a three-player dynamic majoritarian bargaining game

Journal of Economic Theory 116 (2004) 294–322

A three-player dynamic majoritarianbargaining game

Anastassios Kalandrakis�

Department of Political Science, Yale University, P.O. Box 208301, New Haven, CT 06520-8301, USA

Received 24 September 2001; final version received 11 June 2003

Abstract

We analyze an infinitely repeated divide-the-dollar bargaining game with an endogenous

reversion point. In each period a new dollar is divided among three legislators according to the

proposal of a randomly recognized member—if a majority prefer so—or according to previous

period’s allocation otherwise. Although current existence theorems for Markovian equilibria

do not apply for this dynamic game, we fully characterize a Markov equilibrium. The

equilibrium is such that irrespective of the discount factor or the initial division of the dollar,

the proposer eventually extracts the whole dollar in all periods. We also show that proposal

strategies are weakly continuous in the status quo that equilibrium expected utility is not

quasi-concave, and the correspondence of voters’ acceptance set (the set of allocations weakly

preferred over the status quo) fails lower hemicontinuity.

r 2003 Elsevier Inc. All rights reserved.

JEL classification: C72; C73; C78; D72; D78; D79

Keywords: Dynamic games; Endogenous reversion point; Legislative bargaining; Markov perfect Nash

equilibrium; Stage undominated voting strategies; Uncovered set

1. Introduction

Sequential non-cooperative models of legislative bargaining in the tradition ofRomer and Rosenthal [24], Rubinstein [26], Baron and Ferejohn [4], and Banks andDuggan [1,2], have significantly deepened our understanding of the politics oflegislative decision-making. Yet, this literature invariably assumes that legislative

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0022-0531/$ - see front matter r 2003 Elsevier Inc. All rights reserved.

doi:10.1016/S0022-0531(03)00259-X

interaction ceases after a decision is reached. Consequently, the applicability of thesemodels is limited to situations when the reversion point is exogenously given andfixed over time, or to policy areas where legislation cannot be modified after its initialintroduction.

This is unfortunate since most legislatures have constitutional authority tolegislate anew in the bulk of policy jurisdictions. In such policy domains, legislationremains in effect after its promulgation only until or unless the legislature passes anew law. Thus it appears natural to study dynamic bargaining games where (a)policy decisions can be reached in any period, and (b) in the absence of agreementamong the bargaining parties in any given period the status quo policy prevails.

Two reasons account for the paucity of contributions in this area despite therelevance of the above assumptions. The obvious difficulty has to do with thecomplexity of these games. The endogeneity of the reversion point implies thatlegislative decisions in the present have an impact on both the immediate payoff aswell as the future stream of benefits to players. Hence, the strategic calculationsinvolved render the characterization of equilibrium points particularly challenging.

Preceding this obstacle is a more fundamental difficulty. Assuming a continuouspolicy space, these games belong in a class of stochastic games for which existence ofMarkovian equilibria is not guaranteed. Indeed, Harris et al. [16], provide anexample of a dynamic game where subgame perfect (and hence Markov) equilbria donot exist. Chakrabarti [9], where the reader can find a more thorough review of theliterature on stochastic games, establishes existence of Markov Equilibrium pointsfor a subclass of stochastic games. Chakrabarti’s result, though, requires a strongassumption regarding the continuity of transition probabilities in players’ actions1

that rules out the game considered in this paper.Previously, Markov equilibria for infinite-horizon dynamic bargaining games with

endogenous status quo have only been analyzed in [3]. Baron studies the sameinstitutions as in the present analysis and assumes a one-dimensional policy spacewith legislators that have single-peaked stage utilities. He characterizes anequilibrium where outcomes converge to the median’s ideal point from arbitraryinitial decision. Thus, Baron provides a dynamic median voter theorem [8] in anenvironment where the core defined on the basis of the stage preferences oflegislators, is non-empty.

In more than one dimensions, though, the core of the majority rule game isgenerically empty [22,25,27,28] and there are no obvious candidates for the supportof a steady-state distribution of equilibrium decisions in the corresponding dynamicgame (assuming a steady-state distribution exists). Furthermore, in the absence of acore the majority preference relation induces a cycle that encompasses the entirespace of alternatives [18,19] so that more interesting dynamics than those induced inone-dimension become possible. A flavor of such complex dynamics is obtained byBaron and Herron [5], who analyze a game with three players that have Euclidianpreferences over a two-dimensional space of legislative outcomes. The game proves

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1 Specifically, for infinite horizon games the result requires that transition probabilities are norm-

continuous in players’ actions.

A. Kalandrakis / Journal of Economic Theory 116 (2004) 294–322 295

analytically intractable and Baron and Herron provide numerical calculations ofMarkovian equilibria for a finite period\discrete policy space version of the game.

Like Baron and Herron [5], we analyze a three-player game. In each of an infinityof periods one of the three legislators is randomly recognized to make a proposal forthe allocation of a fixed renewable resource (a dollar) among the members of thelegislature. The proposed allocation is implemented if it receives a majority;otherwise, the resource is allocated as it was last period. Legislators’ utility is thediscounted sum of per period payoffs. Although coarse, this model combines amultidimensional policy space with the advantage of (relative) analytical tractability.Thus, our analysis serves as a first step in understanding an important family ofdynamic legislative bargaining models in multidimensional policy spaces.

Indeed, we are able to establish existence and fully characterize a Markovequilibrium such that players condition their behavior only on the status quo. Infact, we refine the equilibrium concept and restrict players to use stage-undominatedvoting strategies [6]. The equilibrium is such that, irrespective of the initial allocationof the dollar or the discount factor, policy outcomes are absorbed with probabilityone in a set that consists of divisions that allocate the whole dollar to the proposer.

Equilibrum dynamics can be motivated through the nature of winning coalitionsthat form along the equilibrium path. In the spirit of Riker [23], coalitions areminimum winning in that at most a bare majority of members receive a positivefraction of the dollar. This is possible in equilibrium because allocations such thatexcluded minorities receive zero share of the dollar reduce the cost of building acoalition in subsequent periods. Due to this externality, players are willing to acceptproposals that exclude other members of the legislature and are even willing toaccept proposals that reduce their own stage payoff compared to that received underthe status quo allocation. Thus, convergence to the equilibrium absorbing set ofpolicy outcomes is fast, with a maximum expected2 time before absorption equal totwo and a half (2.5) periods.

The equilibrium we characterize is not unique, although all additional equilibriawe can find are payoff-equivalent. We show that players’ equilibrium expected utilityis continuous with respect to current period’s decision. We also show thatequilibrium expected utility fails quasi-concavity and has flat areas inducing thickindifference sets. Thus, we cannot rely on the convexity or lower-hemicontinuity ofthe voters’ acceptance set (the set of policies players weakly prefer over the statusquo) in arbitrary dynamic bargaining games of this type, even though the underlyingstage utilities are well-behaved.

Despite the non-continuity of the proposer’s set of feasible (i.e. majority preferred)proposals, our equilibrium has the feature that (mixed) proposal strategies areweakly continuous with respect to the status quo.3 In effect, we obtain equilibriumand continuity of proposal strategies because the pathological (flat) areas of voters’expected utility involve (weakly) suboptimal allocations. As a result these allocationsare never proposed and constitute transient states in the equilibrium-induced

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2 Absorption is not deterministic as it depends on the identity of the proposer recognized in each period.3 Continuity is with respect to the topology of weak convergence.

A. Kalandrakis / Journal of Economic Theory 116 (2004) 294–322296

Markov process of policy outcomes. This may not be true in more general policyspaces, making it harder to establish equilibrium if one exists.

The fact that the proposer obtains the whole dollar in each period after absorptionis in contrast to the convergence to the median result of Baron [3] and thecalculations of Baron and Herron [5] who find that equilibrium legislative decisionsbecome more centrally located with higher discount factor and a longer timehorizon. In the same spirit, we remark that while the uncovered set [15,20] defined onthe basis of stage preferences has full measure in the set of possible allocations, theequilibrium absorbing set of outcomes consists entirely of covered alternatives[12,21]. The above suggest that the properties of the distribution of equilibriumpolicies in these games may depend significantly on the policy area underconsideration.

Among other related contributions, Epple and Riordan [11] study subgame perfectequilibria of a three-player game similar to ours where players alternate makingproposals. They show that at least two subgame perfect equilibria exist for highenough discount factor. These equilibria involve two radically different allocationsof the dollar, suggesting that a folk theorem may apply for the set of subgame perfectequilibria in their game. Also related is the work of Ferejohn et al. [14] who considerthe existence and other properties of a steady-state distribution of policy outcomes ina game where proposals arise randomly from the space of alternatives and voters areimpatient or myopic. Finally, Dixit et al. [10] study the dynamics of compromiseunder efficient subgame perfect equilibria among two parties that alternate in poweraccording to a policy-dependent stochastic rule.

We now proceed to a detailed presentation of the legislative setup. We outlineequilibrium analysis in Section 3 and state the main result in Section 4. We concludein Section 5.

2. Legislative setup and equilibrium notion

The problem involves a set N ¼ f1; 2; 3g of three legislators that choose alegislative outcome xt for each t ¼ 1; 2;y . Possible decisions within each period aredivisions of a dollar among the three legislators, i.e. xt is a triple xt ¼ ðxt

1; xt2; xt

3Þ with

xtiX0 for i ¼ 1; 2; 3 and

P3i¼1 xt

i ¼ 1: Thus, the legislative outcome xt is an element of

the unit simplex in R3 which we denote by D: Legislative interaction is as follows: at

the beginning of each period legislator i ¼ 1; 2; 3 is recognized with probability 13

to

make a proposal zAD: Having observed the proposal legislators vote yes or no: If a

majority of two or more vote yes then xt ¼ z; otherwise xt ¼ xt�1:4 Thus, previous

period’s decision, xt�1; serves as the status quo or reversion point in the currentperiod, t:

Legislators derive von Neumann-Morgenstern stage utility ui : D-R; iAN; fromthe implemented proposal xt: In particular, we assume that legislators are risk-

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4 We assume x0 exogenously given.


neutral and care only about the share of the dollar they receive, i.e. uiðxtÞ ¼ xti :

Legislators discount the future with a common discount factor dA½0; 1Þ; so that the

utility of player i from a sequence of outcomes fxtgþN

t¼1 is given by

ViðfxtgþN

t¼1 Þ ¼XþN

t¼1

dt�1uiðxtÞ: ð1Þ

In general, strategies in this game are functions that map histories5 to the space ofproposals D and voting decisions fyes; nog: In what follows, though, we restrictanalysis to cases when players condition their behavior only on a summary of thehistory of the game that accounts for payoff-relevant effects of past behavior [17].

Specifically, define the state in period t as previous period’s decision xt�1: We denote

the state by sAS so we have s ¼ xt�1 and S ¼ D: We restrict players to Markovstrategies such that they behave identically ex ante (prior to any mixing) in differentperiods with state s; even if that state arises from different histories.

Denote the set of Borel probability measures over D by YðDÞ: In general, a mixed

Markov proposal strategy for legislator i is a function mi : S-YðDÞ: Without delvinginto measurability issues, it is sufficient for the purposes of our analysis to assumethat for every state s; mi has finite support. Thus, we shall use the notation mi½z j s� torepresent the probability that legislator i makes the proposal z when recognizedconditional on the state being s: A Markov voting strategy is an acceptance

correspondence Ai : S7X ; where AiðsÞ represents the allocations for which i votes

yes when the state is s: Then, aMarkov strategy is a mapping si : S-YðDÞ � 2X

where for each s; siðsÞ ¼ ðmi½ j s�;AiðsÞÞ:For a given set of voting strategies, we define the win set of xAD as the set of

proposals that beat x by majority rule:

WðxÞ ¼ yADX3

i¼1

IAiðxÞðyÞX2

��( )

: ð2Þ

Then, for a triple of Markov strategies r ¼ fs1; s2; s3g; we can write the transitionprobability to decision x when the state is s; Q½x j s�; as follows:

Q½x j s� � IW ðsÞðxÞX3

i¼1

1

3mi½x j s�

þ IfsgðxÞX3

i¼1

1

3

Xy:mi ½yjs�40

ID\W ðsÞðyÞmi½y j s�: ð3Þ

The first part of (3) reflects the probability of transition to allocations that areproposed by one of the three players and obtain a majority, while the second part isthe probability that legislative decision is the same as in the last period (i.e. thereversion point or status quo s) because the proposal fails majority passage. Note thatsince we have assumed that mi is a discrete measure, so is Q½x j s�:

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5 A history is a vector that records all proposals as well as all voting decisions that precede an action

(voting or proposing).


Thus, we can recursively define the continuation value, viðsÞ; of legislator i when thestate is s as

viðsÞ ¼X

x:Q½x j s�40

½uiðxÞ þ dviðxÞ�Q½x j s�: ð4Þ

On the basis of (4) write the expected utility of legislator i; UiðxtÞ; solely as a functionof the current decision xt:

UiðxtÞ ¼ uiðxtÞ þ dviðxtÞ; ð5Þ

where it is understood that viðxtÞ—hence UiðxtÞ—are defined for given Markovstrategies r:

Given that legislators are otherwise identical, we focus on Markov proposal andvoting strategies that are symmetric with respect to permutations of the state vector.To be precise, let the one-to-one and onto function l : N-N represent a permutation

of N ¼ f1; 2; 3g and let l̂ðxÞAD denote the corresponding permutation of the vector

xAD so that coordinate l̂ðxÞlðhÞ ¼ xh; h ¼ 1; 2; 3: Then, a symmetric Markov strategy

profile r is such that for all iAN; sAS; and any permutation l:

xAAiðsÞ 3 l̂ðxÞAAlðiÞðl̂ðsÞÞ ð6Þ

and

mi½z j s� ¼ mlðiÞ½l̂ðzÞ j l̂ðsÞ�: ð7Þ

We can now define the equilibrium solution concept as follows:

Definition 1. A Symmetric Markov Perfect Nash Equilibrium in Stage-Undominated

Voting strategies (MPNESUV) is a symmetric Markov strategy profile r� ¼fðm�i ½ j s�;A�

i ðsÞÞg3i¼1; such that for i ¼ 1; 2; 3 and all sAS:

yAA�i ðsÞ 3 UiðyÞXUiðsÞ; ð8Þ

m�i ½z j s�40 ) zAarg maxfUiðxÞ j xAWðsÞg: ð9Þ

Equilibrium condition (8) requires that legislators vote yes if and only if theirexpected utility from the status quo is not larger than their expected utility from theproposal. Such stage-undominated [6] voting strategies rule out uninterestingequilibria where voting decisions constitute best responses solely due to the fact thatlegislators vote unanimously. Equilibrium condition (9) requires that proposersoptimize and play ‘‘no-delay’’ strategies, i.e. their proposals are always approved bya majority.6

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6 Since voting strategies guarantee sAWðsÞ the restriction to ‘‘no-delay’’ proposal strategies in (9) is

consistent with equilibrium.


3. A preview of results

Even in this considerably simplified setup, characterization of a symmetricMPNESUV constitutes a challenging problem due to the cardinality of the statespace that makes it difficult to ascertain the validity of equilibrium conditions (8) and(9). The solution we present arises from an informative guess about the nature of theequilibrium-induced Markov process on policy outcomes defined in (3).

Suppose that proposers build coalitions by allocating a positive amount to at mosttwo legislators. With such proposals, we certainly have si ¼ 0 for some i (possiblydifferent across periods) in all periods but the first. Further suppose that legislator i;with si ¼ 0 does not object to new (optimal) divisions of the dollar z with zi ¼ 0; sothat if jai is recognized in period t þ 1; a coalition of i and j vote yes on a proposalthat allocates the whole dollar to j: But then both legislators i and hai; j receive zero,so that any of the three legislators can successfully form a coalition to extract thewhole dollar in all subsequent periods.

If this conjectured path of play is part of an equilibrium, then it is possible to solvethis game backwards from the period when absorption to the set of outcomes thatgive zero to two legislators takes place, to arbitrary initial allocation of the dollar. Itis by means of this strategy that we demonstrate the advertised result. In this sectionwe offer a brief description of some basic steps that can prove enlightening both as tothe nature of the solution and the process via which it is derived.

Additional notation will be necessary before we can proceed. First, partition thespace of policy outcomes into subsets DyCD; where 0pyo3 indicates the number oflegislators receiving zero share of the dollar:

Dy ¼ xADX3

i¼1

If0gðxiÞ ¼ y

��( )

: ð10Þ

In the following three subsections we will describe equilibrium proposals for thecases y is equal to 0, 1, and 2, respectively. We will illustrate how continuation valuescan be derived on the basis of these proposal strategies. We note that in theremainder of this section we assume (and only prove in the following section) thatthese proposals achieve majority passage and constitute optima for the proposers.

i. Recurrent allocations: sAD2: According to the conjectured equilibrium, D2 is anirreducible absorbing set. In particular, let generic elements of D2 be denoted by

ei ¼ ðei1; ei

2; ei3Þ; with ei

i ¼ 1; eij ¼ 0; jai and assume m�i ½ei j s� ¼ 1; i ¼ 1; 2; 3; and

sAD2; i.e. proposers always obtain the whole dollar when any one of the threelegislators received the whole dollar in the previous period. Since players receive thewhole dollar when recognized and zero otherwise, their expected payoff in each

period is 13; so that the continuation value of player i is:

viðsÞ ¼ %v ¼ 1

3ð1 � dÞ; sAD2; i ¼ 1; 2; 3: ð11Þ

ii. Transient Allocations: sAD1: Moving backwards, consider states s for or prior tothe period of transition to D2; starting with the case sAD1 first, i.e. cases when a

ARTICLE IN PRESSA. Kalandrakis / Journal of Economic Theory 116 (2004) 294–322300

single player received zero in the previous period. As a consequence of the focus onsymmetric MPNESUV, it is sufficient to characterize equilibrium Markov strategiesfor s ¼ ðs1; s2; s3Þ with s1Xs2Xs3—i.e. s3 ¼ 0 in this case. Since s3 ¼ 0; player 1’s and

2’s proposal strategy takes the form m�i ½ei j ðs1; s2; 0Þ� ¼ 1; i ¼ 1; 2: In other words,

players 1 and 2 obtain the consent of player 3 in order to pass a proposal thatallocates them the whole dollar.

With regard to the proposal strategy of player 3, a natural candidate forequilibrium behavior is for this player to form a coalition with player 2 who appearsless ‘expensive’ compared to player 1 since s1Xs2: It turns out that this intuition isnot consistent with equilibrium for some values of s ¼ ðs1; s2; 0Þ; as we show shortly.In particular consider the case, in accordance with this intuition, that player 3 indeedplays a pure proposal strategy and allocates a positive amount to player 2 only. Byinvoking symmetry we deduce that player 2 is indifferent between s ¼ ðs1; s2; 0Þ and aproposal ð0; s2; s1Þ so that according to the above conjecture:

m�3½ð0; s2; s1Þ j ðs1; s2; 0Þ� ¼ 1: ð12Þ

On the basis of the above we can write the continuation values of players asfollows:

v1ðsÞ ¼ 13 ½1 þ d%v� þ 1

3 ½0 þ d%v� þ 13 ½0 þ dv1ð0; s2; s1Þ�; ð13aÞ

v2ðsÞ ¼ 13½0 þ d%v� þ 1

3½1 þ d%v� þ 1

3½s2 þ dv2ð0; s2; s1Þ�; ð13bÞ

v3ðsÞ ¼ 13½0 þ d%v� þ 1

3½0 þ d%v� þ 1

3½s1 þ dv3ð0; s2; s1Þ�: ð13cÞ

Symmetry implies that v1ð0; s2; s1Þ ¼ v3ðsÞ; v2ð0; s2; s1Þ ¼ v2ðsÞ; and v3ð0; s2; s1Þ ¼v1ðsÞ so that after substitution (13b) can be solved for v2ðsÞ and Eqs. (13a) and (13c)can be solved for v3ðsÞ and v1ðsÞ to obtain:

v1ðsÞ ¼1

3ð1 � dÞ �ds2

ð9 � d2Þ; ð14aÞ

v2ðsÞ ¼1

3ð1 � dÞ þs2

ð3 � dÞ; ð14bÞ

v3ðsÞ ¼1

3ð1 � dÞ �3s2

ð9 � d2Þ: ð14cÞ

We will now show that the proposal strategy in Eq. (12) is not optimal for player 3for some values of s1; s2: To see this calculate the amount legislator 1 demands7 fromplayer 3 in order to vote yes on a proposal that excludes legislator 2, assuming thatthe game is subsequently played according to (12). Denote this amount by d1; wehave:

d1 þ dv1ðd1; 0; 1 � d1Þ ¼ s1 þ dv1ðsÞ ð15Þ

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7 A precise definition of demands appears in the following section.


which, substituting for s1 ¼ 1 � s2; v1ðd1; 0; 1 � d1Þ ¼ 13ð1�dÞ þ

d1ð3�dÞ; and v1ðsÞ ¼

13ð1�dÞ �

ds2

ð9�d2Þ from (14b) and (14a), respectively, we can solve for d1 to get d1 ¼ð9�d2Þ�9s2

3ð3þdÞ : This is smaller than the amount demanded by legislator 2, s2; when:

s24ð9 � d2Þ3ð6 þ dÞ ð16Þ

which is less than one-half. As a consequence, when the difference between s1 and s2

is small, i.e. when Eq. (16) holds, legislator 3 has an incentive to mix betweencoalescing with legislators 1 and 2.

Hence, player 3’s proposal strategy can8 take the following form: proposeðq; 0; 1 � qÞ with probability m�3½ðq; 0; 1 � qÞ j s� and ð0; q; 1 � qÞ with probability

m�3½ð0; q; 1 � qÞ j s� ¼ 1 � m�3½ðq; 0; 1 � qÞ j s�; where q ¼ ð9�d2Þ3ð6þdÞ: Using the shorter

notation m�3 ¼ m�3½ð0; q; 1 � qÞ j s�; we can write the continuation values for s such

that Eq. (16) holds as follows:

v1ðsÞ ¼ 13ð1 þ d%vÞ þ 1

3d%v þ 1

3ðð1 � m�3Þðq þ dv1ðq; 0; 1 � qÞÞ

þ m�3dv1ð0; q; 1 � qÞÞ; ð17aÞ

v2ðsÞ ¼ 13 d%v þ 1

3 ð1 þ d%vÞ þ 13 ðð1 � m�3Þdv2ðq; 0; 1 � qÞ

þ m�3ðq þ dv2ð0; q; 1 � qÞÞÞ; ð17bÞ

v3ðsÞ ¼ 13d%v þ 1

3d%v þ 1

3ð1 � q þ dv3ð0; q; 1 � qÞÞ; ð17cÞ

where symmetry is used in (17c). Note that for all sAD1 with s2pð9�d2Þ3ð6þdÞ; the

continuation values of players are given in Eqs. (13a)–(13c). Hence, since q ¼ð9�d2Þ3ð6þdÞo

12; we can use Eqs. (13a), (13b), and (13c) to substitute for v1ð0; q; 1 � qÞ ¼

v2ðq; 0; 1 � qÞ ¼ 13ð1�dÞ � 1

ð6þdÞ; v2ð0; q; 1 � qÞ ¼ v1ðq; 0; 1 � qÞ ¼ 13ð1�dÞ þ

ð3þdÞ3ð6þdÞ; and

v3ð0; q; 1 � qÞ ¼ 13ð1�dÞ � d

3ð6þdÞ: Furthermore, we have:

s1 þ dv1ðsÞ ¼ s2 þ dv2ðsÞ ð18Þfor players 1 and 2 that accept the same amount q: Note that players 1 and 2 have thesame expected payoff in (18), hence their demands are identical for status quo s; eventhough their stage utility from s may differ. Thus, we have formed four linearequations ((17a)–(17c), and (18)) in four unknowns (m�3; and viðsÞ; i ¼ 1; 2; 3) that can

be solved to obtain:

m�3½ð0; q; 1 � qÞ j s� ¼ 9 þ 3dþ d2

ð3 þ 2dÞd � 3ðdþ 6Þs2

ð3 þ 2dÞd ; ð19Þ

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8 There exist other (payoff equivalent) proposal strategies that are consistent with the equilibrium we

characterize in these cases; the one we choose preserves the continuity of proposal strategies with respect to

the status quo.


v1ðsÞ ¼ð15 � dÞ

6ð1 � dÞð6 þ dÞ �1 � 2s2

2d; ð20aÞ

v2ðsÞ ¼ð15 � dÞ

6ð1 � dÞð6 þ dÞ þ1 � 2s2

2d; ð20bÞ

v3ðsÞ ¼1

3ð1 � dÞ �1

ð6 þ dÞ: ð20cÞ

With the above we have described equilibrium strategies and have derivedcontinuation values for states sAD\D0: The Markov process over outcomes induced

by these strategies within this subset of the two-dimensional simplex in R3 is depictedgraphically in Fig. 1. In Fig. 1a we depict the Markov process within the absorbingset D2; while transitions from states in D1 are depicted in Fig. 1b. Fig. 1b depicts astate s with s1Xs24s3 ¼ 0 as we analyzed in this subsection. Notice that transition to

D2 occurs whenever legislators 1 and 2 are recognized, i.e. there is only 13

probability

(when legislator 3 is recognized) that the decision remains in D1 each period the statesAD1:

iii. Transient allocations: t ¼ 1; sAD0: If proposers never allocate a positive fractionof the dollar to more than one other legislator, allocations with all three legislatorshaving a positive amount never prevail except perhaps for the very first period. For

the latter cases, when the game happens to start with a state s ¼ x0AD0; equilibriumproposals are no different in nature than those analyzed so far, except for theadditional complexity introduced by the various combinations of mixed and pureproposal strategies for various subsets of D0 (seven subcases in total). In whatfollows we offer a brief description of equilibrium proposals in these cases.

First, the pattern of mixed proposal strategies described for states sAD1 is also afeature of the equilibrium whenever the difference in allocated amounts between anypair of players under the state sAD0 is small. This is the result of two effects. On theone hand, players with larger amount under s require higher compensation tooverturn the status quo, ceteris paribus. On the other hand, these players are morelikely to be excluded from coalitions in the future if the status quo is preserved,ceteris paribus. The first effect generates an incentive to preserve the status quo,while the second generates an incentive to overturn it. Similar, but opposite,incentives apply for players with small allocations under the status quo.

Thus, for a pair of players with small difference in allocations under s; i.e. when acondition analogous to (16) holds, the combination of the two effects implies thatmixed strategies are required in equilibrium. By mixing in these cases the proposerensures that coalition partners with a less favorable allocation under s; do notbecome too intransigent in their demands because they are guaranteed a position inthe winning coalition under the status quo.

Figs. 2a–d represents the two-dimensional unit simplex in R3 with highlightedareas depicting cases when such mixing between coalition partners takes place foralternative values of the discount factor. Note that this happens for pairs of playerswith nearly equal allocations. From a comparison of these graphs it is apparent that

ARTICLE IN PRESSA. Kalandrakis / Journal of Economic Theory 116 (2004) 294–322 303

mixing takes place in a smaller range of the state space as d decreases. This is a directconsequence of the fact that the weight players put in the future benefit/cost ofpreserving the status quo associated with the probability of inclusion in futurecoalitions diminishes with d: For d ¼ 0 coalition building costs are solely determinedby the share of the dollar under the status quo, and as a result only pure proposalstrategies are played.

Another feature of the equilibrium that is implicit in the above discussion and isillustrated in Fig. 2a–d is the fact that players are willing to vote yes on proposalsthat allocate them a smaller share of the dollar than what they obtain under the states: In fact, there are areas—near the sides of the triangle—in which the player with the

ARTICLE IN PRESS

1

2 3

2s ∈∆

1

( (

1 2, ,0s s

( ), 0,1q −

2 3

q

(0, s2, s1)(0, q,1-q)

1s ∈∆

(a)

(b)

Fig. 1. Equilibrium-induced Markov process—sAD\D0: Key: (a) D2 is an irreducible absorbing set; (b)

Player 3 mixes between ð0; q; 1 � qÞ and ðq; 0; 1 � qÞ; q ¼ ð9�d2Þ3ð6þdÞ; when difference between s1 and s2 is small

(allocations marked with ). Sources: Constructed by author on basis of Proposition 1.


smallest share of the dollar accepts proposals that allocate her zero and the wholedollar to the proposer. In other words, there is the possibility of direct transition toD2 from states in D0:

In these cases, players take into account both the immediate loss in accepting asmaller amount compared to what they obtain under the status quo, s; as well as theexternality that such proposals generate through a reduction in their coalitionbuilding costs in the future. By accepting such proposals these players are able toextract the whole dollar if recognized in the next period, while they would have toallocate a positive amount to one of the other players had they rejected the proposaland preserved the status quo. As is the case for mixed proposal strategies, the area of

ARTICLE IN PRESS

0= .3=

a

b

c

d

e

g f

.5= .9=

δ δ

δ δ

(a) (b)

(c) (d)

Fig. 2. Demands and equilibrium proposal strategies vs. discount factor. Key: Allocations for which

mixed proposal strategies are played and/or legislators demand zero, expand with larger discount factor.

Areas corresponding to Cases a–g of Proposition 1 are labeled in (d). Legislator with min. amount

demands zero & plays mixed proposal strategy (Proposition 1, Case b). Proposer(s) play mixed

proposal strategies (Proposition 1, Cases d–g). Legislator with min. amount demands zero. Proposers

play pure strategies (Proposition 1, Case a). & All legislators demand positive amount. Proposers play

pure strategies (Proposition 1, case c). Sources: Constructed by author on basis of Proposition 1.


direct absorption to D2 contracts with d; since the value of future reduction incoalition building costs diminishes as well.

The equilibrium-induced Markov process described above is depicted graphicallyin Fig. 3. While Proposition 1 in the next section contains the exact statement of theequilibrium, we provide here a brief summary that can serve as a guide for theanalysis to follow.

Summary 1. For any dA½0; 1Þ there exists a symmetric MPNESUV that induces a

Markov process over outcomes such that:

* D2 is an irreducible absorbing set.* For any state sAD1 there is probability 2

3of transition to D2; and 1

3of remaining in D1:

* For some states sAD0 there is probability 23

of transition to D2 by a majority formed

by the proposer and the player with minimum amount in s; and probability 13

of

transition to D1:* In the remaining cases of states sAD0 there is probability 1 of transition to D1:* For sAD0;D1; proposers mix between possible coalition partners that have positive

and nearly equal allocation under the state s:

4. Equilibrium

In this section we fully specify the proposal and voting strategies described above,and we prove that they constitute part of a symmetric MPNESUV. Note that in the

ARTICLE IN PRESS

0∆ 0∆

1 13

23

13 1∆

23

1 2∆ Irreducible Absorbing Set

′ ′′

Fig. 3. Equilibrium-induced Markov process. Key: D00 is the set containing states in Cases c–g of

Proposition 1 with s340: D000 ; is the set containing states in Cases a and b of Proposition 1 with s340:

Sources: Constructed by author on basis of Proposition 1.


previous section we derived continuation values for the conjectured equilibrium forstates sAD\D0; so that we have a closed form expression for the expected utility ofplayers for these legislative outcomes. It is critical in establishing the equilibrium thatthis expected utility function satisfies the following continuity and monotonicityproperties:

Lemma 1. Consider a symmetric Markov Perfect strategy profile with expected

utility UiðsÞ; sAD\D0; determined by the continuation values in Eq. (11) for sAD2;

Eqs. (14a)–(14c) when s2pð9�d2Þ3ð6þdÞ; and (20a)–(20c) when s24

ð9�d2Þ3ð6þdÞ: Then, for all x ¼

ðx; 1 � x; 0ÞAD (a) UiðxÞ; i ¼ 1; 2; 3 is continuous and piece-wise differentiable

with respect to x; (b) U1ðxÞ does not decrease with x; while U2ðxÞ does not increase

with x:

Proof. From symmetry and Eqs. (11), (14a)–(14c), and (20a)–(20c) we have

U1ðx; 1 � x; 0Þ ¼

1 þ d3ð1 � dÞ if x ¼ 1;

x þ d1

3ð1 � dÞ �dð1 � xÞð9 � d2Þ

!if xA 1 � ð9 � d2Þ

3ð6 þ dÞ; 1�

;

1

2þ dð15 � dÞ

6ð1 � dÞð6 þ dÞ if xAð9 � d2Þ3ð6 þ dÞ;

1 � ð9 � d2Þ3ð6 þ dÞ

;

x þ d1

3 ð1 � dÞ þx

ð3 � dÞ

if xA 0;

ð9 � d2Þ3ð6 þ dÞ

�;

d3ð1 � dÞ if x ¼ 0;

8>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>:

ð21Þ

hence limx-1 x þ d 13ð1�dÞ �

dð1�xÞð9�d2Þ

� �h i¼ 1 þ d

3ð1�dÞ; limx-0 x þ d 13ð1�dÞ þ x

ð3�dÞ

� �h i¼

d3ð1�dÞ; and U1 1 � ð9�d2Þ

3ð6þdÞ;ð9�d2Þ3ð6þdÞ; 0

� �¼ U1

ð9�d2Þ3ð6þdÞ; 1 � ð9�d2Þ

3ð6þdÞ; 0� �

¼ 12þ dð15�dÞ

6ð1�dÞð6þdÞ; which

proves (a) for i ¼ 1; 2 by symmetry. Continuity for i ¼ 3 follows sinceP3

i¼1 UiðxÞ ¼1

1�d: For (b), we have dU1ðxÞdx

¼ 1 þ d2

ð9�d2Þ40 for xA 1 � ð9�d2Þ3ð6þdÞ; 1

h �; dU1ðxÞ

dx¼ 0 for

xA ð9�d2Þ3ð6þdÞ; 1 � ð9�d2Þ

3ð6þdÞ

� �; and dU1ðxÞ

dx¼ 1 þ d

ð3�dÞ40 for xA 0; ð9�d2Þ3ð6þdÞ

� i; and symmetry

completes the proof for U2ðx; 1 � x; 0Þ: &

The significance of Lemma 1 lies with the fact that we seek to establish anequilibrium with proposals that allocate a positive amount to at most two legislators.We now state the definition of the equilibrium demand of a legislator:


Definition 2. For a symmetric MPNESUV, legislator i0s demand when the state is s isthe minimum amount diðsÞA½0; 1� such that for a proposal xAD\D0 with xi ¼diðsÞ; xj ¼ 1 � diðsÞ; jai; we have

UiðxÞXUiðsÞ: ð22Þ

For the specific equilibrium we characterize, the continuity of the expected utilityfunction in (21) established in Lemma 1 ensures that the demand of legislator i for a

state sAD is well defined as long as UiðsÞp1 þ d3ð1�dÞ: If the latter condition fails, then

legislator i always prefers the status quo s over any proposal in D\D0:9 Note that as a

result of symmetry the demand does not depend on which of the two possiblelegislators jai receives xj ¼ 1 � diðsÞ:

Assuming a demand diðsÞA½0; 1� exists, we define a minimum-winning-consistent

proposal as follows:

Definition 3. For a symmetric MPNESUV, a minimum winning consistent (mwc-)proposal, xð j; i; diðsÞÞ; for proposer j and legislator iaj with demand diðsÞA½0; 1�is an allocation xð j; i; diðsÞÞAD\D0 with coordinates xið j; i; diðsÞÞ ¼ diðsÞ;xjð j; i; diðsÞÞ ¼ 1 � diðsÞ; and xhð j; i; diðsÞÞ ¼ 0; haj; i:

With a mwc-proposal xð j; i; diðsÞÞ; the proposer j allocates the demand,diðsÞ; to legislator iaj and retains the rest of the dollar. Notice that the definitionof demands allows for strict inequality in (22), which in the equilibrium wecharacterize may occur for demands equal to zero. Hence, in equilibrium we mayhave a mwc-proposal xð j; i; diðsÞÞAD2; i.e. such that the proposer retains the wholedollar.

By the definition of demands and the monotonicity established in part (b) ofLemma 1 we immediately deduce:

Lemma 2. Consider a symmetric Markov Perfect strategy profile with expected

utility, UiðxÞ; for xAD\D0 given by (21). Every mwc-proposal xð j; i; diðsÞÞ; jai is

such that

xð j; i; diðsÞÞAarg maxfUjðxÞ j xAD\D0;UiðxÞXUiðsÞg: ð23Þ

Thus, according to the conjectured equilibrium, proposals for any state sAD takethe form of mwc-proposals. In what follows, we will make use of the notion ofequilibrium demands and mwc-proposals in order to state and establish theequilibrium. To reduce the notational burden, we omit the dependence of demandsand mwc-proposals on the state s and write di and xð j; i; diÞ instead, unless otherwisenecessary. We prove the following:

ARTICLE IN PRESS

9 This is never the case in the equilibrium we characterize.


Proposition 1. There exists a symmetric MPNESUV with the following demands,proposal strategies, and continuation values for all sAD; where s1Xs2Xs3:

Case a: s3p 3dð9�d2Þ s2; s3p1 � 3ð6þdÞ

ð9�d2Þ s2:

d1 ¼1 � s2 �

9 � d2

9s3 if s3o

3

ð6 þ dÞ �9s2

ð9 � d2Þ;

3 � d3

� 9s2 þ ð9 � d2Þs3

3ð3 þ dÞ if s3X3

ð6 þ dÞ �9s2

ð9 � d2Þ;

8>>><>>>:

ð24Þ

d2 ¼ s2; d3 ¼ 0; ð25Þ

m�i ½ei j s� ¼ m�3½xð3; 2; d2Þ j s� ¼ 1; i ¼ 1; 2; ð26Þ

v1ðsÞ ¼1

3ð1 � dÞ �ds2

ð9 � d2Þ; ð27aÞ

v2ðsÞ ¼1

3ð1 � dÞ þs2

ð3 � dÞ; ð27bÞ

v3ðsÞ ¼1

3ð1 � dÞ �3s2

ð9 � d2Þ: ð27cÞ

Case b: s3p d2ð3þdÞ; s341 � 3ð6þdÞ

ð9�d2Þ s2:

d1 ¼ d2 ¼ ðs1 þ s2Þð9 � d2Þ3ð6 þ dÞ ; d3 ¼ 0; ð28Þ

m�i ½ei j s� ¼ 1; i ¼ 1; 2; ð29Þ

m�3½xð3; 2; d2Þ j s� ¼ 1 � m�3½xð3; 1; d1Þ j s� ¼9 þ dð3 þ dÞdð3 þ 2dÞ

� 3ð6 þ dÞs2

dð3 þ 2dÞð1 � s3Þ; ð30Þ

v1ðsÞ ¼1

3ð1 � dÞ þð3 þ dÞs2 � 3s1

dð6 þ dÞ ; ð31aÞ

v2ðsÞ ¼1

3ð1 � dÞ þð3 þ dÞs1 � 3s2

dð6 þ dÞ ; ð31bÞ

v3ðsÞ ¼1

3ð1 � dÞ �ðs1 þ s2Þð6 þ dÞ : ð31cÞ


Case c: s34 3dð9�d2Þs2; s3p

ð9�2d2Þ3ð3þdÞ s2; s2p

27�2dð9þ3d�d2Þ3ð18�9d�4d2Þ � 3ð1�dÞð3þdÞ

ð18�9d�4d2Þs3:

d1 ¼

1 � ð81 � 54d� 27d2 þ 6d3 þ 2d4Þs2

3ð27 � 18d� 6d2 þ 2d3Þif

27 � 18d� 6d2 þ 2d3

ð9 � 2d2 � 3dÞð6 þ dÞ

�ð81 � 27d� 27d2 þ 3d3 þ 2d4Þs3

3ð27 � 18d� 6d2 þ 2d3Þ�ð9 � 6d� 2d2Þð9 � 3d� 2d2Þ

s24s3;

ð3 � dÞ3

� ð9 � 3d� 2d2Þð3 � dÞð27 � 18d� 6d2 þ 2d3Þ

s3 if27 � 18d� 6d2 þ 2d3

ð9 � 2d2 � 3dÞð6 þ dÞ

� ð9 � 6d� 2d2Þð3 � dÞð27 � 18d� 6d2 þ 2d3Þ

s2 �ð9 � 6d� 2d2Þð9 � 3d� 2d2Þ

s2ps3;

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð32aÞ

d2 ¼ ð9 � d2Þðð3 � 2dÞs2 � ds3Þ27 � 2dð9 þ 3d� d2Þ

; ð32bÞ

d3 ¼ ð3 � dÞðð9 � d2Þs3 � 3ds2Þ27 � 2dð9 þ 3d� d2Þ

; ð32cÞ

m�1½xð1; 3; d3Þ j s� ¼ m�2½xð2; 3; d3Þ j s� ¼ m�3½xð3; 2; d2Þ j s� ¼ 1; ð33Þ

v1ðsÞ ¼1

3ð1 � dÞ þ2d2s2 � ð9 � 2d2Þs3

27 � 2dð9 þ 3d� d2Þ; ð34aÞ

v2ðsÞ ¼1

3ð1 � dÞ þð9 � 2d2Þs2 � 3ð3 þ dÞs3

27 � 2dð9 þ 3d� d2Þ; ð34bÞ

v3ðsÞ ¼1

3ð1 � dÞ þð18 þ 3d� 2d2Þs3 � 9s2

27 � 2dð9 þ 3d� d2Þ: ð34cÞ

Case d: s34 d2ð3þdÞ; s3p

ð9�2d2Þ9ð3þdÞ ; s24

27�2dð9þ3d�d2Þ3ð18�9d�4d2Þ � 3ð1�dÞð3þdÞ

ð18�9d�4d2Þs3:

d1 ¼ d2 ¼ ð2d� 3 þ 3s3Þð9 � d2Þ3ð4d2 þ 9d� 18Þ

; ð35aÞ

d3 ¼ dð3 � dÞ � 2ð9 � d2Þs3

ð4d2 þ 9d� 18Þ; ð35bÞ

m�i ½xði; 3; d3Þ j s� ¼ 1; i ¼ 1; 2; ð36Þ


m�3½xð3; 2; d2Þ j s� ¼ 1 � m�3½xð3; 1; d1Þ j s�

¼ 27 � dð9 þ 6dþ 2d2Þdð3 þ 2dÞð3 � 3s3 � 2dÞ

� 3ð18 � 9d� 4d2Þs2 þ 3ð9 � 3d� d2Þs3

dð3 þ 2dÞð3 � 3s3 � 2dÞ ; ð37Þ

v1ðsÞ ¼1

3ð1 � dÞ �ð9 � 6d� 2d2Þ � ð9 � 12d� 4d2Þs3

dð18 � 9d� 4d2Þþ s2

d; ð38aÞ

v2ðsÞ ¼1

3ð1 � dÞ þð9 � 3d� 2d2Þ � 3ð3 þ dÞs3

dð18 � 9d� 4d2Þ� s2

d; ð38bÞ

v3ðsÞ ¼1

3ð1 � dÞ �3 � ð15 þ 4dÞs3

ð18 � 9d� 4d2Þ: ð38cÞ

Case e: s34ð9�2d2Þ3ð3þdÞ s2; s1X

9þ2dð3þdÞ9ð3þdÞ :

d1 ¼

1 þ ð2d3 þ 6d2 � 18d� 54Þs2

3ð18 þ 3d� 2d2Þif s3o

ð18 þ 3d� 2d2Þ2ð3 þ dÞð6 þ dÞ � s2;

þð6d2 þ 2d3 � 18d� 54Þs3

3ð18 þ 3d� 2d2Þð3 � dÞ

3� ðs2 þ s3Þ2ð9 � d2Þ

ð18 þ 3d� 2d2Þif

18 þ 3d� 2d2

2ð3 þ dÞð6 þ dÞ � s2ps3;

8>>>>>>>>><>>>>>>>>>:

ð39aÞ

d2 ¼ d3 ¼ ð9 � d2Þðs2 þ s3Þð18 þ 3d� 2d2Þ

; ð39bÞ

m�1½xð1; 3; d3Þ j s� ¼ 1 � m�1½xð1; 2; d2Þ j s� ¼3ð3 þ dÞs2 � ð9 � 2d2Þs3

dð3 þ 2dÞðs2 þ s3Þ; ð40Þ

m�2½xð2; 3; d3Þ j s� ¼ m�3½xð3; 2; d2Þ j s� ¼ 1; ð41Þ

v1ðsÞ ¼1

3ð1 � dÞ �ð3 þ 2dÞðs2 þ s3Þð18 þ 3d� 2d2Þ

; ð42aÞ

v2ðsÞ ¼1

3ð1 � dÞ þ3ð3 þ dÞs3 � ð9 � 2d2Þs2

dð18 þ 3d� 2d2Þ; ð42bÞ

v3ðsÞ ¼1

3ð1 � dÞ þ3ð3 þ dÞs2 � ð9 � 2d2Þs3

dð18 þ 3d� 2d2Þ: ð42cÞ


Case f: s2p13; s1o

9þ2dð3þdÞ9ð3þdÞ :

di ¼3 � d

9; i ¼ 1; 2; 3; ð43Þ

m�1½xð1; 3; d3Þ j s� ¼ 1 � m�1½xð1; 2; d2Þ j s� ¼ 1 � 3ð3 þ dÞð1 � 3s2Þdð3 þ 2dÞ ; ð44Þ

m�2½xð2; 3; d3Þ j s� ¼ 1 � m�2½xð2; 1; d1Þ j s� ¼3ð3 þ dÞð3s1 � 1Þ

dð3 þ 2dÞ ; ð45Þ

m�3½xð3; 2; d2Þ j s� ¼ 1; ð46Þ

viðsÞ ¼1

3ð1 � dÞ þ1 � 3si

3d; i ¼ 1; 2; 3: ð47Þ

Case g: s2413; s34

ð9�2d2Þ9ð3þdÞ :

di ¼3 � d

9; i ¼ 1; 2; 3; ð48Þ

m�1½xð1; 3; d3Þ j s� ¼ 1; ð49Þ

m�2½xð2; 3; d3Þ j s� ¼ 1 � m�2½xð2; 1; d1Þ j s� ¼3ð3 þ dÞð1 � 3s3Þ

dð3 þ 2dÞ ; ð50Þ

m�3½xð3; 2; d2Þ j s� ¼ 1 � m�3½xð3; 1; d1Þ j s� ¼ 1 þ 3ð3 þ dÞð1 � 3s2Þdð3 þ 2dÞ ; ð51Þ

viðsÞ ¼1

3ð1 � dÞ þ1 � 3si

3d; i ¼ 1; 2; 3: ð52Þ

Proof. Observe that Cases a and b of Proposition 1 subsume the cases with statesAD\D0 we analyzed in Sections 3.i–ii. It is tedious but straightforward to verify bydirect application of (4) that if players play the proposal strategies in Cases a–g andthese proposals obtain a majority, then continuation values are as reported inProposition 1. Then, on the basis of the definition in (5), we obtain players’ expectedutility functions, Ui; and additional algebraic manipulation shows that the reporteddemands are in accordance with Definition 2. Furthermore, all reported non-degenerate mixing probabilities are well defined.10

On the basis of the expected utility functions, Ui; we can then constructequilibrium voting strategies A�

i ðsÞ ¼ fx j UiðxÞXUiðsÞg; i ¼ 1; 2; 3; for all sAD:These voting strategies obviously satisfy equilibrium condition (8). Then, to proveProposition 1 it suffices to verify equilibrium condition (9). To do so, we make use ofthree additional lemmas. Lemma 3 establishes some properties of equilibriumdemands, i.e. that they sum to less than unity and that they are (weakly) ordered in

ARTICLE IN PRESS

10 All the above-mentioned calculations are available upon request.


accordance with the ordering of allocations under the state s: Lemma 4 thenestablishes that the proposal strategies for legislators i ¼ 1; 2; 3 in Proposition 1maximize UiðxÞ over all xAWðsÞ\D0; these proposals would then maximize UiðxÞover all xAWðsÞ if there is no xAWðsÞ-D0 that accrues i higher utility. We establishthat this is indeed the case in Lemma 5.

Lemma 3. For all sAD; the demands reported in Proposition 1 are such that (a)P3i¼1 diðsÞp1; and (b) siXsj ) diðsÞXdjðsÞ:

Proof. See the appendix. &

We now show that equilibrium proposals are optima over feasible alternatives inD\D0:

Lemma 4. mi½z j s�40 ) zAarg maxfUiðxÞ j xAWðsÞ\D0g; for all z; sAD:

Proof. All equilibrium proposals take the form of mwc-proposals xði; j; djÞ: Also,

whenever mi½xði; j; djÞ j s�40 and mi½xði; h; dhÞ j s�40; haj we have dh ¼ dj so that

Uiðxði; j; djÞÞ ¼ Uiðxði; h; dhÞÞ: Thus, in view of Lemma 2 it suffices to show that if

mi½xði; j; djÞ j s� ¼ 1; jai; then Uiðxði; j; djÞÞXUiðxði; h; dhÞÞ; hai; j; i.e. proposer i has

no incentive to coalesce with player h instead of j: To showUiðxði; j; djÞÞXUiðxði; h; dhÞÞ; haiaj it suffices to show dhXdj by part (b) of Lemma

1. We can trivially check that dhXdj is true in Proposition 1 since we have s1Xs2Xs3

and (by part (b) of Lemma 3) d1Xd2Xd3 and: when d2ad3 we havem1½xð1; 3; d3Þ j s� ¼ 1; when d1ad3 we have m2½xð2; 3; d3Þ j s� ¼ 1; and when d1ad2

we have m3½xð3; 2; d2Þ j s� ¼ 1: &

We conclude the proof by showing that optimum proposal strategiescannot belong in D0: In particular, we show that if an alternative in D0 beatsthe status quo by majority rule, then for any player i we can find anotheralternative in D\D0 that is also majority preferred to the status quo and improves i’sutility.

Lemma 5. Assume xAWðsÞ-D0; then for any i ¼ 1; 2; 3 there exists yAWðsÞ\D0 such

that UiðyÞXUiðxÞ:

Proof. Let xAWðsÞ-D0 and consider any i: Consider first the case xAA�i ðsÞ: Then, x

is weakly preferred to s by a majority of (at least) i and some jai: Now set y ¼xði; j; djðxÞÞ; where djðxÞ is the applicable demand from Proposition 1. We have

Ujðxði; j; djðxÞÞÞXUjðxÞ; by the definition of demand. From part (a) of Lemma 3 we

have diðxÞ þ djðxÞp1 and as a result xiði; j; djðxÞÞ ¼ 1 � djðxÞXdiðxÞ; hence,

Uiðxði; j; djðxÞÞÞX UiðxÞ; which follows from the weak monotonicity in part (b) of

Lemma 1. Thus, y ¼ xði; j; djðxÞÞAWðsÞ by a majority of i and j; and we have

completed the proof for this case. Now consider the case xeA�i ðsÞ; i.e. UiðsÞ4UiðxÞ:


Part (a) of Lemma 3 ensures that diðsÞ þ djðsÞp1; hence proposal y ¼ xði; j; djðsÞÞhas xiði; j; djðsÞÞXdiðsÞ: Then, UiðyÞXUiðsÞ4UiðxÞ; UjðyÞXUjðsÞ; and yAWðsÞ\D0:

The above hold for arbitrary i; and we have completed the proof. &

As a result of Lemmas 4 and 5, equilibrium proposals are optima over the entirerange of feasible alternatives. It then follows that proposal strategies in Cases a–g ofProposition 1 satisfy equilibrium condition (9) which completes the proof. &

It is easy to show that the MPNESUV in Proposition 1 is not unique. Themultiplicity of equilibria arises from the fact that the expected utility function

Uiðx; 1 � x; 0Þ in Eq. (21) is constant for xA ð9�d2Þ3ð6þdÞ; 1 � ð9�d2Þ

3ð6þdÞ

h i; i ¼ 1; 2: Thus,

proposer h and coalition partner j are indifferent among all proposals with xh ¼1 � z; xj ¼ z where zA ð9�d2Þ

3ð6þdÞ; 1 � ð9�d2Þ3ð6þdÞ

h i; whenever the demand of j is dj ¼ ð9�d2Þ

3ð6þdÞ: But

all the additional equilibria we obtain exploiting this feature of equilibriumexpected utility are payoff equivalent to the one we establish in Proposition 1. Thus,it remains an open question whether the class of MPNESUV for this gameare payoff equivalent, in analogy to the result of Eraslan [13] for the Baron–Ferejohnmodel.

Among the multiple payoff equivalent equilibria we can establish, the one wereport in Proposition 1 has the added feature that proposal strategies, mi½ j s�; areweakly continuous in the status quo, s: To be precise, we show that:

Proposition 2. The equilibrium proposal strategies m�i : S-YðDÞ in Proposition 1 are

such that for every sAD and every sequence snAD with sn-s; m�i ½ j sn� converges weakly

to m�i ½ j s�:

Proof. See the appendix. &

Thus, in equilibrium a small change in the status quo implies a small change inproposal strategies m�i ½ j s� and, by extension, to the equilibrium transition

probabilities Q½x j s� defined in Eq. (3). An immediate implication of the continuityof transition probabilities is the fact that continuation functions viðxÞ and expectedutility UiðxÞ are continuous. It is interesting to ask whether UiðxÞ inherits otherproperties of the stage utility function uiðxÞ: Unfortunately, the answer is negative:

Proposition 3. The expected utility function, Ui; induced by the equilibrium in

Proposition 1, (a) is continuous, (b) is not quasi-concave, and (c) if d40; Ui is constant

on a non-empty, open subset of D:

Proof. (a) follows from Proposition 2. To show (b), consider x1 ¼ ð13; 2

3; 0Þ; x2 ¼

ð13; 0; 2

3Þ; and define convex combinations xðlÞ ¼ lx1 þ ð1 � lÞx2: Since 1

3oð9�d2Þ

3ð6þdÞ; we

have U1ðx1Þ ¼ U1ðx2Þ ¼ 13þ dð 1

3ð1�dÞ þ 13ð3�dÞÞ ¼ 1

3ð1�dÞ þ d3ð3�dÞ: Also, U1ðxð1

2ÞÞ ¼


U1ð13; 1

3; 1

3Þ ¼ 1

3þ d 1

3ð1�dÞ ¼ 13ð1�dÞ: Clearly, U1ðxð1

2ÞÞo1

2U1ðx1Þ þ 1

2U1ðx2Þ: Finally, for (c)

UiðxÞ ¼ 13ð1�dÞ; i ¼ 1; 2; 3 for x in Cases f and g of Proposition 1. &

In Fig. 4 we depict the indifference contours induced by the equilibrium expectedutility UiðxÞ for various values of the discount factor d: Fig. 4a corresponds to the casethe discount factor is zero, and effectively depicts the indifference contours for thestage utility of players, uiðxÞ: For positive discount factors the indifference mapsbecome non-standard inducing acceptance sets, A�

i ðxÞ; that are non-convex. Also,

UiðxÞ; takes constant values and induces ‘thick’ indifference contours in Cases f and g

of Proposition 1. These are represented in Figs. 4b–d by the hexagon that forms partof the indifference contour at the center of the unit simplex. Thus, our analysis rulesout convexity of acceptance sets, A�

i ðxÞ; or lower-hemicontinuity of the acceptance sets

correspondence, A�i ðxÞ; as general properties of such bargaining games.

ARTICLE IN PRESS

Fig. 4. Indifference contours of equilibrium expected utility vs. discount factor. Key: Graphs (a)–(d) map

the equilibrium expected utility, UiðxÞ; of the player whose stage utility satiation point corresponds to the

top corner of each triangle. Arrows indicate direction of increasing utility. Successive indifference contours

represent an increase in utility by 1/15-th. The hexagons are part of ‘thick’ indifference contour (Cases f

and g of Proposition 1). Equilibrium expected utility fails quasi-concavity and induces non-convex

acceptance sets. Sources: Constructed by author on basis of Proposition 1.


We conclude this section with a brief discussion of two additional features of theequilibrium. First, we note that the equilibrium absorbing set, D2; is the only subset ofthe entire simplex D that does not belong to the uncovered set.11 Despite the fact thatthe set of covered alternatives has measure zero in the space of possible allocations D;equilibrium outcomes fall in that set with probability one in the long run.

Second, convergence of the equilibrium distribution of legislative decisions is fast.The long-run distribution of policy outcomes is a natural focus in such dynamicgames, but this focus is less justified if convergence to the steady-state distribution isslow. If that were the case for our equilibrium—and depending on the initialallocation of the dollar—legislative policy decisions might concentrate in an area ofrelatively equitable allocations for a significant period of time before eventualabsorption. This is not the case, since (except perhaps for the very first period) there

is probability 23

of absorption in D2; from any equilibrium allocation not in that set.

As a result, it is straightforward to show that the maximum expected time beforeabsorption to D2 is 2.5 periods.

5. Conclusions

We analyzed a three-player majority rule bargaining game with a recurringdecision over a divide-the-dollar policy space and an endogenous reversion point ordefault alternative. We provided a complete characterization of a Markov PerfectNash equilibrium for this dynamic game. The equilibrium is such that in the long runand irrespective of the discount factor or the initial allocation of the dollar theproposer obtains the whole dollar with probability one.

Besides establishing existence, we showed that the equilibrium expected utility ofplayers is continuous in current period’s decision while proposal strategies areweakly continuous in the state variable, i.e. the status quo. While equilibrium is wellbehaved in that respect, players’s expected utility is not quasi-concave and inducesthick indifference contours.

The non-standard form of equilibrium expected utility implies that we cannot relyon the lower-hemicontinuity of the proposers’ feasible set in the equilibrium analysisof this class of games. It remains an open question whether it is possible to showexistence of Markov equilibrium in these dynamic legislative bargaining games withmore general policy spaces.

Acknowledgments

I thank David Baron, John Duggan, David Epstein, Dino Gerardi, Greg Huber,Adam Meirowitz, Thomas Palfrey, Maggie Penn, and George Tsebelis for theircomments in various formats and occasions. I also thank seminar participants of the

ARTICLE IN PRESS

11 We use the following definition of the covering relation: y covers x if ygx and zgy ) zgx , where

g is the (strong) majority preference relation. See [12,21].


2001 APSA Conference and the 2001 Wallis Conference on Political Economy, andan associate editor of this journal. All errors are mine.

Appendix

In this appendix we prove Lemma 3 and Proposition 2. We start with the lemma:

Proof of Lemma 3. By symmetry it suffices to consider s such that s1Xs2Xs3; whencepart (b) reduces to showing d1Xd2Xd3: We have the following cases:

Case a: The two possible values of d1 are ranked: we have 1 � s2 � 9�d2

9s3X

3�d3

�9s2þð9�d2Þs3

3ð3þdÞ 39ð1 � s2 � s3Þ þ d2s3 þ 3dX0; which is true. As a result, to show (a) it

suffices to consider d1 ¼ 1 � s2 � 9�d2

9s3 and we have

P3i¼1 diðsÞ ¼ 1 � s2 � 9�d2

9s3 þ

s2p13� 9�d2

9s3p0; which clearly holds. For (b), we first have that d2 ¼ s2Xd3 ¼

0; so it remains to show d1Xd2: By the ranking of the possible values of d1 it suffices

to show d1 ¼ 3�d3

� 9s2þð9�d2Þs33ð3þdÞ Xd2; which is equivalent to 3�d

3� 9s2þð9�d2Þs3

3ð3þdÞ Xs2; which

is equivalent to s3p1 � 3ð6þdÞð9�d2Þ s2; which holds in Case a, hence we have completed

part (b).

Case b:P3

i¼1 diðsÞ ¼ 2 ðs1þs2Þð9�d2Þ3ð6þdÞ p13s1 þ s2p

ð18þ3dÞð18�2d2Þ; and the latter is clearly

true since ð18þ3dÞð18�2d2ÞX1; hence (a) holds. For (b), we clearly have d1 ¼ d2Xd3 ¼ 0:

Case c: The two possible values of d1 are ranked: we have

1 � ð81�54d�27d2þ6d3þ2d4Þs2

3ð27�18d�6d2þ2d3Þ � ð81�27d�27d2þ3d3þ2d4Þs3

3ð27�18d�6d2þ2d3Þ

Xð3�dÞ

3� ð9�3d�2d2Þð3�dÞ

ð27�18d�6d2þ2d3Þs3 � ð9�6d�2d2Þð3�dÞð27�18d�6d2þ2d3Þs2

39ds2 � 6d2s2 � 3d2s3 þ ð27 � 18d� 6d2 þ 2d3Þð1 � s2 � s3ÞX0;

which is true since both ð9ds2 � 6d2s2 � 3d2s3Þ and the remaining term are positive.Thus, to show (a) it suffices to consider

d1 ¼ 1 � ð81�54d�27d2þ6d3þ2d4Þs2


3ð27�18d�6d2þ2d3Þ :

We have

X3

i¼1diðsÞ ¼ 1 � ð81�54d�27d2þ6d3þ2d4Þs2


3ð27�18d�6d2þ2d3Þ

þ ð9�d2Þðð3�2dÞs2�ds3Þ27�2dð9þ3d�d2Þ þ ð3�dÞðð9�d2Þs3�3ds2Þ

27�2dð9þ3d�d2Þ

p13� 13ðð27�18d�3d2þ2d3Þdð27�18d�6d2þ2d3Þ s3 þ dð27�27dþ2d3Þ

ð27�18d�6d2þ2d3Þ s2Þp0;


which is true hence (a) holds. For (b) it suffices to consider

d1 ¼ ð3�dÞ3 � ð9�3d�2d2Þð3�dÞ

ð27�18d�6d2þ2d3Þ s3 � ð9�6d�2d2Þð3�dÞð27�18d�6d2þ2d3Þ s2;

we have

d1Xd233�d

3þ ð3�dÞð3ds2�ð9�3d�2d2Þðs2þs3ÞÞ

27�2dð9þ3d�d2Þ Xð9�d2Þðð3�2dÞs2�ds3Þ

27�2dð9þ3d�d2Þ

3 s2p27�2dð9þ3d�d2Þ

3ð18�9d�4d2Þ � 3ð1�dÞð3þdÞð18�9d�4d2Þ s3;

which is true in Case c: Likewise,

d2Xd33ð9�d2Þðð3�2dÞs2�ds3Þ

27�2dð9þ3d�d2Þ Xð3�dÞðð9�d2Þs3�3ds2Þ

27�2dð9þ3d�d2Þ 3s3pð9�2d2Þ3ð3þdÞ s2;

which is also true.

Case d:P3

i¼1 diðsÞ ¼ 2 ð2d�3þ3s3Þð9�d2Þ3ð4d2þ9d�18Þ þ dð3�dÞ�2ð9�d2Þs3

ð4d2þ9d�18Þ ¼ 1 � d3p13dX0; which is

true and establishes (a). For (b) we have

d1 ¼ d2Xd33ð2d�3þ3s3Þð9�d2Þ

3ð4d2þ9d�18Þ Xdð3�dÞ�2ð9�d2Þs3

ð4d2þ9d�18Þ 3s3p9�2d2

9ð3þdÞ;

which holds in Case d:

Case e: The two possible values of d1 are ranked: we have 1 þ ð2d3þ6d2�18d�54Þs2

3ð18þ3d�2d2Þ þð6d2þ2d3�18d�54Þs3

3ð18þ3d�2d2Þ Xð3�dÞ

3� ðs2þs3Þ2ð9�d2Þ

ð18þ3d�2d2Þ 3ð18 � 2d2Þð1 � s2 � s3Þ þ 3dX0; which

clearly holds. Thus to show (a) it suffices to consider d1 ¼ 1 þ ð2d3þ6d2�18d�54Þs2

3ð18þ3d�2d2Þ þð6d2þ2d3�18d�54Þs3

3ð18þ3d�2d2Þ : We then haveP3

i¼1 diðsÞ ¼ 1 þ ð2d3þ6d2�18d�54Þs2

3ð18þ3d�2d2Þ þ ð6d2þ2d3�18d�54Þs3

3ð18þ3d�2d2Þ þ

2 ð9�d2Þðs2þs3Þð18þ3d�2d2Þp13ð18d� 2d3Þðs2 þ s3ÞX0; which is true and proves (a). For (b) it

suffices to consider d1 ¼ ð3�dÞ3 � ðs2þs3Þ2ð9�d2Þ

ð18þ3d�2d2Þ and we have d1Xd2 ¼ d333�d

3 �2ð9�d2Þðs2þs3Þð18þ3d�2d2Þ X

ð9�d2Þðs2þs3Þð18þ3d�2d2Þ3

ð18þ3d�2d2Þ9ð3þdÞ Xs2 þ s331 � ð18þ3d�2d2Þ

9ð3þdÞ p1 � s2 �

s33s1X9þ2dð3þdÞ

9ð3þdÞ ; which holds in Case e:

Cases f, g: We haveP3

i¼1 diðsÞ ¼ 3 3�d9

¼ 1 � d3p1; that is true and establishes (a),

while (b) holds trivially. &

We now proceed to the proof of Proposition 2:

Proof of Proposition 2. The equilibrium is such that m�i ½ j s� has mass on at most two

points, xði; j; djðsÞÞ and xði; h; dhðsÞÞ; iaj; h; jah: It suffices to show that these

proposals (when played with positive probability) and associated mixing probabil-ities are continuous in s: Then clearly sn-s impliesX

jai

m�i ½xði; j; djðsnÞÞ j sn�f ðxði; j; djðsnÞÞÞ-Xjai

m�i ½xði; j; djðsÞÞ j s�f ðxði; j; djðsÞÞÞ


for any bounded, continuous f ; which establishes weak convergence [7]. Continuityholds in the interior of Cases a–g in Proposition 1, so it remains to check theboundaries of these cases. In order to distinguish the various applicable functionalforms we shall write dw

h and m�wi ½ j s� where wAfa; b; c; d; e; f ; gg identifies the case for

which the respective functional form applies and where the argument from demandsdw

h ðsÞ is dropped throughout for compactness. We have:

* Boundary of Cases a and b: At the boundary we have s3 ¼ 1 � 3ð6þdÞð9�d2Þ s2; then

m�b3 ½xð3; 2; d2Þ j s� ¼ 9þdð3þdÞ

dð3þ2dÞ � 3ð6þdÞs2dð3þ2dÞð1�s3Þ

¼ 1 ¼ m�a3 ½xð3; 2; d2Þ j s�: Also db

2 ¼ðs1þs2Þð9�d2Þ

3ð6þdÞ ¼ s2 ¼ da2 : Clearly m�a

i ½xði; 3; da3 Þ j s� ¼ m�b

i ½xði; 3; db3 Þ j s� ¼ 1 and da

3 ¼db

3 ¼ 0:

* Boundary of Cases a and c: At the boundary we have s3 ¼ 3dð9�d2Þ s2: Then dc

3 ¼ð3�dÞðð9�d2Þs3�3ds2Þ

27�2dð9þ3d�d2Þ ¼ 0 ¼ da3 : Also dc

2 ¼ ð9�d2Þðð3�2dÞs2�ds3Þ27�2dð9þ3d�d2Þ ¼ s2 ¼ da

2 :

* Boundary of Cases b and d: At the boundary we have s3 ¼ d2ð3þdÞ: Thus

m�b3 ½xð3; 2; d2Þ j s� ¼ 9þdð3þdÞ

dð3þ2dÞ � 3ð6þdÞs2

dð3þ2dÞð1� d2ð3þdÞÞ

¼ ð9þ3dþd2Þ�3ð6þ2dÞs2dð3þ2dÞ :

Likewise,

m�d3 ½xð3; 2; d2Þ j s� ¼ 27�dð9þ6dþ2d2Þ

dð3þ2dÞð3�3d

2ð3þdÞ�2dÞ�

3ð18�9d�4d2Þs2þ3ð9�3d�d2Þ d2ð3þdÞ

dð3þ2dÞð3�3d

2ð3þdÞ�2dÞ¼ ð9þ3dþd2Þ�3ð6þ2dÞs2

dð3þ2dÞ :

With regard to demands, we have

db2 ¼ ðs1þs2Þð9�d2Þ

3ð6þdÞ ¼ð1� d

2ð3þdÞÞð9�d2Þ3ð6þdÞ ¼ 3�d

6; and dd

2 ¼ð2d�3þ3

d2ð3þdÞÞð9�d2Þ

3ð4d2þ9d�18Þ ¼ 3�d6:

Also

dd3 ¼ dð3�dÞ�2ð9�d2Þs3

ð4d2þ9d�18Þ ¼dð3�dÞ�2ð9�d2Þ d

2ð3þdÞð4d2þ9d�18Þ ¼ 0 ¼ db

3 :

* Boundary of Cases c and d: At the boundary s2 ¼ 27�2dð9þ3d�d2Þ3ð18�9d�4d2Þ � 3ð1�dÞð3þdÞ

ð18�9d�4d2Þs3: We

have


dð3þ2dÞð3�3s3�2dÞ �3ð18�9d�4d2Þs2þ3ð9�3d�d2Þs3

dð3þ2dÞð3�3s3�2dÞ ¼ 1:

Also

dd2 ¼ ð2d�3þ3s3Þð9�d2Þ

3ð4d2þ9d�18Þ while dc2 ¼ ð9�d2Þðð3�2dÞs2�ds3Þ

27�2dð9þ3d�d2Þ ¼ ð2d�3þ3s3Þð9�d2Þ3ð4d2þ9d�18Þ :


Finally,


ð4d2þ9d�18Þ while dc3 ¼ ð3�dÞðð9�d2Þs3�3ds2Þ

27�2dð9þ3d�d2Þ ¼ dð3�dÞ�2ð9�d2Þs3

ð4d2þ9d�18Þ :

* Boundary of Cases c and e: At the boundary we have s3 ¼ ð9�2d2Þ3ð3þdÞ s2: Then

m�e1 ½xð1; 3; d3Þ j s� ¼ 3ð3þdÞs2�ð9�2d2Þs3

dð3þ2dÞðs2þs3Þ¼ 1: Also dc

3 ¼ ð3�dÞðð9�d2Þs3�3ds2Þ27�2dð9þ3d�d2Þ ¼ ð3�dÞ

3s2; and

dc2 ¼ ð9�d2Þðð3�2dÞs2�ds3Þ

27�2dð9þ3d�d2Þ ¼ ð3�dÞ3

s2; while de2 ¼ de

3 ¼ ð9�d2Þðs2þs3Þð18þ3d�2d2Þ ¼ ð3�dÞ

3s2:

* Boundary of Cases d and g: At the boundary we have s3 ¼ ð9�2d2Þ9ð3þdÞ : Then,


dð3þ2dÞð3�3s3�2dÞ �3ð18�9d�4d2Þs2þ3ð9�3d�d2Þs3

dð3þ2dÞð3�3s3�2dÞ

¼ 1 þ 3ð3þdÞð1�3s2Þdð3þ2dÞ ¼ m�g

3 ½xð3; 2; d2Þ j s�:

Also,

m�g2 ½xð2; 3; d3Þ j s� ¼ 3ð3þdÞð1�3s3Þ

dð3þ2dÞ ¼ 1:

Finally,

dd2 ¼ ð2d�3þ3s3Þð9�d2Þ

3ð4d2þ9d�18Þ ¼ð2d�3þ3

ð9�2d2Þ9ð3þdÞ Þð9�d2Þ

3ð4d2þ9d�18Þ ¼ 3�d9

¼ dg2 ;

and


ð4d2þ9d�18Þ ¼ 3�d9

¼ dg3 :

* Boundary of Cases e and f : At the boundary we have s1 ¼ 9þ2dð3þdÞ9ð3þdÞ : Then,

m�e1 ½xð1; 3; d3Þ j s� ¼ 3ð3þdÞs2�ð9�2d2Þs3

dð3þ2dÞðs2þs3Þ¼

3ð3þdÞs2�ð9�2d2Þð1�s2�9þ2dð3þdÞ

9ð3þdÞ Þ

dð3þ2dÞð1�9þ2dð3þdÞ9ð3þdÞ Þ

¼ 1 � 3ð3þdÞð1�3s2Þdð3þ2dÞ ;

while

m�f1 ½xð; 3; d3Þ j s� ¼ 1 � 3ð3þdÞð1�3s2Þ

dð3þ2dÞ :

Also,

m�f2 ½xð2; 3; d3Þ j s� ¼ 3ð3þdÞð3s1�1Þ

dð3þ2dÞ ¼ 1:

Lastly, de2 ¼ de

3 ¼ ð9�d2Þðs2þs3Þð18þ3d�2d2Þ ¼

ð9�d2Þð1�9þ2dð3þdÞ9ð3þdÞ Þ

ð18þ3d�2d2Þ ¼ 3�d9

¼ df

2 ¼ df

3 :


* Boundary of Cases f and g: At the boundary we have s2 ¼ 13: Clearly d

fi ¼ d

gi :

With regard to mixing probabilities we have

m�f1 ½xð1; 3; d3Þ j s� ¼ 1 � 3ð3þdÞð1�3s2Þ

dð3þ2dÞ ¼ 1:

Also,

m�f2 ½xð2; 3; d3Þ j s� ¼ 3ð3þdÞð3s1�1Þ

dð3þ2dÞ and m�g2 ½xð2; 3; d3Þ j s�

¼ 3ð3þdÞð1�3s3Þdð3þ2dÞ ¼ 3ð3þdÞð1�3ð1�s1�

13ÞÞ

dð3þ2dÞ ¼ 3ð3þdÞð3s1�1Þdð3þ2dÞ :

Finally,

m�g3 ½xð3; 2; d2Þ j s� ¼ 1 þ 3ð3þdÞð1�3s2Þ

dð3þ2dÞ ¼ 1: &

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a three-player dynamic majoritarian bargaining game

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