� K2.1 � �/ja� � ajjb� � bj �K1jb� � bj2;
and therefore,jb� � bj � .K2=K1/ .1 � �/ja� � aj:
(a), (c) & (d)) Condition 1. Under assumptions (a) and (c), there is a unique b� 2 B.a�/. Then,assumption (d) implies that a� is not a best-reply to b�. Moreover, under assumption (c) there is no a ¤ a�with �.a; b�/ D �.a�; b�/, hence Condition 1 follows. �
D Appendix for Section 7
Throughout Appendix D, we maintain Conditions 1 and 4. We write U and L to designate the upper andlower boundaries of the correspondence E , respectively, i.e.
U.�/defD sup E.�/ ; L.�/
defD inf E.�/; for all � 2 Œ0; 1�:
Proposition D.1. The upper boundary U is a viscosity sub-solution of the upper optimality equation on.0; 1/.
Proof. If U is not a sub-solution, there exists q 2 .0; 1/ and a C2-function V W .0; 1/ ! R such that0 D .V � U �/.q/ < .V � U �/.�/ for all � 2 .0; 1/ n fqg, and
H�.q; V .q/; V 0.q// D H�.q; U �.q/; V 0.q// > V 00.q/ :
Since H� is lower semi-continuous, U � is upper semi-continuous and V > U � on .0; 1/ n fqg, there exist" and ı > 0 small enough such that for all � 2 Œq � "; q C "�,
H.�; V .�/ � ı; V 0.�// > V 00.�/; (56)
V.q � "/ � ı > U �.q � "/ � U.q � "/ and V.q C "/ � ı > U �.q C "/ � U.q C "/: (57)
Figure 7 displays the configuration of functions U � and V � ı. Fix a pair .�0; W0/ 2 Graph E with�0 2 .q � "; q C "/ and W0 > V.�0/ � ı. (Such .�0; W0/ exists because V.q/ D U �.q/ and U � isu.s.c..) Let .at ; Nbt ; �t / be a sequential equilibrium that attains the pair .�0; W0/. Denoting by .Wt / thecontinuation value of the normal type, we have
dWt D r.Wt � g.at ; Nbt // dt C rˇt � .dXt � �.at ; Nbt / dt/
for some ˇ 2 L. Next, we will show that, with positive probability, eventually Wt becomes greater thanU.�t /, leading to a contradiction since U is the upper boundary of E .
60
*U
( )V ! "#
$#q q $+qBeliefs
Payoffs
*U
( )V !
0 0( , )W!
Figure 7: A viscosity sub-solution.
Let Dt D Wt � .V .�t / � ı/. By Itô’s formula,
dV.�t / D j t j2.V00.�t /2
� V0.�t /
1 � �t/ dt C tV
0.�t / dZnt ;
where t D .at ; Nbt ; �t /, and hence,
dDt D�rDtCr.V .�t /�ı/�rg.at ; Nbt /�j t j2.V
00.�t /2� V
0.�t /1 � �t
/�dt C �
rˇt�. Nbt /� tV 0.�t /�dZn
t :
Therefore, so long as Dt � D0=2,
(a) �t cannot exit the interval Œq � "; q C "� by (57), and
(b) there exists � > 0 such that either the drift of Dt is greater than rD0=2 or the norm of the volatil-ity of Dt is greater than �; because of inequality (56) using an argument similar to the proof ofLemma C.8.39
This implies that with positive probability Dt stays above D0=2 and eventually reaches any arbitrarilylarge level. Since payoffs are bounded, this leads to a contradiction. We conclude that U must be asub-solution of the upper optimality equation. �
39While the required argument is similar to the one used in the proof of Lemma C.8, there are important differences, so weoutline them here. First, the functions d and f from Lemma C.8 must be re-defined, with the test function V replacing U in thenew definition. Second, the definition of the compact set ˆ also requires change: ˆ would now be the set of all .a; Nb; �; ˇ/ with� 2 Œq� "; qC "� and jˇj �M such that the incentive constraints (42) are satisfied and d.a; Nb; �/ � 0. Since � is bounded awayfrom 0 and 1, the boundary conditions will play no role here. Finally, note that U is assumed to satisfy the optimality equation inthe lemma, while here V satisfies the strict inequality (56). Accordingly, we need to modify the last part of that proof as follows:f .a; Nb; �; ˇ/ D 0 implies d.a; Nb; �/ > 0, and therefore we have � > 0.
61
The next lemma is an auxiliary result used in the proof of Proposition D.2 below.
Lemma D.1. The correspondence E of public sequential equilibrium payoffs is convex-valued and has anarc-connected graph.
Proof. First, we show that E is convex-valued. Fix p 2 .0; 1/, Nw, w 2 E.p/ with Nw > w, and v 2 .w; Nw/.Let us show that v 2 E.p/. Consider the set V D f.�; w/ jw D ˛j� � pj C vg where ˛ > 0 is chosenlarge enough so that ˛j� � p j C v > U.�/ for all � sufficiently close to 0 and 1. Let .�t ; Wt /t�0 be thebelief / continuation value process of a public sequential equilibrium that yields the normal type a payoffof Nw. Let � defD infft > 0 j.�t ; Wt / 2 V g. By Theorem 8 specialized to the case with a single behavioraltype, we have limt!1 �t D 0 with probability 1 � p, and limt!1 �t D 1 with probability p, hence� < 1 almost surely.40 If with positive probability �� D p, then W� D v and hence v 2 E.p/, which isthe desired result. For the case when �� ¤ p almost surely, the proof proceeds is two steps. In Step 1, weconstruct a pair of continuous curves NC , C � Graph Ej.0;1/ such that their projection on the �-coordinateis the whole .0; 1/ and
inf fw j .p;w/ 2 NCg > v > sup fw j .p;w/ 2 Cg:In Step 2, we use these curves to construct a sequential equilibrium for prior p under which the normaltype receives a payoff of v, concluding the proof that E is convex valued.
Step 1. If �� ¤ p almost surely, then both �� < p and �� > p must happen with positive probabilityby the martingale property. Hence, there exists a continuous curve C � graph E with endpoints .p1; w1/and .p2; w2/, with p1 < p < p2, such that for all .�; w/ 2 C we have w > v and � 2 .p1; p2/. Fix0 < " < p � p1. We will now construct a continuous curve C 0 � graph Ej.0;p1C"� that has .p1; w1/as an endpoint and satisfies inf f� j 9w s.t. .�; w/ 2 C 0g D 0. Fix a public sequential equilibrium of thedynamic game with prior p1 that yields the normal type a payoff of w1. Let Pn denote the probabilitymeasure over the sample paths of X induced by the strategy of the normal type. By Theorem 8 we have�t ! 0 Pn-almost surely. Moreover, since .�t / is a supermartingale under Pn, the maximal inequality fornon-negative supermartingales yields:
Pn
"supt�0
�t � p1 C "#� 1 � p1
p1 C "> 0 :
Therefore, there exists a sample path . N�t ; NWt /t�0 with the property that N�t � p1C" < p for all t and N�t !0 as t !1. Thus, the curve C 0 � graph E , defined as the image of the path t 7! . N�t ; NWt /, has .p1; w1/ asan end-point and satisfies inf f� j 9w s.t. .�; w/ 2 C 0g D 0. Similarly, we can construct a continuous curveC 00 � Graph EjŒp2�";1/ that has .p2; w2/ as an endpoint and satisfies sup f� j 9w s.t. .�; w/ 2 C 00g D 1.
We have thus constructed a continuous curve NC defDC 0 [ C [ C 00 � Graph Ej.0;1/ which projectsonto .0; 1/ and satisfies inf fw j .p;w/ 2 NCg > v. A similar construction yields a continuous curveC � Graph Ej.0;1/ which projects onto .0; 1/ and satisfies sup fw j .p;w/ 2 Cg < v.
Step 2. We will now construct a sequential equilibrium for prior p under which the normal typereceives a payoff of v. Let � 7! .a.�/; Nb.�// be a measurable selection from the correspondence of static
40To be precise, Theorem 8 does not state what happens conditional on the behavioral type. However, in the particular case ofa single behavioral type, it is easy to adapt the proof of that theorem (cf. Appendix E) to show that limt!1 �t D 1 under thebehavioral type.
62
Bayesian Nash equilibrium. Define .�t /t�0 as the unique weak solution of
d�t D �j .a.�t /; Nb.�t /; �t /j2=.1 � �t /C .a.�t /; Nb.�t /; �t / � dZnt
with initial condition �0 D p.41 Next, let .Wt /t�0 be the unique solution of the pathwise deterministicODE
dWt D r.Wt � g.a.�t /; Nb.�t /// dtwith initial conditionW0 D v, up to the stopping time T > 0when .�t ; Wt / first hits either NC or C . Definethe strategy profile .at ; Nbt /t�0 as follows: for t < T , set .at ; Nbt / D .a.�t /; Nb.�t //; from t D T onwards.at ; Nbt / follows a sequential equilibrium of the game with prior �T . By Theorem 2 .at ; Nbt ; �t /t�0 mustbe a sequential equilibrium that yields the normal type a payoff of v. We have thus shown that v 2 E.p/,concluding the proof that E is convex-valued.
We now show that Graph.E/ is arc-connected. Fix p < q, v 2 E.p/ and w 2 E.q/. Consider asequential equilibrium of the game with prior q that yields the normal type a payoff of w. Since �t ! 0
under the normal type, there exists a continuous curve C � graph E with endpoints .q; w/ and .p; v0/ forsome v0 2 E.p/. Since E is convex-valued, the straight line C 0 connecting .p; v/ to .p; v0/ is containedin the graph of E . Hence C [ C 0 is a continuous curve connecting .q; w/ and .p; v/ which is contained inthe graph of E , concluding the proof that E has an arc-connected graph. �
Proposition D.2. The upper boundary U is a viscosity super-solution of the upper optimality equation on.0; 1/.
Proof. If U is not a super-solution, there exists q 2 .0; 1/ and a C2-function V W .0; 1/ ! R such that0 D .U� � V /.q/ < .U� � V /.�/ for all � 2 .0; 1/ n fqg, and
H�.q; V .q/; V 0.q// D H�.q; U�.q/; V 0.q// < V 00.q/ :
Since H� is upper semi-continuous, U� is lower semi-continuous and U� > V on .0; 1/ n fqg, there exist", ı > 0 small enough such that for all � 2 Œq � "; q C "�,
H.�; V .�/C ı; V 0.�// < V 00.�/; (58)
V.q � "/C ı < U.q � "/ and V.q C "/C ı < U.q C "/: (59)
Figure 8 displays the configuration of functions U� and V Cı. Fix a pair .�0; W0/ with �0 2 .q�"; qC"/and U.�0/ < W0 < V.�0/C ı.
We will now construct a sequential equilibrium that attains .�0; W0/, and this will lead to a contradic-tion since U.�0/ < W0 and U is the upper boundary of E .
Let � 7! .a.�/; Nb.�// 2 N .�; �.1 � �/V 0.�/=r/ be a measurable selection of action profiles thatminimize
rV .�/ � rg.a; Nb/ � j .a; Nb; �/j2.V00.�/2� V
0.�/1 � � / (60)
41Condition 1 ensures that .a.�t /; Nb.�t /; �t / is bounded away from zero when .a.�t /; Nb.�t // 2 N .�t ; 0/, hence standardresults for existence and uniqueness of weak solutions apply (Karatzas and Shreve, 1991, p. 327).
63
*U
( )V ! "+
#$q q
( , )W% %!
#+q
( , ( ))q U q# #$ $
C
Beliefs
Payoffs( , ( ))q U q# #+ +
*U
( )V !
0 0( , )W!
Figure 8: A viscosity super-solution.
over all .a; Nb/ 2 N .�; �.1��/V 0.�/=r/, for each � 2 .0; 1/. Define .�t /t�0 as the unique weak solutionof
d�t D �j .a.�t /; Nb.�t /; �t /j2=.1 � �t / dt C .a.�t /; Nb.�t /; �t / � dZnt (61)
on the interval Œq � "; q C "�, with initial condition �0.42 Next, let .Wt /t�0 be the unique strong solutionof
dWt D r.Wt � g.a.�t /; Nb.�t /// dt C .a.�t /; Nb.�t /; �t /V 0.�t / � dZnt ; (62)
with initial conditionW0, until the stopping time when �t first exits Œq�"; qC"�.43 Then, by Itô’s formula,the process Dt D Wt � V.�t / � ı has zero volatility and drift given by
rDt C rV .�t / � rg.a.�t /; Nb.�t // � j .a.�t /; Nb.�t //; �t /j2�V 00.�t /2
� V0.�t /
1 � �t
�:
By (58) and the definition of .a.�/; Nb.�//, the drift of Dt is strictly negative so long as Dt � 0 and�t 2 Œq � "; q C "�. (Note that D0 < 0.) Therefore, the process .�t ; Wt /t�0 remains under the curve.�t ; V .�t /C ı/ from time zero onwards so long as �t 2 Œq � "; q C "�.
By Lemma D.1 there exists a continuous curve C � EjŒq�";qC"� that connects the points .q� "; U.q�"// and .q C "; U.q C "//. By (59), the path C and the function V C ı bound a connected region inŒq � "; q C "� � R that contains .�0; W0/, as shown in Figure 8. Since the drift of Dt is strictly negativewhile �t 2 Œq � "; q C "�, the pair .�t ; Wt / eventually hits the path C at a stopping time � <1 before �texits the interval Œq � "; q C "�.
42Existence and uniqueness of a weak solution on a closed sub-interval of .0; 1/ follows from the fact that V 0 must be boundedon such subinterval, and therefore must be bounded away from zero by Lemma B.1. See Karatzas and Shreve (1991, p. 327).
43Existence and uniqueness of a strong solution follows from the Lipschitz and linear growth conditions in W , and the bound-edness of .a.�/; Nb.�/; �/V 0.�/ on Œq � "; q C "�.
64
We will now construct a sequential equilibrium for prior �0 under which the normal type receives apayoff of W0. Consider the strategy profile and belief process that coincides with .a.�t /; Nb.�t /; �t / upto time � , and follows a sequential equilibrium of the game with prior �� at all times after � . Since Wtis bounded, .a.�t /; Nb.�t // 2 N .�t ; �t .1 � �t /V 0.�t /=r/ and the processes .�t /t�0 and .Wt /t�0 follow(61) and (62) respectively, Theorem 2 implies that the strategy profile .a.�t /; Nb.�t //t�0 and the beliefprocess .�t /t�0 form a sequential equilibrium of the game with prior �0. It follows that W0 2 E.�0/,and this is a contradiction since W0 > U.�0/. Thus, U must be a super-solution of the upper optimalityequation. �
Lemma D.2. Every bounded viscosity solution of the upper optimality equation is locally Lipschitz con-tinuous.
Proof. En route to a contradiction, suppose that U is a bounded viscosity solution that is not locallyLipschitz. That is, for some p 2 .0; 1/ and " 2 .0; 1
2/ satisfying Œp�2"; pC2"� � .0; 1/ the restriction of
U to Œp � "; p C "� is not Lipschitz continuous. Let M D sup jU j. By Corollary B.1 there exists K > 0
such that for all .�; u; u0/ 2 Œp � 2"; p C 2"� � Œ�M;M� �R,
jH�.�; u; u0/j � K.1C ju0j2/ ; (63)
Since the restriction of U to Œp � "; pC "� is not Lipschitz continuous, there exist �0, �1 2 Œp � "; pC "�such that jU�.�1/ � U�.�0/j
j�1 � �0j� maxf1; exp.2M.4K C 1="//g : (64)
Hereafter we assume that �1 > �0 and U�.�1/ > U�.�0/. The proof for the reciprocal case is similarand will be omitted for brevity. Let V W J ! R be the solution of the differential equation
V 00.�/ D 2K.1C V 0.�/2/ ; (65)
with initial conditions given by
V.�1/ D U�.�1/ and V 0.�1/ D U�.�1/ � U�.�0/�1 � �0
; (66)
where J is the maximal interval of existence around �1.We claim that V has the following two properties:
(a) There exists �� 2 J \.p�2"; pC2"/ such that V.��/ D �M and �� < �0. In particular, �0 2 J .
(b) V.�0/ > U�.�0/.
We first prove property (a). For all � 2 J such that V 0.�/ > 1, we have V 00.�/ < 4KV 0.�/2 or,equivalently, .logV 0/0.�/ < 4KV 0.�/, which implies
V. O�/ � V. Q�/ > 1
4K.log.V 0. O�// � log.V 0. Q�/// ; 8 O�; Q� 2 J such that V 0. O�/ > V 0. Q�/ > 1 : (67)
By (64) and (66), we have 14K
log.V 0.�1// > 2M , and therefore a unique Q� 2 J exists such that
1
4K.log.V 0.�1// � log.V 0. Q�/// D 2M : (68)
65
Since V 0. Q�/ > 1, it follows from (67) that V.�1/ � V. Q�/ > 2M and so V. Q�/ < �M . Since V.�1/ >U.�0/ � �M , there exists some �� 2 . Q�; �1/ such that V.��/ D �M . Moreover �� must belong to.p � 2"; p C 2"/, because the strict convexity of V implies
�1 � �� < V.�1/ � V.��/V 0.��/
<2M
V 0. O�/ <2M
log.V 0. O�// D2M
log.V 0.�1// � 8KM< " ;
where the equality follows from (68) and the rightmost inequality follows from (64). Finally, we have�� < �0, otherwise the inequality V.��/ � U�.�0/ and the initial conditions (66) would imply
V 0.�1/ D U�.�1/ � U�.�0/�1 � �0 � V.�1/ � V.��/
�1 � �� ;
which would violate the strict convexity of V . This concludes the proof of property (a).Turning to property (b), the strict convexity of V and the initial conditions (66) imply
U�.�1/ � V.�0/�1 � �0
D V.�1/ � V.�0/�1 � �0
< V 0.�1/ D U�.�1/ � U�.�0/�1 � �0
;
and therefore V.�0/ > U�.�0/, as claimed.Now define L D max fV.�/ � U�.�/ j� 2 Œ��; �1�g. By property (b), we must have L > 0. Let O�
be a point at which the maximum L is attained. Since V.��/ D �M and V.�1/ D U�.�1/, we must haveO� 2 .��; �1/ and therefore V � L is a test function satisfying
U�. O�/ D V. O�/ � L and U�.�/ � V.�/ � L for each � 2 .��; �1/ :
Since U is a viscosity supersolution,
V 00. O�/ � H�. O�; V . O�/ � L; V 0. O�// ;
hence, by (63),V 00. O�/ � K.1C V 0. O�/2/ < 2K.1C V 0. O�/2//;
and this is a contradiction, since by construction V satisfies equation (65). �
Lemma D.3. Every bounded viscosity solution of the upper optimality equation is continuously differen-tiable with absolutely continuous derivatives.
Proof. Let U W .0; 1/! R be a bounded solution of the upper optimality equation. By Lemma D.2, U islocally Lipschitz and hence differentiable almost everywhere. We will now show that U is differentiableeverywhere. Fix � 2 .0; 1/. Since U is locally Lipschitz, there exist ı > 0 and k > 0 such that for everyp 2 .� � ı; � C ı/ and every smooth test function V W .� � ı; � C ı/! R satisfying V.p/ D U.p/ andV � U we have
jV 0.p/j � k :It follows from Corollary B.1 that there exists some M > 0 such that
jH.p;U.p/; V 0.p//j �M
for every p 2 .� � ı; � C ı/ and every smooth test function V satisfying V � U and V.p/ D U.p/.
66
Let us now show that for all " 2 .0; ı/ and "0 2 .0; "/
�M"0." � "0/ < U.� C "0/ ��"0
"U.� C "/C " � "0
"U.�/
�< M"0." � "0/: (69)
If not, for example if the second inequality fails, then we can choose K > 0 such that the C 2 function (aparabola)
"0 7! f .� C "0/ D�"0
"U.� C "/C " � "0
"U.�/
�CM"0." � "0/CK
is strictly above U.�C "0/ over .0; "/, except for a tangency point at some "00 2 .0; "/. But this contradictsthe fact that U is a viscosity subsolution, since f 00.�C"00/ D �2M < H.�C"00; U.pC"00/; U 0.�C"00//.This contradictions proves inequalities (69).
It follows from (69) that for all 0 < "0 < " < ı,ˇ̌̌̌U.� C "0/ � U.�/
"0� U.� C "/ � U.�/
"
ˇ̌̌̌�M":
Thus, as " converges to 0 from above, .U.� C "/ � U.�//=" converges to a limit U 0.�C/. Similarly, if" converges to 0 from below, the quotient above converges to a limit U 0.��/. We claim that U 0.�C/ DU 0.��/. Otherwise, if for example U 0.�C/ > U 0.��/; then the function
"0 7! f1.� C "0/ D U.�/C "0U0.��/C U 0.�C/
2CM"02
is below U in a neighborhood of �; except for a tangency point at �. But this leads to a contradiction,because f 001 .�/ D 2M > H.�;U.�/; U 0.�// and U is a super-solution. Therefore U 0.�C/ D U 0.��/and we conclude that U is differentiable at every � 2 .0; 1/.
It remains to show that U 0 is locally Lipschitz. Fix � 2 .0; 1/ and, arguing just as above, choose ı > 0and M > 0 so that
jH.p;U.p/; V 0.p//j �Mfor every p 2 .� � ı; �C ı/ and every smooth test function V satisfying V.p/ D U.p/ and either V � Uor V � U . We affirm that for any p 2 .� � ı; � C ı/ and " 2 .0; ı/
jU 0.p/ � U 0.p C "/j � 2M":
Otherwise, e.g. if U 0.p C "/ > U 0.p/C 2M" for some p 2 .� � ı; � C ı/ and " 2 .0; ı/, then the testfunction
"0 7! f2.p C "0/ D "0
"U.p C "/C " � "0
"U.p/ �M"0." � "0/
must be above U at some "0 2 .0; "/ (since f 02.pC "/�f 02.p/ D 2M".) Therefore, there exists a constantK > 0 such that f2.p C "0/ �K stays below U for "0 2 Œ0; "�; except for a tangency at some "00 2 .0; "/.But then
f 002 .� C "00/ D 2M > H.� C "00; U.� C "00/; U 0.� C "00//;contradicting the assumption that U is a viscosity super-solution. �
67
Proposition D.3. The upper boundary U is a continuously differentiable function, with absolutely contin-uous derivatives. Moreover, U is the greatest bounded solution of the differential inclusion
U 00.�/ 2 ŒH.�; U.�/; U 0.�// ; H�.�; U.�/; U 0.�//� a.e. : (70)
Proof. First, by Propositions D.1, and D.2 and Lemma D.3, the upper boundary U is a differentiablefunction with absolutely continuous derivative that solves the differential inclusion (70). If U is not thegreatest bounded solution of (70), then there exists another bounded solution V which is strictly greaterthan U at some p 2 .0; 1/. Choose " > 0 such that V.p/ � " > U.p/. We will show that V.p/ � " is thepayoff of a public sequential equilibrium, which is a contradiction since U is the upper boundary.
From the inequalityV 00.�/ � H.�; V .�/; V 0.�// a.e.
it follows that a measurable selection � 7! .a.�/; Nb.�// 2 N .�; �.1 � �/V 0.�/=r/ exists such that
rV .�/ � rg.a.�/; Nb.�/; �/ � j .a.�/; Nb.�/; �/j2.V00.�/2� V
0.�/1 � � / � 0 ; (71)
for almost every � 2 .0; 1/. Let .�t / be the unique weak solution of
d�t D �j .a.�t /; Nb.�t /; �t /j2=.1 � �t /C .a.�t /; Nb.�t /; �t / dZnt ;
with initial condition �0 D p. Let .Wt / be the unique strong solution of
dWt D r.Wt � g.a.�t /; Nb.�t /// dt C V 0.�t / .a.�t /; Nb.�t /; �t / dZnt ;
with initial condition W0 D V.p/ � ". Consider the process Dt D Wt � V.�t /. It follows from Itô’sformula for differentiable functions with absolutely continuous derivatives that:
dDt
dtD rDt C rV .�t / � rg.a.�t /; Nb.�t /; �t / � j .a.�t /; Nb.�t /; �t /j2.V
00.�t /2
� V0.�t /
1 � �t/ :
Therefore, by (71) we havedDt
dt� rDt ;
and since D0 D �" < 0 it follows that Wt & �1. Let � be the first time that .�t ; Wt / hits the graph ofU . Consider a strategy profile / belief process that coincides with .at ; Nbt ; �t / up to time � and, after that,follows a public sequential equilibrium of the game with prior �� with value U.�� /. Theorem 2 impliesthat the strategy profile / belief process thus constructed is a sequential equilibrium that yields the largeplayer payoff V.p/ � " > U.p/, a contradiction. �
E Proof of Theorem 8
Consider the closed set
IC D f.a; Nb; �; ˇ/ 2 A ��.B/ ��K �Rd W conditions (11) and (12) are satisfiedg:
68
Then, the image of this set under
.a; Nb; �; ˇ/ 7! .�0; �0�. Nb/�1.�.a; Nb/ � ��.a; Nb//; rˇ>�. Nb//
is a closed set that does not intersect the line segment .0; 1/ � f0g � f0g by Condition 10. Thus, for any" > 0 there exist constants C > 0 and M > 0 such that for all .a; Nb; �; ˇ/ 2 IC with �0 2 Œ"; 1 � "�,either
j�0�. Nb/�1.�.a; Nb/ � ��.a; Nb//j � C or jrˇ>�. Nb/j �M:Fix a public sequential equilibrium .at ; Nbt ; �t /t�0 with continuation values .Wt /t�0 for the normal
type and consider the evolution of
exp.K1.Wt � g//CK2�20;twhile �0;t 2 Œ"; 1 � "�, where the constants K1 and K2 > 0 will be determined later. By Itô’s formula,under the probability measure generated by the normal type the process exp.K1.Wt � g// has drift
K1 exp.K1.Wt � g//r.Wt � g.at ; Nbt //CK21 exp.K1.Wt � g//r2jˇ>t �. Nbt /j2=2:
Thus, we can guarantee that the drift of exp.K1.Wt�g// is greater than or equal to 1 whenever jrˇ>t �. Nbt /j �M by choosing K1 > 0 such that
�K1r. Ng � g/CK21M 2=2 � 1:
Moreover, the drift of K2�20;t must always be non-negative, since under the normal type �0;t is a sub-martingale, and hence K2�20;t is also a submartingale. Now, even when jrˇ>t �. Nbt /j < M , the drift ofexp.K1.Wt � g// is still greater than or equal to
�K1 exp.K1. Ng � g//r. Ng � g/:
But in this case we have j�0;t�. Nbt /�1.�.at ; Nbt /���.at ; Nbt //j � C , so the drift of K2�20;t is greater thanor equal to K2C 2. Thus, by choosing K2 large enough so that
K2C2 �K1 exp.K1. Ng � g//r. Ng � g/ � 1;
we can ensure that the drift of exp.K1.Wt � g//CK2�20;t is always greater than 1 while �0;t 2 Œ"; 1� "�.But since exp.K1.Wt � g// C K2�20;t must be bounded in a sequential equilibrium, it follows that �0;tmust eventually exit the interval Œ"; 1 � "� with probability 1 in any sequential equilibrium. Since " > 0 isarbitrary, it follows that the bounded submartingale .�0;t /t�0 must converge to 0 or 1 almost surely, andit cannot converge to 0 with positive probability under the probability measure generated by the normaltype. �
F Appendix for Section 9
We begin with the following monotonicity lemma, which will be used throughout this appendix.
Lemma F.1. Fix .�; �/, .�0; �0/ 2 Œ0; 1��R, .a; b/ 2M.�; �/ and .a0; b0/ 2M.�0; �0/. If �0 � �, �0 � �and �0 � 0 then a0 � a.
69
Proof. In three steps:Step 1. The best-reply bBR of the small players, which is single-valued and defined by the fixed point
conditionbBR.a; �/ D arg max
b0�h.a�; b0; bBR.a; �//C .1 � �/h.a�; b0; bBR.a; �//;
is increasing in a and �.For each .a; �/ 2 A� Œ0; 1�, any pure action b 2 B which is a best-reply for the small players to .a; �/
must satisfy the first-order condition
.�h2.a�; b; b/C .1 � �/h2.a; b; b//. Ob � b/ � 0 8 Ob 2 B:
Since h22 C h23 < 0 along the diagonal of the small players’ actions, for each fixed .a; �/ the functionb 7! �h2.a
�; b; b/C .1� �/h2.a; b; b/ is strictly decreasing, hence the best-reply of the small players isunique for all .a; �/, as claimed.
To see that bBR is increasing, let a0 � a, �0 � � and suppose, towards a contradiction, that b0 DbBR.a0; �0/ < bBR.a; �/ D b. By the first-order conditions,
.�h2.a�; b; b/C .1 � �/h2.a; b; b//.b0 � b„ƒ‚…
<0
/ � 0
and.�0h2.a�; b0; b0/C .1 � �0/h2.a; b0; b0//.b � b0„ƒ‚…
>0
/ � 0;
hence
�h2.a�; b; b/C .1 � �/h2.a; b; b/ � 0 � �0h2.a�; b0; b0/C .1 � �0/h2.a; b0; b0/;
and this is a contradiction since we have assumed h12 � 0 and h22C h23 < 0 along the diagonal of smallplayers’ actions.
Step 2. For each .b; �/ 2 B�R define BR.b; �/defD arg maxa0 g.a
0; b/C��.a0/.44 Then, for all .a; b; �/and .a0; b0; �0/ 2 A � B �R with a 2 BR.b; �/ and a0 2 BR.b0; �0/,
Œb0 � b; �0 � � and �0 � 0� H) a0 � a:If a 2 BR.b; �/ and a0 2 BR.b0; �0/ then the first-order conditions imply
.g1.a; b/C ��0.a// .a0 � a/ � 0 and .g1.a0; b0/C �0�0.a0//.a � a0/ � 0:
Towards a contradiction, suppose b0 � b, �0 � maxf0; �g and a0 < a. Then, the inequalities above imply
g1.a; b/C ��0.a/ � 0 � g1.a0; b0/C �0�0.a0/:But this is a contradiction, since we have g1.a0; b0/ > g1.a; b/ by g11 < 0 and g12 � 0, and also�0�0.a0/ � �0�0.a/ � ��0.a/ by �00 � 0, �0 > 0 and �0 � 0.
Step 3. If .a; b/ 2M.�; �/, .a0; b0/ 2M.�0; �0/, �0 � � and �0 � maxf0; �g then a0 � a.Suppose not, i.e. .a; b/ 2 M.�; �/, .a0; b0/ 2 M.�0; �0/, �0 � �, �0 � maxf0; �g and a0 < a. Then,
we must have b0 � b by Step 1 and b0 > b by Step 2, a contradiction. Therefore a0 � a, as claimed. �
44Note that BR may not be single-valued when � < 0.
70
We are now ready to prove Lemma 2, the continuity lemma used in the proof of Theorem 10.
Proof of Lemma 2. Fix an arbitrary constantM > 0. Consider the setˆ of all tuples .a; b; �/ 2 A�B�R
satisfying
a 2 arg maxa02A
g.a0; b/C ��.a0/; b 2 arg maxb02B
h.a; b0; b/; g.a; b/C ��.a/ � g.aN ; bN /C "; (72)
and � �M .We claim that for all .a; b; �/ 2 ˆ we have � > 0. Otherwise, if � � 0 for some .a; b; �/ 2 ˆ, then
a � aN by Lemma F.1 and therefore b � bN , since the small players’ best reply is increasing in the largeplayer’s action, as shown in Step 1 of the proof of Lemma F.1. But since .aN ; bN / is a Nash equilibriumand g2 � 0, it follows that g.a; b/C ��.a/ � g.a; b/ � g.a; bN / � g.aN ; bN / < g.aN ; bN /C ", andthis is a contradiction since we have assumed that .a; b; �/ satisfy (72).
Thus, ˆ is a compact set and the continuous function .a; b; �/ 7! � achieves its minimum, �0, onˆ. Moreover, we must have �0 > 0 by the argument in the previous paragraph. It follows that � �ı
defDmin fM; �0g for any .a; b; �/ satisfying conditions (72). �
Turning to the proof of Proposition 9, since it is is similar to the proof of Proposition 4, we only providea sketch.
Proof of Proposition 9 (sketch). As in Appendix C.6, we use a first-order condition to characterize theaction profile .a; b/ 2 M.�; .V .� C ��.a// � V.�//=r/ for each .�; V / 2 .0; 1/ � C inc.Œ0; 1�/. Toexpress this condition, we define a function G W A � B � Œ0; 1� � C inc.Œ0; 1�/! R2 as follows:
G.a; b; �; V /defD�g1.a; b/C�0.a/
�V.�C��.a//�V.�/�=r; � h2.a�; b; b/C .1��/ h2.a; b; b/�: (73)
Thus, the first-order necessary and sufficient condition for .a; b/ 2M.�; .V .�C��.a//�V.�//=r/ canbe stated as:
G.a; b; �; V / � . Oa � a; Ob � b/ � 0 8. Oa; Ob/ 2 A � B:Next, using an argument very similar to the proof of Lemma C.11, we can show:
.A/ 9L > 0 such that 8.a; b; �/ 2 A � B 2 .0; 1/ and V , OV 2 C inc.Œ0; 1�/,
jG.a; b; �; OV / �G.a; b; �; V /j � Ld1. OV ; V /;
where d1 is the supremum distance on C inc.Œ0; 1�/;
.B/ 9M > 0 such that 8.�; V / 2 .0; 1/ � C inc.Œ0; 1�/ and .a; b/, . Oa; Ob/ 2 A � B ,�G. Oa; Ob; �; V / �G.a; b; �; V /� � . Oa � a; Ob � b/ � �M .j Oa � aj2 C j Ob � bj2/:
Finally, with conditions .A/ and .B/ above in place, we can follow the steps of the proof of Proposition 4(with the mapping G replacing F ) and prove the desired result. �
Next, towards the proof of Theorem 11, we first show that the optimality equation has a boundedincreasing solution.
71
Proposition F.1. There exists a bounded increasing continuous function U W .0; 1/! R which solves theoptimality equation (36) on .0; 1/.
The proof of Proposition F.1 relies on a series of lemmas.
Lemma F.2. For every " > 0 and every compact set K � C inc.Œ"; 1�/ there exists c > 0 such that for all.�; V / 2 Œ"; 1 � "� �K,
��.a.�; V // � c:
Proof. By Proposition 9 the function .�; V / 7! ��.a.�; V // is continuous on the domain Œ"; 1 � "� �C inc.Œ"; 1�/, and hence it achieves its minimum on the compact set Œ"; 1 � "� � K at some point .�0; V0/.Thus, letting c defD��0.a.�0; V0// yields the desired result. Indeed, we must have c � 0 since � is in-creasing. Moreover, c could be zero only if a.�0; V0/ D a�, in which case �V0.�0/ D 0 and hence a�would have to be part of a static Nash equilibrium of the complete information game, which is ruled outby assumption (c). �
For ease of notation, for each continuous and bounded function V W .0; 1/ ! R and � 2 .0; 1/ letH.�; V / denote the right-hand side of the optimality equation, i.e.
H.�; V /defD rg
�a.�; V /; b.�; V /
�C ��a.�; V /��V.�/ � rV .�/��.a.�; V //��.a.�; V //
:
Next, for each ˛ 2 Œg; Ng� and " > 0 consider the following initial value problem for a suitably modifiedversion of the optimality equation:
Problem IVP."; ˛/. Find a real-valued, continuous function U defined on an interval Œ�˛; 1�, with �˛ <1 � ", such that45
U 0.�/ D maxf0;H.�; U /g 8� 2 Œ�˛; 1 � "/; (74)
U.�/ D ˛ 8� 2 Œ1 � "; 1�:
With this definition in place we have:
Lemma F.3. For every " > 0 and ˛ 2 Œg; Ng� a unique solution of the IVP."; ˛/ exists on an interval Œ�˛; 1�,with �˛ < 1 � ". Moreover, if for some ˛0 2 Œg; Ng� a unique solution of the IVP."; ˛0/ exists on Œ�0; 1�,then for every ˛ in a neighborhood of ˛0 a unique solution U";˛ of the IVP."; ˛/ exists on the same intervalŒ�0; 1� and ˛ 7! U";˛ is a continuous function under the supremum metric.
Proof. We will apply Theorems 2.2 and 2.3 from Hale and Lunel (1993, pp. 43–44), which provide suffi-cient conditions under which an initial value problem for a retarded functional differential equation locallyadmits a unique solution, which is continuous in initial conditions.
45 Here our convention is that U 0.�/ is the left-derivative of U at �. But since the right-hand side is continuous in �, a solutioncan fail to be differentiable only at � D 1 � ", at which point the left-derivative can be different from the right-derivative, whichis identically zero given the initial condition.
72
By Proposition 9 and the fact that ��.a.�; V // > 0 for all .�; V / 2 Œ"; 1 � "� � C inc.Œ"; 1�/ thefunction maxf0;H g is continuous on Œ"; 1� "��C inc.Œ"; 1�/. Moreover, by Proposition 9 and Lemma F.2,for every compact set K � C inc.Œ"; 1�/ and every � 2 Œ"; 1 � "� the function V 7! maxf0;H.�; V /g isLipschitz continuous on K with a Lipschitz constant which is uniform in � 2 Œ"; 1 � "�. We have thusshown that maxf0;H g, the right-hand side of the retarded functional differential equation (74), satisfies allthe conditions of Theorems 2.2 and 2.3 from Hale and Lunel (1993, pg. 43-44), and therefore the IVP."; ˛/admits a local solution which is unique and continuous in the initial condition ˛. �
Next, we show that for a suitable choice of the initial condition ˛, the unique solution to the IVP."; ˛/exists on the entire interval Œ"; 1�, takes values in the set Œg; Ng� and solves the optimality equation onŒ"; 1 � "�.
Lemma F.4. For every " > 0 there exists a continuous increasing function U W Œ"; 1�! Œg; Ng� that solvesthe optimality equation (36) on Œ"; 1 � "� and is constant on Œ1 � "; 1�.
Proof. Fix " > 0. For each ˛ 2 Œg; Ng� write U˛ to designate the unique solution of the IVP."; ˛/, and letI˛ � .0; 1� be its maximal interval of existence. Consider the following disjoint sets:
J1 Dn˛ 2 Œg; Ng� W U˛.�/ � g for some � 2 I˛ \ . "
2; 1 � "�
oand
J2 D
8̂<̂:˛ 2 Œg; Ng� W
I˛ � . "2 ; 1�;U˛.�/ > g for all � 2 . "
2; 1�;
H.�; U˛/ < 0 for some � 2 . "2; 1 � "�
9>=>; :We will show that J1 and J2 are nonempty open sets relative to Œg; Ng�. Since J1 and J2 are disjoint wewill conclude that there exists some ˛ 2 Œg; Ng� n .J1 [ J2/. For this ˛, the solution to the IVP."; ˛/, U˛, iswell-defined on . "
2; 1�, takes values in Œg; Ng� and satisfies H.�;U˛/ � 0 for all � 2 . "
2; 1 � "�. The latter
implies that U˛ solves the optimality equation on . "2; 1 � "�.
Let us show that J1 and J2 are open relative to Œg; Ng�. Clearly, J2 is open in Œg; Ng� because ˛ 7! U˛
and ˛ 7! H.�; U˛/ are continuous functions by Proposition 9 and Lemma F.3. Also J1 is open in Œg; Ng�,because for each ˛ 2 J1 we have U˛.�0/ � g for some �0 2 I˛ \ ."; 1 � "�, and therefore
H.�0; U˛/ Dr
> 0‚ …„ ƒ�g.a.�0; U˛/; b.�0; U˛// � U˛.�0/
�C��a.�0; U˛/� � 0‚ …„ ƒ�U˛.�0/
��.a.�0; U˛//��.a.�0; U˛//„ ƒ‚ …> 0
> 0;
which implies U 0̨ .�0/ > 0 by (74). Hence, there exists �1 2 . "2 ; �0/ with U˛.�1/ < g, and so, by thecontinuity of Q̨ 7! U Q̨ we must have Q̨ 2 J1 for all Q̨ in a neighborhood of ˛, as was to be shown.
It remains to show that J1 and J2 are non-empty sets. It is clear that g 2 J1. As for J2, note that theconstant function NU D Ng solves the equation with initial condition Ng. To see this, observe that � NU D 0
identically and that g.a.�; NU/; b.�; NU// < Ng, so H.�; NU/ < 0, hence NU 0.�/ D 0 D max f0;H.�; NU/g.Therefore, Ng 2 J2. �
73
Lemma F.5. For every V 2 C inc.Œ0; 1�/ and every � and �0 2 .0; 1/ with �0 < �,
��0.a.�0; V // < ��.a.�; V //C � � �0:
Proof. Fix V 2 C inc.Œ0; 1�/ and �, �0 2 .0; 1/ with �0 < �. Suppose, towards a contradiction, that�0C��0.a.�0; V // � �C��.a.�; V //. Since � is strictly increasing, the function . O�; Oa/ 7! O�C� O�. Oa/is strictly increasing in O� and strictly decreasing in Oa, and therefore we must have a.�0; V / < a.�; V /. Itfollows from Lemma F.1 that �V.�0/ < �V.�/. But since V is increasing we have V.�0/ � V.�/ andV.�0 C��0.a.�0; V /// � V.� C��.a.�; V ///, and therefore �V.�0/ � �V.�/, a contradiction. �
Lemma F.6. For every " > 0 there exists K > 0 such that for every continuous, increasing functionV W Œ"; 1� ! Œg; Ng� and every �, �0 2 Œ"; 1 � "� with �0 < �, if a.�; V / ¤ minA and a.�0; V / ¤ minAthen
rg.a.�0; V /; b.�0; V //C �.a.�0; V //�V.�0/� rg.a.�; V /; b.�; V //C �.a.�; V //�V.�/ �K .� � �0/:
Proof. Let " > 0 be fixed. The proof proceeds in four steps:Step 1. There exists a constant Ka > 0 such that for all V 2 C inc.Œ0; 1�/ and all �, �0 2 Œ"; 1� "� with
�0 < �,a.�0; V / � a.�; V / �Ka.� � �0/:
By the definition of ��.a/ and Lemma F.5, for all " � � < �0 � 1 � ",
�.a.�; V // � �.a.�0; V // D ��0
.a.�0; V //��0.a.�0; V //�0.1 � �0/ � �
�.a.�; V //��.a.�; V //�.1 � �/
D
� N�‚ …„ ƒ�.1 � �/��0.a.�0; V //.
� ���0 by Lemma F.5‚ …„ ƒ��0.a.�0; V // ���.a.�; V ///
�0.1 � �0/�.1 � �/„ ƒ‚ …� "4
C
C
� N�‚ …„ ƒ��.a.�; V //��.a.�; V //.
� ���0‚ …„ ƒ�.1 � �/ � �0.1 � �0//
�0.1 � �0/�.1 � �/„ ƒ‚ …� "4
C
�1‚ …„ ƒ�.1 � �/��.a.�; V //.
� �.a.�0;V //��.a.�;V //‚ …„ ƒ��0
.a.�0; V // � ��.a.�; V ///�0.1 � �0/�.1 � �/„ ƒ‚ …
� "4
� 2 N�"�4.� � �0/ � "�4.�.a.�; V // � �.a.�0; V ///;
where N� defDmaxa2A �.a/. Hence,
�.a.�; V // � �.a.�0; V // � 2 N�.1C "4/�1.� � �0/;
74
and therefore, by the mean value theorem,
a.�; V / � a.�0; V / � 2 N� .� � �0/mina �0.a/.1C "4/
;
which proves the desired inequality with KadefD 2 N�
mina �0.a/.1C"4/.
Step 2. There exists a constant Kb > 0 such that for all V 2 C inc.Œ0; 1�/ and all �, �0 2 Œ"; 1� "� with�0 < �,
b.�0; V / � b.�; V / �Kb.� � �0/:
First, note that the best-reply of the small players, .a; �/ 7! bBR.a; �/, is increasing and Lipschitzcontinuous.46 Therefore, there exist constants c1 and c2 > 0 such that for all �, �0 2 .0; 1/ with �0 � �and all a, a0 2 A with a0 � a,
0 � bBR.a; �/ � bBR.a0; �0/ � c1.a � a0/C c2.� � �0/:
Hence, for all V 2 C inc.Œ0; 1�/ and all �, �0 2 Œ"; 1 � "� with �0 < �, if a.�; V / � a.�0; V / � 0 then
b.�; V / � b.�0; V / � c1.a.�; V / � a.�0; V //C c2.� � �0/� c1Ka.� � �0/C c2.� � �0/D .c1Ka C c2/.� � �0/;
where Ka is the constant from Step 1. Moreover, if a.�; V / � a.�0; V / < 0 then
b.�; V / � b.�0; V / � bBR.a.�0; V /; �/ � bBR.a.�0; V /; �0/� c2.� � �0/ � .c1Ka C c2/.� � �0/;
which thus proves the desired inequality with KbdefD c1Ka C c2.
Step 3. For each .a; b/ 2 A�B and each � � 0, if a D arg maxa0 g.a0; b/C��.a0/ 2 .minA;maxA/
then � D �g1.a; b/=�0.a/.This follows directly from the first-order condition when a 2 .minA;maxA/.
Step 4. There exists K > 0 such that for every � and �0 2 Œ"; 1 � "� with �0 < � and every V 2C inc.Œ"; 1�/, if .a; b/ D .a.�; V /; b.�; V // with a ¤ minA, and .a0; b0/ D .a.�0; V /; b.�0; V // witha0 ¤ minA, then
rg.a; b/C �.a/�V.�/ � rg.a0; b0/C �.a0/�V.�0/ � K.� � �0/:
First note that a ¤ a� D maxA, otherwise we would have�V.�/ D 0, and therefore .a�; b/would bea static Nash equilibrium of the complete information game, which is ruled out by condition (d). Likewise,
46The monotonicity of bBR is Step 2 from the proof of Lemma F.1. The Lipschitz continuity is a straightforward implicationof the first-order condition and the facts that h12.a; b; b/ is bounded from above and .h22 C h23/.a; b; b/ is bounded away fromzero.
75
a0 ¤ a�. Since by assumption we also have a ¤ minA and a0 ¤ minA, we can apply Step 3 to concludethat
�V.�/=r D �.a; b/ and �V.�0/=r D �.a0; b0/;where the function � W A � B ! .0;1/ is such that
�. Oa; Ob/ D �g1. Oa; Ob/=�0. Oa/ 8. Oa; Ob/ 2 A � B;
so it satisfies �1 > 0 and �2 � 0, since g11 < 0, g12 � 0, �0 > 0 and �00 � 0.Since .a0; b0/ 2M.�0; �.a0; b0// we have
g.a; b0/C �.a/�.a0; b0/ � g.a0; b0/C �.a0/�.a0; b0/;
and therefore,
g.a; b/C �.a/�.a; b/ � �g.a0; b0/C �.a0/�.a0; b0/�
D g.a; b/C �.a/�.a; b/ � �g.a; b0/C �.a/�.a0; b0/�
C g.a; b0/C �.a/�.a0; b0/ � �g.a0; b0/C �.a0/�.a0; b0/�
� g.a; b/C �.a/�.a; b/ � �g.a; b0/C �.a/�.a0; b0/�
D g.a; b/C �.a/�.a; b/ � �g.a; b0/C �.a/�.a; b0/�
C �.a/.�.a; b0/ � �.a0; b0//:
Hence, for some . Na; Nb/ we have
g.a; b/C �.a/�.a; b/ � �g.a0; b0/C �.a0/�.a0; b0/�
� .g2.a; Nb/C �.a/�2.a; Nb// .b � b0/ C �.a/�1. Na; b0/ .a � a0/� c3 .b � b0/C c4 .a � a0/;
where c3defDmax
. Oa; Ob/ g2. Oa; Ob/ C �. Oa/�2. Oa; Ob/ > 0 and c4defDmax
. Oa; Ob/ �. Oa/�1. Oa; Ob/ > 0. It followsfrom Steps 1 and 2 that
g.a; b/C �.a/�.a; b/ � g.a0; b0/C �.a0/�.a0; b0/ C .c3Kb C c4Ka/.� � �0/;
which is the desired result with K defD r.c3Kb C c4Ka/. �
Lemma F.7. For every " > 0 there exists R > 0 such that for every continuous, increasing functionU W Œ"; 1�! Œg; Ng� that solves the optimality equation (36) on Œ"; 1 � "� and is constant on Œ1 � "; 1�,
jU 0.�/j � R 8� 2 Œ"; 1 � "�:
Proof. Fix " > 0. We begin with the definition of the upper bound R. Since .a�; b/ … M.�; 0/ for all.b; �/ 2 B� Œ0; 1�, there exists 0 < � < a��minA such that for all .a; b; �; �/ 2 A�B� Œ0; 1�� Œ0; Ng�g�with .a; b/ 2M.�; �/,
a � a� � � H) � � �: (75)
76
Let K > 0 designate the constant from Lemma F.6 and define � defDmina �.a/. Next choose 0 < ı <��2K
small enough that for all � 2 Œ"; 1 � "� and a 2 A,
��.a/ � ı H) a � a� � �; (76)
Let C defD.r C N�/. Ng � g/=� and choose R > 2C=ı large enough that
��
2 N�.log ı � log.C
R// > Ng � g: (77)
Now suppose the thesis of the lemma were false, i.e. max�2Œ";1�"� U 0.�/ > R for some continuous,increasing function U W Œ"; 1� ! Œg; Ng� which solves the optimality equation on Œ"; 1 � "� and is constanton Œ1 � "; 1�. Let �0 2 arg max�2Œ";1�"� U 0.�/. By the optimality equation,
R < U 0.�0/ �.r C N�/. Ng � g/���0.a.�0; U //
D C
��0.a.�0; U //;
which yields
��0.a.�0; U // <C
R<ı
2; (78)
hence a.�0; U / > a� � � by (76), which implies
�U.�0/ � �; (79)
by (75).To conclude the proof it is enough to show that
U 0.�/ � ��
2 N�.CRC �0 � �/
8� 2 Œ�0 � ı; �0�; (80)
for this implies
U.�0/ � U.�0 � ı/ �Z �0
�0�ı�� d�
2 N�.CRC �0 � �/
D ��
2 N�.log.C
RC ı/ � log.
C
R// > Ng � g;
by (77), and this is a contradiction since U takes values in Œg; Ng�.To prove inequality (80) first recall that U solves the optimality equation on Œ"; 1 � "�, that is,
U 0.�/ D rg.a.�; U /; b.�; U //C �.a.�; U //�U.�/ � rU.�/��.a.�; U //��.a.�; U //
; 8� 2 Œ"; 1 � "�: (81)
Hence, by Lemmas F.5 and F.6 and inequality (78), for all � 2 Œ�0 � ı; �0�,
U 0.�/ �
� ���0.a.�0;U //U0.�0/; by (81)‚ …„ ƒ
rg.a.�0; U /; b.�0; U //C �.a.�0; U //�U.�0/ � rU.�0/ �� Kı < ��=2‚ …„ ƒK��0 � �
�N� � .��0.a.�0; U //„ ƒ‚ …
� C=RC�0 � � /
� �U 0.�0/��0.a.�0; U // � ��2
N� .CRC �0 � �/
:
77
Thus, to prove inequality (80) and conclude the proof of the lemma, it suffices to show that
U 0.�0/��0.a.�0; U // � �:
Indeed, by the mean value theorem and the fact that U is constant on Œ1 � "; 1�, we must have �U.�0/ �U 0. N�/��0.a.�0; U // for some N� 2 Œ�0; 1�"�. Hence,�U.�0/ � U 0.�0/��0.a.�0; U // sinceU 0.�0/ Dmax fU 0.�/ W � 2 Œ"; 1 � "�g, and therefore U 0.�0/��0.a.�0; U /// � � follows from (79). �
We are now ready to prove Proposition F.1.
Proof of Proposition F.1. Our method of proof is to construct a solution U W .0; 1/ ! Œg; Ng� as a limitof a sequence of solutions on expanding closed subintervals of .0; 1/. Using Lemma F.4, for each n � 1there exists a continuous, increasing function Un W Œ1n ; 1� ! Œg; Ng� that solves the optimality equation onŒ1n; 1 � 1
n�. Since for m � n the restriction of Um to Œ1
n; 1� solves the optimality equation on Œ1
n; 1 � 1
n�,
by Lemma F.7 the derivative of Um is uniformly bounded for m � n, and so the sequence .Um/m�nis uniformly bounded and equicontinuous over the domain Œ1
n; 1 � 1
n�. By the Arzelà-Ascoli Theorem,
for every n there exists a subsequence of .Um/m�n that converges uniformly on Œ1n; 1 � 1
n�. Hence, by
a standard diagonalization argument, we can find a subsequence .Unk/k�1 that converges pointwise to a
continuous, increasing functionU W .0; 1/! Œg; Ng� such that the convergence is uniform on every compactsubset of .0; 1/.
It remains to show that U solves the equation on .0; 1/. If we show that U 0nk.�/! H.�;U / uniformly
on any closed subinterval Œ�0; �1� � .0; 1/, it will then follow that U is differentiable and U 0.�/ DH.�;U /. First, note that from any � 2 Œ�0; �1� the posterior cannot jump above �2 D �1C�1.1��1/ N���� .Since Proposition 9 and Lemma F.2 imply that V 7! H.�; V / is Lipschitz continuous on the compact setfU jŒ�0; �2�g [ fUnk
jŒ�0; �2� W k � 1g with a Lipschitz constant that is uniform in � 2 Œ�0; �1�, and thesequence .Unk
/k�1 converges to U uniformly on Œ�0; �2�, it follows that U 0nk.�/ D H.�;Unk
/ convergesto H.�;U / uniformly on Œ�0; �1�, as required. �
The following lemma concerns boundary conditions.
Lemma F.8. Every bounded increasing solution of the optimality equation U W .0; 1/ ! R satisfies thefollowing conditions at p 2 f0; 1g:
lim�!p U.�/ D g.M.p; 0// and lim
�!p �.1 � �/U0.�/ D 0:
Proof. Let us show that lim�!0 U.�/ D g.M.0; 0//. First, by continuity, as � ! 0 we have �U.�/ DU.� C��.a.�; U ///� U.�/! 0, and hence .a.�; U /; b.�; U //!M.0; 0/ D .aN ; bN /. Therefore, iflim�!0 U.�/ < g.aN ; bN / then the optimality equation implies
lim�!0 �U
0.�/ D r.g.aN ; bN / � lim�!0 U.�//�.a�/ � �.aN / > 0:
Thus, for 0 < c < r.g.aN ; bN / � lim�!0 U.�//=.�.a�/ � �.aN // we must have U 0.�/ > c=� forall � sufficiently close to 0. But then U cannot be bounded, since the anti-derivative of c=�, whichis c log �, tends to �1 as � ! 0, and this is a contradiction. Thus, we must have lim�!0 U.�/ �g.aN ; bN /. Moreover, if lim�!0 U.�/ > g.aN ; bN / then a similar argument can be used to show that
78
lim�!0 �U 0.�/ < 0, which is impossible sinceU is increasing. We have thus shown that lim�!0 U.�/ Dg.M.0; 0//. The proof for the � ! 1 case is analogous.
Finally, the condition lim�!p �.1 � �/U 0.�/ D 0 follows from the optimality equation and theboundary condition lim�!p U.�/ D g.M.p; 0//. �
Proposition F.2. The optimality equation (36) has a unique bounded increasing solution on .0; 1/.
Proof. By Proposition F.1 the optimality equation has at least one bounded increasing solution. SupposeUand V are two such solutions. Assuming thatU.�/ > V.�/ for some � 2 .0; 1/, let �0 2 .0; 1/ be the pointwhere the difference U � V is maximized, which is well-defined because lim�!p U.�/ D lim�!p V.�/for p 2 f0; 1g by Lemma F.8. Thus, we have U.�0/ � V.�0/ > 0 and U 0.�0/ � V 0.�0/ D 0.
Let
�U.�0/defD U.�0 C��0.a.�0; U /// � U.�0/;
�V.�0/defD V.�0 C��0.a.�0; V /// � V.�0/:
We claim that �U.�0/ > �V.�0/. Otherwise, if �U.�0/ � �V.�0/, then a.�0; U / � a.�0; V / byLemma F.1, and hence b.�0; U / � b.�0; V / by Step 1 of the proof of Lemma F.1. Therefore,
rg.a.�0; U /; b.�0; U //C �.a.�0; U //�U.�0/ � rg.a.�0; U /; b.�0; V //C �.a.�0; U //�V.�0/� rg.a.�0; V /; b.�0; V //C �.a.�0; V //�V.�0/;
where the first inequality uses g2 � 0 and the second inequality follows from the fact that .a.�0; V /; b.�0; V // 2M.�0; �V.�0/=r/. Then, the optimality equation implies
U 0.�0/ D rg.a.�0; U /; b.�0; U //C �.a.�0; U //�U.�0/ � rU.�0/�0.1 � �0/.�.a�/ � �.a.�0; U ///
<rg.a.�0; V /; b.�0; V //C �.a.�0; V //�V.�0/ � rV .�0/
�0.1 � �0/.�.a�/ � �.a.�0; V /// D V 0.�0/;
and this is a contradiction. Thus, �U.�0/ > �V.�0/, as claimed.It follows from�U.�0/ > �V.�0/ that a.�0; U / � a.�0; V /, by Lemma F.1. Hence,��0.a.�0; U // �
��0.a.�0; V //, and since V is increasing,
U.�0 C��0.a.�0; U /// � V.�0 C��0.a.�0; U /// �U.�0 C��0.a.�0; U /// � V.�0 C��0.a.�0; V /// > U.�0/ � V.�0/;
where the last inequality follows from �U.�0/ > �V.�0/. But this is a contradiction, for we picked �0to be the point where the difference U � V is maximized. Thus, the optimality equation has a uniquebounded increasing solution. �
We are now ready to prove Theorem 11.
Proof of Theorem 11. By Proposition F.2, the optimality equation has a unique bounded solution U W.0; 1/ ! R. By Lemma F.8 such solution satisfies the boundary conditions. The proof proceeds in twosteps. In Step 1 we show that, given any prior �0 2 .0; 1/ on the behavioral type, there is no sequential
79
equilibrium that yields a payoff different fromU.�0/ to the normal type. In Step 2 we show that the Marko-vian strategy profile .am
t ; bmt /
defD.a.�t�; U /; b.�t�; U // is the unique sequential equilibrium yielding thepayoff U.�0/ to the normal type.
Step 1. Fix a prior �0 2 .0; 1/ and a sequential equilibrium .at ; Nbt ; �t /t�0. By Theorem 9 the beliefprocess .�t /t�0 solves
d�t D ���t�.at /��t�.at / dt C ��t�.at / dNt ;
and the process .Wt /t�0 of continuation values of the normal type solves
dWt D r.Wt� � g.at ; Nbt / � �t�.at // dt C r�t dNt ; (82)
for some predictable process .�t /t�0 such that
.at ; Nbt / 2M.�t�; �t / almost everywhere: (83)
Thus, the process UtdefDU.�t / satisfies
dUt D ���t�.at /��t�.at /U 0.�t�/ dt C �Ut dNt ; (84)
where�Ut
defDU.�t� C��t�.at // � U.�t�/:We will now demonstrate that D0
defDW0 � U.�0/ D 0 by showing that if D0 ¤ 0 then the processDt
defDWt �U.�t /must eventually grow arbitrarily large with positive probability, which is a contradictionsince U and W are both bounded. Without loss of generality, we assume D0 > 0. It follows from (82),(84) and the optimality equation (36) that the process .Dt /t�0 jumps by
r�t ��Ut
when a Poisson event arrives, and has drift given by
rDt� C f .amt ; b
mt ; �U
mt ; �t�/ � f .at ; Nbt ; �t ; �t�/;
where�Um
tdefDU.�t� C��t�.am
t // � U.�t�/;and for each .a; b; �; �/ 2 A � B � Œ0; Ng � g� � Œ0; 1�,
f .a; b; �; �/defD rg.a; b/C r��.a/C �.1 � �/U 0.�/�.a/:
Claim. There exists " > 0 such that, so long asDt � D0=2, either the drift ofDt is greater than or equalto rD0=4 or Dt jumps up by more than " upon the arrival of a Poisson event.
To prove this claim, note that by Lemma F.1, if r�t � �Umt � 0 then am
t � at , and hence bmt � bt
since the small players’ best reply in M is increasing in the large player’s action, as shown in Step 1 of the
80
proof of Lemma F.1. Therefore, r�t ��Umt � 0 implies
f .amt ; b
mt ; �U
mt ; �t�/ � f .at ; Nbt ; �t ; �t�/ D
� rg.at ;bmt /C�Um
t �.at /‚ …„ ƒrg.am
t ; bmt /C�Um
t �.amt /�rg.at ; Nbt / � r�t�.at /
C �t�.1 � �t�/U 0.�t�/.�.amt / � �.at /„ ƒ‚ …� 0 by �0 > 0
/
� rg.at ; bmt / � rg.at ; Nbt /„ ƒ‚ …
� 0 by g2 � 0C.�Um
t � r�t„ ƒ‚ …� 0
/�.at /:
Thus,r�t ��Um
t � 0 H) f .amt ; b
mt ; �U
mt ; �t�/ � f .at ; Nbt ; �t ; �t�/ � 0:
By a continuity/compactness argument similar to the one used in the proof of Lemma C.8, there exists" > 0 such that for all t and after all public histories,
r�t ��Umt � " H) f .am
t ; bmt ; �U
mt ; �t�/ � f .at ; Nbt ; �t ; �t�/ � �rD0=4;
hence, so long as Dt � D0=2,
r�t ��Umt � " H) drift of Dt is greater than or equal to rD0=4:
Thus, so long as Dt � D0=2, if the drift of Dt is less than rD0=4, then r�t � �Umt > ", which implies
that at � amt by Lemma F.1, and hence that �Ut � �Um
t since U is increasing; therefore, r�t � �Ut ,the jump in Dt upon the arrival of a Poisson event, must be greater than or equal to r�t � �Um
t > ", asclaimed.
Since we have assumed that D0 > 0, the claim above readily implies that with positive probabilitythe process Dt D Wt � U.�t / must eventually grow arbitrarily large, which is impossible since .Wt /t�0and .U.�t //t�0 are bounded processes. We have thus shown that no sequential equilibrium can yield apayoff greater than U.�0/. A similar argument proves that payoffs below U.�0/ also cannot be achievedin sequential equilibria.
Step 2. First, let us show that the Markovian strategy profile .a.�t ; U /; b.�t ; U //t�0 is a sequentialequilibrium profile yielding the normal type a payoff of U.�0/. Let .�t /t�0 be the solution of
d�t D ���t�.a.�t�; U //��t�.a.�t�; U // dt C ��t�.a.�t�; U // dNt ;
with initial condition �0. Thus, using the optimality equation,
dUmt D .rUm
t� � rg.amt ; b
mt / � �.am
t /�Umt / dt C�Um
t dNt ;
where .amt ; b
mt /
defD.a.�t�; U /; b.�t�; U //,Umt
defDU.�t / and�Umt
defDU.�t�C��t�.amt //�U.�t�/. Since
.Umt /t�0 is bounded and .am
t ; bmt / 2 M.�t�; �Um
t /, Theorem 9 implies that .amt ; b
mt /t�0 is a sequential
equilibrium profile in which the normal type receives a payoff of U.�0/.It remains to show that .a.�t ; U /; b.�t ; U //t�0 is the unique sequential equilibrium. Indeed, if
.at ; Nbt /t�0 is an arbitrary sequential equilibrium, then the associated belief/continuation value pair .�t ; Wt /must stay in the graph of U , by Step 1. Then, by (82) and (84), we must have .at ; Nbt / 2M.�t�; �t / a.e.,where r�t D U.�t�C��t�.at //�U.�t�/. Therefore, by Proposition 9, .at ; Nbt / D .a.�t�; U /; b.�t�; U //a.e., as was to be shown. �
81
References
Abreu, Dilip, Paul Milgrom, and David Pearce (1991), “Information and timing in repeated partnerships.”Econometrica, 59, 1713–1733.
Abreu, Dilip, David Pearce, and Ennio Stacchetti (1990), “Toward a theory of discounted repeated gameswith imperfect monitoring.” Econometrica, 58, 1041–1063.
Aubin, J.P. and A. Cellina (1984), Differential Inclusions. Springer.
Barro, Robert J. (1986), “Reputation in a model of monetary policy with incomplete information.” Journalof Monetary Economics, 17, 3–20.
Billingsley, Patrick (1999), Convergence of Probability Measures. John Wiley & Sons, Inc., second edition.
Board, Simon and Moritz Meyer-ter-Vehn (2010), “Reputation for quality.” mimeo., UCLA.
Bolton, Patrick and Christopher Harris (1999), “Strategic experimentation.” Econometrica, 67, 349–374.
Brémaud, Pierre (1981), Point Processes and Queues: Martingale Dynamics. Springer-Verlag, first edi-tion.
Celentani, M. and W. Pesendorfer (1996), “Reputation in dynamic games.” J. Econ. Theory, 70, 109–132.
Chari, Varadarajan V. and Patrick J. Kehoe (1993), “Sustainable plans and debt.” J. Econ. Theory, 61,230–261.
Cole, Harold L., James Dow, and William B. English (1995), “Default, settlement, and signalling: Lendingresumption in a reputational model of sovereign debt.” International Economic Review, 36, 365–85.
Crandall, Michael G., Hitoshi Ishii, and Pierre-Louis Lions (1992), “User’s guide to viscosity solutions ofsecond order differential equations.” Bulletin of the American Mathematical Society, 27, 1–67.
Cripps, Martin, George J. Mailath, and Larry Samuelson (2004), “Imperfect monitoring and impermanentreputations.” Econometrica, 72, 407–432.
Cukierman, Alex and Allan Meltzer (1986), “A theory of ambiguity, credibility and inflation under discre-tion and asymmetric information.” Econometrica, 54, 1099–1128.
de Coster, Collette and Patrick Habets (2006), Two-Point Boundary Value Problems: Lower and UpperSolutions, volume 205 of Mathematics in Science and Engineering. Elsevier, first edition.
Diamond, Douglas W. (1989), “Reputation acquisition in debt markets.” Journal of Political Economy, 97,828–62.
Ely, Jeffrey C., Drew Fudenberg, and David K. Levine (2008), “When is reputation bad?” Games andEconomic Behavior, 63, 498–526.
Ely, Jeffrey C. and Juuso Valimaki (2003), “Bad reputation.” The Quarterly Journal of Economics, 118,785–81.
82
Faingold, Eduardo (2008), “Building a reputation under frequent decisions.” mimeo., Yale University.
Faingold, Eduardo and Yuliy Sannikov (2010), “Reputation in continuous-time games, supplementarymaterial: public randomization.” mimeo., Yale University.
Fudenberg, Drew and David K. Levine (1992), “Maintaining a reputation when strategies are imperfectlyobserved.” Review of Economic Studies, 59, 561–579.
Fudenberg, Drew and David K. Levine (1994), “Efficiency and observability with long-run and short-runplayers.” Journal of Economic Theory, 62, 103–135.
Fudenberg, Drew and David K. Levine (2007), “Continuous-time models of repeated games with imperfectpublic monitoring.” Review of Economic Dynamics, 10, 173–192.
Fudenberg, Drew and David K. Levine (2009), “Repeated games with frequent signals.” Quarterly Journalof Economics, 124, 233–265.
Fudenberg, Drew, David K. Levine, and Eric Maskin (1994), “The folk theorem with imperfect publicinformation.” Econometrica, 62, 997–1039.
Harris, Christopher, Philip Reny, and Arthur Robson (1995), “The existence of subgame-perfect equilib-rium in continuous games with almost perfect information: a case for public randomization.” Econo-metrica, 63, 507–44.
Holmstrom, Bengt and Paul Migrom (1991), “Multitask principal-agent analyses: incentive contracts,ownership and job design.” Journal of Law, Economics and Organization, 7, 24–52.
Karatzas, Ioannis and Steven E. Shreve (1991), Brownian Motion and Stochastic Calculus. Springer, sec-ond edition.
Keller, Godfrey and Sven Rady (1999), “Optimal experimentation in a changing environment.” The Reviewof Economic Studies, 66, 475–507.
Klein, Benjamin and Keith B. Leffler (1981), “The role of market forces in assuring contractual perfor-mance.” Journal of Political Economy, 89, 615–41.
Kreps, David and Robert Wilson (1982), “Reputation and imperfect information.” Journal of EconomicTheory, 27, 253–279.
Kydland, F. and E. Prescott (1977), “Rules rather than discretion: the inconsistency of optimal plans.”Journal of Political Economy, 85, 473–492.
Liptser, Robert S. and Albert N. Shiryaev (1977), Statistics of Random Processes: General Theory, vol-ume I. New York: Springer-Verlag.
Liu, Qingmin and Andrzej Skrzypacz (2010), “Limited records and reputation.” mimeo., University ofPennsylvania.
83
Mailath, George J. and Larry Samuelson (2006), Repeated Games and Reputations: Long-run Relation-ships. Oxford University Press, first edition.
Maskin, Eric and Jean Tirole (2001), “Markov perfect equilibrium: I. Observable actions.” Journal ofEconomic Theory, 100, 191–219.
Milgrom, Paul and John Roberts (1982), “Predation, reputation and entry deterrence.” Journal of EconomicTheory.
Moscarini, Giuseppe and Lones Smith (2001), “The optimal level of experimentation.” Econometrica, 69,1629–1644.
Sannikov, Yuliy (2007), “Games with imperfectly observable actions in continuous time.” Econometrica,75, 1285–1329.
Sannikov, Yuliy (2008), “A continuous-time version of the principal-agent problem.” The Review of Eco-nomic Studies, 75, 957–984.
Sannikov, Yuliy and Andrzej Skrzypacz (2007), “Impossibility of collusion under imperfect monitoringwith flexible production.” American Economic Review, 97, 1794–1823.
Sannikov, Yuliy and Andrzej Skrzypacz (2010), “The role of information in repeated games with frequentactions.” Econometrica, 78, 847–882.
84