equivalence of optimal noisy-ranking contests and tullock contests

9
Journal of Mathematical Economics 47 (2011) 740–748 Contents lists available at SciVerse ScienceDirect Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco Equivalence of optimal noisy-ranking contests and Tullock contests Yohan Pelosse GATE Lyon Saint-Etienne, CNRS UMR 5824 - Université Lyon 2, 93 chemin des Mouilles - 69131 - ECULLY cedex, France article info Article history: Received 7 January 2011 Received in revised form 2 August 2011 Accepted 23 October 2011 Available online 30 October 2011 Keywords: Contest success function Equivalent contests Probabilistic choice abstract We analyze a noisy-ranking contest in which participants compete in several dimensions. The organizer randomly samples a number of dimensions and awards a prize to the most productive agent. When the contest is optimally designed, we establish a structural equivalence between this family of noisy-ranking contests and contests built upon Tullock contest success functions. Our result also shows that in this class of noisy-contests, the profit-maximization problem of the planner can be turned into a stochastic choice problem for a planner who has some deterministic preferences over the contestants’ win probabilities. © 2011 Elsevier B.V. All rights reserved. 1. Introduction A contest is a game in which players exert effort to win a certain prize. Lobbying activities, research and development races, and competitions for promotions, to name a few, all have this property. To model a contest requires specification of each player’s contest success function (CSF), which describes the relationship between each player’s effort level and his probability of winning. There, a widely adopted approach is the lottery contest model. One of its most popular special case is the Tullock contest model (see Tullock, 1980) where the probability that a contestant wins is given by the ratio of the output of his effort with respect to the total output supplied by all contestants. A handful of scholarly papers, Dasgupta and Nti (1998), Epstein and Nitzan (2006) and Corchón and Dahm (2011) bring a mechanism design perspective to the study of contests in order to provide some decision theoretic justification for this particular class of CSFs. 1 Much of this recent literature is based on the assumption that win probabilities are part of the strategic choice variables of the designer. By postulating suitably chosen preferences for the contest organizer, 2 this literature provides a general approach to view popular CSFs, like the Tullock form, as the result of the optimal choice of a designer. Tel.: +33 0 4 72 86 60 60; fax: +33 0 4 72 86 60 90. E-mail address: [email protected]. 1 Other alternative microeconomic foundations for CSFs are provided by Dahm and Porteiro (2005) and Corchón and Dahm (2010). Blavatskyy (2010), Münster (2008) and Rai and Sarin (2009) offer axiomatic characterizations of contest success functions following the seminal paper by Skaperdas (1996). 2 We use the terms contest administrator, contest organizer, designer and planner interchangeably. In all these studies, the designer’s payoff is influenced by the contestants’ efforts or performance, but not by the rent or prize of the contest. This class of models is therefore in stark contrast to the dominant view of mediated contests (e.g., Lazear and Rosen, 1981; Nalebuff and Stiglitz, 1983). In these models, the realized output of each agent is a stochastic function of his effort and the organizer’s objective function is the sum of the outputs of all agents minus the prize paid to the winner. In the economic literature, such noisy-ranking contests are conventionally termed ‘‘rank- order tournaments’’. This literature has notably shown how prizes based on rank-orders of performance can be effectively used to provide incentives in internal labor markets. In addition, the labor tournament model has been extended to analyze political lobbying and research contests. 3 A natural question then is whether there exists a structural equivalence between the family of noisy-ranking contests designed as an optimally incentive mechanism and the class of Tullock Contests. Otherwise stated, can we derive win probabilities of the Tullock form when the designer cares about the total expected output but also values the prize he hands out to the winner? To address this question, we adopt an extended version of the standard noisy-ranking contest in which participants compete in several dimensions, as suggested by Dubey and Wu (2001). 4 Con- testants’ efforts give rise to a stochastic output in each dimension that the organizer values. The organizer chooses randomly a sam- ple of dimensions and allocates the prize through comparing the 3 For example, Che and Gale (1998) use an all-pay auction with complete information to model political campaigns, and show how a cap on individual political contribution may actually increase aggregate expenditures. 4 Our model is based on the ‘‘low-information’’ case of the multi-period tournament model developed by these authors. 0304-4068/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2011.10.005

Upload: yohan-pelosse

Post on 05-Sep-2016

223 views

Category:

Documents


3 download

TRANSCRIPT

Page 1: Equivalence of optimal noisy-ranking contests and Tullock contests

Journal of Mathematical Economics 47 (2011) 740–748

Contents lists available at SciVerse ScienceDirect

Journal of Mathematical Economics

journal homepage: www.elsevier.com/locate/jmateco

Equivalence of optimal noisy-ranking contests and Tullock contestsYohan Pelosse ∗

GATE Lyon Saint-Etienne, CNRS UMR 5824 - Université Lyon 2, 93 chemin des Mouilles - 69131 - ECULLY cedex, France

a r t i c l e i n f o

Article history:Received 7 January 2011Received in revised form2 August 2011Accepted 23 October 2011Available online 30 October 2011

Keywords:Contest success functionEquivalent contestsProbabilistic choice

a b s t r a c t

We analyze a noisy-ranking contest in which participants compete in several dimensions. The organizerrandomly samples a number of dimensions and awards a prize to the most productive agent. When thecontest is optimally designed, we establish a structural equivalence between this family of noisy-rankingcontests and contests built upon Tullock contest success functions. Our result also shows that in this classof noisy-contests, the profit-maximization problem of the planner can be turned into a stochastic choiceproblem for a planner who has some deterministic preferences over the contestants’ win probabilities.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

A contest is a game inwhich players exert effort towin a certainprize. Lobbying activities, research and development races, andcompetitions for promotions, to name a few, all have this property.To model a contest requires specification of each player’s contestsuccess function (CSF), which describes the relationship betweeneach player’s effort level and his probability of winning. There, awidely adopted approach is the lottery contest model. One of itsmost popular special case is the Tullock contestmodel (see Tullock,1980) where the probability that a contestant wins is given by theratio of the output of his effort with respect to the total outputsupplied by all contestants.

A handful of scholarly papers, Dasgupta and Nti (1998),Epstein and Nitzan (2006) and Corchón and Dahm (2011) bringa mechanism design perspective to the study of contests inorder to provide some decision theoretic justification for thisparticular class of CSFs.1 Much of this recent literature is basedon the assumption that win probabilities are part of the strategicchoice variables of the designer. By postulating suitably chosenpreferences for the contest organizer,2 this literature provides ageneral approach to view popular CSFs, like the Tullock form, asthe result of the optimal choice of a designer.

∗ Tel.: +33 0 4 72 86 60 60; fax: +33 0 4 72 86 60 90.E-mail address: [email protected].

1 Other alternative microeconomic foundations for CSFs are provided by Dahmand Porteiro (2005) and Corchón and Dahm (2010). Blavatskyy (2010), Münster(2008) and Rai and Sarin (2009) offer axiomatic characterizations of contest successfunctions following the seminal paper by Skaperdas (1996).2 We use the terms contest administrator, contest organizer, designer and

planner interchangeably.

0304-4068/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.jmateco.2011.10.005

In all these studies, the designer’s payoff is influenced by thecontestants’ efforts or performance, but not by the rent or prize ofthe contest. This class of models is therefore in stark contrast tothe dominant view of mediated contests (e.g., Lazear and Rosen,1981; Nalebuff and Stiglitz, 1983). In these models, the realizedoutput of each agent is a stochastic function of his effort and theorganizer’s objective function is the sumof the outputs of all agentsminus the prize paid to the winner. In the economic literature,such noisy-ranking contests are conventionally termed ‘‘rank-order tournaments’’. This literature has notably shown how prizesbased on rank-orders of performance can be effectively used toprovide incentives in internal labor markets. In addition, the labortournamentmodel has been extended to analyze political lobbyingand research contests.3 A natural question then is whether thereexists a structural equivalence between the family of noisy-rankingcontests designed as an optimally incentive mechanism and theclass of Tullock Contests. Otherwise stated, can we derive winprobabilities of the Tullock formwhen the designer cares about thetotal expected output but also values the prize he hands out to thewinner?

To address this question, we adopt an extended version of thestandard noisy-ranking contest in which participants compete inseveral dimensions, as suggested by Dubey and Wu (2001).4 Con-testants’ efforts give rise to a stochastic output in each dimensionthat the organizer values. The organizer chooses randomly a sam-ple of dimensions and allocates the prize through comparing the

3 For example, Che and Gale (1998) use an all-pay auction with completeinformation to model political campaigns, and show how a cap on individualpolitical contribution may actually increase aggregate expenditures.4 Our model is based on the ‘‘low-information’’ case of the multi-period

tournament model developed by these authors.

Page 2: Equivalence of optimal noisy-ranking contests and Tullock contests

Y. Pelosse / Journal of Mathematical Economics 47 (2011) 740–748 741

resulting sum of perceived outputs of each contestant. As contes-tants’ outputs accrue to the benefits of the contest organizer, heallocates the prize to the contestant who obtains the highest per-ceived output on the sample. The aim of the designer is to elicit acertain profile of effort levels spread over the different dimensionsof the contest from the two contestants, at the least cost. Whencontestants compete in several dimensions, the designer can steerthe win probabilities in a particular direction by selecting the sizeof a random sample of dimensions that are relevant towin the con-test. This framework therefore extends the standard noisy-rankingcontest model of tournaments with an indivisible prize (see Lazearand Rosen, 1981; Green and Stokey, 1983; Nalebuff and Stiglitz,1983; Rosen, 1986) to a situation where the organizer running thecontest can decide on themagnitude of the prize aswell as the pre-cision with which to observe the contestants’ outputs. To motivatethis framework, here are two examples.(1) Tournaments in internal labor markets: A manager decides to

award a bonus or a promotion to a higher echelon, to the bestsalesman of the year. To motivate the salesmen, he announcesa bonus to be bestowed as follows. He will perform an annualspot-check of sales by picking a random sample of 1 ≤ n ≤ Kdays and reward the salesmanwith the highest volume of salesin the sample.

(2) Rent-seeking contests with a committee: Our setup also fits sit-uations where there is a single dimension but the winnerof the contest is not chosen by a single designer but by acommittee.5 In this case, efforts are therefore interpreted asthe amount of resources devoted to persuading each com-mittee member (think of two researchers who compete for agrant). Here, the K dimensions coincide with the K potentialcommittee members. The composition of the committee de-pends on the designer’s preferences, (think of an anonymousresearch grants board together with the referees) and contes-tants do not know the actual committee members.6

We consider contestants with heterogeneous skills and focuson contests in which the organizer awards a prize only tothe best-performing agent on the sample. By definition, thissimple aggregation rule follows the time-honored tradition ofone-dimensional contests.7 Hence, this is in line with our mainobjective, which is to bridge noisy-ranking contests where the ruleis precisely of the ‘‘all-pay auctions’’ type, with Tullock contests.8Furthermore, a more complicated (weighted) rule would a prioricontradict the assumption that dimensions are substitutes.

An implication of making the prize endogenous is that thedesigner must take into account the welfare effects of his decisionon contestants, which in a contest depend on contestants’ disutilityfor effort and prize valuations. In this sense, our analysis bridgesthe gap between the contest literature and the recent literatureon political economy in which the decider takes into account thewelfare of rent-seekers (see Grossman and Helpman, 2001). Ourmain result delineates the class of noisy-ranking contests thathappen to boil down to a Tullock contest when the number ofdimensions n has been optimally chosen by the planner. Hence,in equilibrium, it is as if contestants were playing a rent-seekinggame with a specified Tullock-form CSF, thereby establishing an

5 As suggested by Nitzan (1994) and Amegashie (2002), the decision of awardinga rent is often made by a committee and not by an individual rent setter.6 Alternatively, the present setup is also analytically consistent with a

(randomized) voting rule (see e.g. Congleton (1984) andWeibull and Laslier (2009)),according to which all the K members of the committee simultaneously cast theirvotes, whereafter a random sample of 1 ≤ n ≤ K of these votes is drawn and thecollective decision is made.7 See e.g. Konrad (2009).8 A complete analysis of the optimal aggregation rules is out of the scope of this

paper.

equivalence result between the two seemingly unrelated familiesof noisy-ranking contests and Tullock contests. As a result, thetwo structures cannot be distinguished from each other as longas we are concerned with the optimal design of the noisy-rankingcontest.

In order to construct our isomorphism between these twofamilies of contests, we confine attention to the benchmark caseof a planner who wants to implement symmetric equilibria withthe particular property that the same effort is spread out alongall dimensions, at the least cost (i.e. the prize implementingthis profile is minimum).9 This is consistent with the planner’sobjectives, since he values agents’ effort along each dimensionequally and independently.10

Our equivalence result builds on the following set of observa-tions.When the sample size increases, the information of the plan-ner about the skill of each agent increases. Our characterizationdelineates the conditions where the prize happens to be a truemeasure of this information (in terms of information theory). Thisway, we can formally think of the value of the prize implement-ing a particular profile as the amount of information obtainedby the planner when he observes a given number of dimensions.When the information increases (n increases), the probability thatthe planner fails to take notice of the superior skill of the best-performing agent decreases. Conversely, if the planner’s choice ofa sample size is too small, this handicaps the best-skilled agent,thereby increasing the uncertainty of the outcome of the contest.In this context, the planner’s problem can be equivalently reformu-lated as the problem of finding the optimal quantity of informationhe has to acquire about the contestants during the competition.This results in a probabilistic choice model which has as a specialcase win probabilities of the multinomial logit form.

This structural equivalence is the result of an underlyingstochastic choice model, based on deterministic (and known)preferences over lotteries (win probabilities) which turns out tobe similar to Machina (1985). The bulk of our proofs is precisely todelineate and characterize the class of contests where such a classof decision problems arises for the planner. More precisely, in ourmodel the planner faces a set of alternatives, A = {1wins, 2wins}.However, he also has to incur the cost of implementing a profile asan equilibrium, a cost that depends on the sample size n. In fact,even if the planner is only concerned about the implementation ofa certain profile at the least cost (he is not interested in promotinga particular agent), he must optimally handicap the best skilled-agent (by choosing a sufficiently small sample size) in order toachieve these objectives. As the win probabilities are induced bythe size of the random sample, the planner has some preferencesover these lotteries.Related Literature

We are not aware of any work studying a structural connectionbetween Tullock contests and mediated contests that mirrors astandard moral hazard setting. However, this paper is closelyrelated to the literature that attempts to bridge different contestmodeling approaches.

Baye and Hoppe (2003) reveal the strategic equivalence ofresearch tournament models (Fullerton and McAfee, 1999), PatentRace models (Dasgupta and Stiglitz, 1980) and winner-take-allTullockContestmodels. Another strandof the literature establishes

9 In the presentmulti-dimensional contest, a thorough characterization of the setof pure NE is out of the scope of this paper. The case where the planner implementsthe maximal effort profile corresponds to the situation analyzed by Dubey and Wu(2001) in their ‘‘low-information’’ case.10 Moreover, if the planner has decreasingmarginal benefits from the contestants’sum of efforts in each dimension, then hewill induce them to assign the same effortlevel to each dimension.

Page 3: Equivalence of optimal noisy-ranking contests and Tullock contests

742 Y. Pelosse / Journal of Mathematical Economics 47 (2011) 740–748

some stochastic equivalence between certain families of noisy-ranking contest models and the family of Tullock contests, whichgives a stochastic foundation to the ratio-form. In this literature,noise enters additively in performance and is distributed asthe extreme value distribution to obtain the logit specification(McFadden, 1974). The idea of deriving the ratio form of CSFs fromsome distributional assumptions was first proposed by Hirshleiferand Riley (1992). This was extended by Jia (2008) to an arbitrarynumber of players.11 In related work, Baye and Hoppe (2003) offermicrofoundations for a subset of CSFs of the ratio-form in thecontext of innovation tournaments and patent races. Our workis also related to Fu and Lu (2008) who establish a stochasticequivalence between noisy ranking contests and Tullock contests.

What is new in our approach is that our equivalence is notstochastic but structural. It is based on the fact that the searchfor an optimal profit for the planner corresponds to a class ofstochastic decision problems as in Machina (1985). Our approachdepends on the presence of a planner and is thus very muchin line with the burgeoning literature in which CSFs are derivedas an optimal choice of a contest administrator (Corchón andDahm, 2011; Dasgupta and Nti, 1998; Epstein and Nitzan, 2006).Epstein and Nitzan (2006) partially rationalize CSFs by analyzinghow a contest administrator rationally decides whether to havea contest and if a contest takes place how he chooses among afixed set of CSFs. The main motivation for their work and oursare thus similar. However, in most of these papers, the main aimis to provide a foundation for popular CSFs, while we explorethe common thread that connects seemingly disparate familiesof contests within the framework of Tullock form contest successfunctions. Our approach ismost closely related to Dasgupta andNti(1998) who find some conditions where the classical ratio-formCSF is optimal in a situation, where the contest designer valuesthe prize. Albeit in a quite different setting, we pursue their lineof inquiry by giving the planner the power to determine the size ofthe prize.

This paper is also related to the small literature on multi-dimensional contests. Clark and Konrad (2007) extend thestandard Tullock model to several dimensions. Hence, from thisperspective, our equivalence result is thus a way to justify thisclass of models. Our setup builds upon the multi-period contestanalyzed by Dubey and Wu (2001).12 Dubey and Wu demonstratethe need for a reduced scrutiny (optimal sample sizes of theprincipal must be small) as agents acquire more information abouteach other as the game unfolds over time. One of the differenceswith their initial setup is that we obtain a similar phenomenon byallowing for sufficient variability in the skills of agents.

The remainder of the paper proceeds as follows. We set up themodel and present general existence and uniqueness conditions inSection 2. In Section 3, we present the main result of the paper. Allproofs are relegated to Appendix.

2. Model

2.1. Notation and setup

We analyze a generalized version of a mediated noisy-rankingcontest model in the spirit of Lazear and Rosen (1981). Twocontestants 1 and 2 take part in a contest that is designed by a

11 Formally Jia (2008) shows that if the error term is multiplicative with anexponential distribution, the CSF then has the Tullock form.12 In a companion paper, Dubey andHaimanko (2003) complements the approachof Dubey and Wu (2001) by fixing the prize and examining the variable behaviorinduced by the prize. The presentmodel corresponds to the ‘‘low-information’’ caseof their setup, when agents know only the time period they are in.

planner. This contest is comprised of a finite number of K ≥ 2dimensions. In each dimension, k = 1, . . . , K , contestant i (=1, 2)chooses an effort level xik in a finite set X ⊂ R+ of effort levels. Apure strategy for contestant i is a K -tuple xi = (xi1, . . . , xiK ). Thefiniteness of the set of effort levels is for simplicity of exposition.13Each contestant’s effort outlay xik in dimension k is not directlyobservable to the contest organizer: the administrator perceives anoisy output (or signal) yik about contestant i’s effort in dimensionk. The planner randomly samples n dimensions out of these Koutput realizations. For simplicity, we assume that the planneruses the time-honored all-pay auction rule: he selects the agentwho obtains the largest sum of output on the sample. In the eventof a tie, each agent wins with probability 1/2.14

Contestants know that exactly n out of the K dimensionsare required to win the contest. However, the n dimensions aresampled at random by the planner. So contestants exert effortwithout knowing which sample of size 1 ≤ n ≤ K is relevant towin the contest.

Formally, let θn ⊆ {1, . . . , K} be a sample of size n. Given therealization of outputs (yik)Kk=1 and the choice of a (random) sampleθn, contestant i (=1, 2) obtains a total output of−k∈θn

yik ≡ gi(xi, θn).

The prize is awarded to the contestant for whom gi(xi, θn) is thelargest.15

2.2. Payoffs

Let (X)K × (X)K ≡ X be the set of all pure strategy profiles.Let xi denote the pure strategy wherein contestant i expendsthe lowest effort in every dimension k while xi represents thestrategy wherein contestant i exerts the maximal effort level ineach dimension k. Fix two strategies, xi = (xi1, . . . , xiK ) andxi =

(xi1, . . . ,xiK ), in (X)K . In the remainder of the paper, we say thatxi ≻xi iff ∑k xik >

∑kxik for any xi,xi ∈ (X)K . As in any standard

noisy-ranking contest, contestants value the prize b. For simplicity,we assume that the valuation for the prize b is the same for bothplayers. Thereafter, let v(b) denote the (positive and finite) value ofthe prize b to player i (=1, 2)with v(0) = 0. We further postulatethat the valuation v : R+ → R+ is a continuous, strictly increasingmonotonic function, and contestants value the prize sufficiently sothat v(b) → ∞ when b → ∞.

Players also incur a cost from effort described by a strictlyincreasing (with respect to the order ≻), bounded and sufficientlyconvex function, ci : (X)K → R+ with ci(xi) = 016 for i (=1, 2).Thus, contestant i’s overall payoff equals, v(b) − ci(xi), if he winsand expends a stream of effort levels xi over the K dimensions, and−ci(xi) otherwise.

When the administrator chooses a random sample of size n andthe strategy profile is x = (x1, x2), contestant i’s expected payoffwrites,

pi(i wins | x; n)v(b)− ci(xi) ≡ Πni (x),

with pi(i wins | x; n) the ex ante probability for contestant i towin the prize b under strategy profile x and (random) sample of

13 Our analysis could be extended to a continuum set of effort levels by extendingsuitably the regularity conditions on payoffs.14 Nevertheless, the present model is robust to more general schemes. A proof ofthis statement is available upon request.15 Remark that when the planner does not sample the contestants’ outputs, themodel turns into a standard noisy all-pay auction (with several dimensions).16 This condition ensures that contestants’ equilibrium expected payoffs arepositive.

Page 4: Equivalence of optimal noisy-ranking contests and Tullock contests

Y. Pelosse / Journal of Mathematical Economics 47 (2011) 740–748 743

size n. As in any standard one-dimensional contests, we assumethat player i’s chances of winning are increasing in his own efforti.e. pi(i wins | (xi, x−i); n) is strictly increasing in xi with respectto the order ≻.17

2.3. Equilibrium concept

Let Πn≡ (Πn

1 ,Πn2 ) be the pair of expected payoff functions

under an arbitrary prize for contestants 1 and 2 when theadministrator’s sample size is n. Since we hold the total numberof possible dimensions K fixed throughout, Γ (Πn, b) denotes thenormal form game induced by a pair (b, n).

Each Γ (Πn, b) is a simultaneous-move game and the solutionconcept we use throughout the paper is that of a pure-strategyNash equilibrium (NE) of this game. Given x = (xi, x−i) with xi =

(xi1, . . . , xiK ), letx\xi ≡ (xi, x−i)be a strategyprofilewhenplayer iunilaterally deviates towards an arbitrary strategyxi ∈ (X)K . Thus,when the administrator considers n dimensions, we say that x is aNash equilibrium (NE) of Γ (Πn, b) if

Πni (x) ≥ Πn

i (x \xi) for any xi ∈ (X)K and i = 1, 2.

We shall focus on the implementation of symmetric equilibria inpure strategies, (x1, x2) ∈ X i.e. x1 = x2 = xi, with the propertythat the same effort is spread out along all dimensions of thecontest18 i.e. xik = xil, ∀k = l. Let the set of those symmetric purestrategy profiles be denoted by XSym. We say that a sample size n isimplementable for x ∈ XSym if the prize b required to induce x as anNE of Γ (Πn, b) is finite. For each x ∈ XSym, define, if it exists, theprize, b(n | x) ≡ min {b : x is an NE of Γ (Πn, b)}.

As we focus attention on the implementation of profileswherein both players put in the same effort level in eachdimension, the following result should come as no surprise.19

Lemma 1. Any effort strategy profile x ∈ XSym is implementable inany Γ (Πn, b(n | x)) of dimension 1 ≤ n ≤ K .

2.4. Contests with heterogeneous contestants

We first aim at delineating the class of such noisy-rankingcontests where the existence and the uniqueness of an optimalsample size is guaranteed.

Throughout, we assume the following.A.1 For every x ∈ XSym, pi(i wins | x; n) ≡ p1(n | x) is strictlymonotonically increasing in the sample size n (resp. p2(n | x) isstrictly monotonically decreasing).

A.1 means that contestant 1 is more skilled than contestant2 (think of the agents being drawn from diverse populations)and captures the idea that the review process of the planner isinformative.20

Let (X)KSym be the set of strategies xi = xi wherein contestanti (=1, 2) exerts the same effort level in each dimension. Fix aprofile x = (x1, x2) ∈ XSym and let

pi(i wins | x; n)− pi(i wins | x \xi; n) ≡ 1pi(i wins | x \xi; n)17 Notice that this is consistentwith the property that if contestant i exerts a largereffort xik > xik in a given dimension k, then his output y goes up, in the sense offirst-order stochastic dominance. In Dubey’s and Wu (2001), first-order stochasticdominance is imposed for the maximal effort level strategy.18 This is analogous to the procedure adopted by Klumpp and Polborn (2006) thatthey call the Symmetric Uniform Campaign Equilibrium.19 Indeed, the symmetry in effort brings about a situation that is similar to anyone-dimensional noisy-ranking contests that is ‘‘well-behaved’’ (see e.g. Lazear andRosen, 1981).20 Formally, A.1 is a way to stay away from the troublesome situations where winprobabilities fail to be strictly monotonic with respect to the sample size n.

withxi = xi, denote the increment in i’s probability of winningresulting from changes in own strategy (while keeping fixed hisopponent’s strategy at x−i).

Consider pi(., x−i) : (X)KSym × {1, . . . , K} → [0, 1] and saythat pi(., x−i) satisfies strict increasing differences (resp. strict de-creasing differences) in (xi, n) if1pi(i wins | x \xi; n) > (resp. <)1pi(i wins | x \xi; n′) whenever xi ≻ xi and n > n′,∀(xi; n) ∈

(X)KSym × {1, . . . , K} and (xi; n′) ∈ (X)KSym × {1, . . . , K}.A.2 For contestant 1, p1(., x2) has increasing differences while forcontestant 2, p2(., x1) has decreasing differences for all profiles(x1, x2) ∈ XSym.

In words, A.2 essentially says that contestant 1 (2) always(never) benefits from a larger sample size.

Let xi ∈ (X)K \ {xi} be the strategy of i which minimizes i’sincentive to put in xi and1ci(xi) ≡ ci(xi)−ci(xi) the correspondingincrement in i’s disutility of effort. We postulate the followingassumption.

A.3 1p2(2 wins|x\x2;1)1p1(1 wins|x\x1;1) > 1c2(x2)

1c1(x1)and 1p2(2 wins|x\x2;K)

1p1(1 wins|x\x1;K) < 1c2(x2)1c1(x1)

, for allx ∈ XSym.

This assumption entails that the contestants’ heterogeneity,measured via1c1(x1) and1c2(x2) cannot outweigh the variabilityin the productivities of the contestants as stated by A.1 and A.2.A.1–A.3 are fairly intuitive and allow to characterize the uniqueoptimal sample size n∗ (see Proposition 1) as a vector of winprobabilities that corresponds to the choice of an optimal samplesize n∗ (see Lemma 3).

2.5. The probabilistic choice of the planner

For our purpose, we essentially need to write the prize as afunction of the vector of win probabilities. To do so, we first definethe map

pi(· | x) : {1, . . . , K} → {pi(i wins | x; 1), . . . , pi(i wins | x; K)}

and say that p ≡ p(n | x) with p(n | x) = (p1(n | x), p2(n | x)),is an NE-implementable distribution whenever n NE-implementsx. Second, we reformulate the planner’s optimization problem interms of the win probabilities. Formally, we consider the incentiveprize for each i (=1, 2), bIi(pi) ≡ bi(n | x) and define the I-prize asmax

bI1(p1), bI2(p2)

≡ bI(p). This represents the minimal prize

that has to be handed out to the winner to implement x as an NEof Γ (bI(p), p). This way, the administrator’s problem can be recastas,

p∗= arg min

p∈∆NE2 (x)

bI(p),

where ∆NE2 (x) denotes the set of NE-implementable distributions

for profile x. In the sequel, we will refer to this problem asthe probabilistic choice problem of the planner. In accordancewith the previous definitions, we call p∗

∈ ∆NE2 (x) the NE-optimal

distribution of x.

2.6. Existence and uniqueness of an NE-optimal distribution

By pursuing his own objectives, the optimal solution of theplanner induces the contest to have certain desirable fairnessproperties. It is therefore instructive to formulate the optimalchoice of the planner in terms of the ‘‘rent’’ or the contestants’valuation of the prize that has to be given up to the contestants.

Define the valuation of agent i associated with i (=1, 2)’s(minimal) incentive prize, bIi(pi) ≡ bi(n | x). A rent for player icorresponds to the induced valuation, v(bIi(pi)) ≡ v∗

i (pi). There-fore, max

v∗

1(p1), v∗

2(p2)

≡ v∗(p) is the minimal incentive rentthat has to be given up to the contestants in order to implement xas an NE of Γ (bI(p), p).

Page 5: Equivalence of optimal noisy-ranking contests and Tullock contests

744 Y. Pelosse / Journal of Mathematical Economics 47 (2011) 740–748

Proposition 1. Given any x ∈ XSym, let {Γ (bI(p(n | x)), p(n | x)) :

1 ≤ n ≤ K} satisfy hypotheses A.1– A.3. Then, there exists a uniqueNE-optimal probability distribution (p∗

1, p∗

2) ∈ ∆NE2 (x) such that

v∗

1(p∗

1) = v∗

2(p∗

2).

Proof. See Appendix A. �

For a given sample size n inducing a distribution p(n | x), theworse-off contestant requires a larger prize if one wants him toexert xi. Thus, the requirement that the worse-off contestant becompensated through a higher prize, leads the well-off contestantto extract a rent. This leads the administrator to allocate the rentaccording to the ‘‘egalitarian solution’’ given in Proposition 1.

2.7. Equivalent noisy-ranking contests and Tullock contests

Our notion of equivalence between an optimal noisy-rankingcontest and the family of Tullock contests is defined as follows.

Definition 1. We say thatΓ (bI(p(n∗| x)), p(n∗

| x)) is equivalentto a Tullock contest if for any x ∈ XSym there is a uniqueNE-optimaldistribution, p(n∗

| x) ≡ Ψ (x), such that

Ψi(x) =fi(x) exp(hi(x)/δ)

f1(x) exp(h1(x)/δ)+ f2(x) exp(h2(x)/δ),

for any i (=1, 2)where fi(x) and hi(x) are two real positiveweightsfor contestant i and δ is a positive constant.

In words, this definition says that a noisy-ranking contest isequivalent to a Tullock contest if the optimal implementation ofevery symmetric pure NE strategy profile in XSym of the noisy-ranking contest generates win probabilities of the Tullock-form.This allows to understand the vectorΨ = (Ψ1,Ψ2) as a real TullockCSF whereΨ associates to each vector of efforts x ∈ XSym, a lotteryspecifying for each agent a probability Ψi(x) of getting the prize.Moreover, as the set (X)KSym, with typical element xi = (x, . . . , x),is isomorphic21 to a set of effort levels E ⊂ R+, which has the samecardinality as (X)KSym, with typical element x, one can understand(Ψ1,Ψ2) as the CSF of a standard one-dimensional Tullock contestwhere the efforts of both players are in E.22

Before we state our main result, we need to narrow downthe class of noisy-contest models by way of imposing thefollowing additional assumption. Let ∆2 be the one-dimensionalunit simplex and let int(∆2) be the relative interior of∆2 in R2.A.4 (c1(x1), c2(x2)) ∈ int(∆2) for all (x1, x2) ∈ XSym.

Clearly, A.4 is a normalization concerning the disutilities fromwork. In order to state our equivalence result, we also need towritethe I-prize in terms of the contestants’ win probabilities. To do so,we have to define the participation prize function which ensuresthat each contestant derives a positive utility in equilibrium for anyp ∈ ∆NE

2 (x). Formally, let bPi be the prize function for contestant idefined as

bPi (pi) := min {b : piv(b)− ci(xi) > 0} .

Since the administrator wants all contestants to participate, wemust have bP(p) = max

bP1(p1), bP2(p2)

. Hence, bP is a well-

defined ∨-shaped function which we call the P-prize.The following lemma establishes an important relationship

between the I-prize and the P-prize.

21 That is, we consider the equivalence relation, ∼, with x ∼ xi for i (=1, 2),whenever xi = (x, . . . , x).22 Notice that our definition includes the well-known class of (multinomial) CSFsof the logit form (see McFadden, 1974) used in the literature of contests (seee.g., Rosen, 1986; Dixit, 1987; Skaperdas and Gan, 1995).

Lemma 2. Under hypotheses A.1 and A.2 , there exists a strictlyincreasing function φ : R++ → R++ such that bI = φ ◦ bP .

Proof. See Appendix B. �

Finally, we need to impose the following technical assumption onthe I-prize.A.5 φ is a concave function and φ′

+(0) ≤ δ with δ a positive

constant.23

3. Main results

Our main result is the following theorem.

Theorem 1. SupposeΓ (bI(p(n | x), p(n | x))) : 1 ≤ n ≤ K

sat-

isfies hypotheses A.1– A.5 for all x ∈ XSym. Then, we may withoutloss of generality set

(i) bI(p) = δbP(p) −∑

i ai(x)pri(x)i −

∑i βi for ai(x) = 0, ri(x)

= 0;(ii) if r1(x) = r2(x) = 1 for all x ∈ XSym, then the win probability

Ψ of a contestant i (=1, 2) is such that

Ψi(x) =

ci(xi) exp

ai(x)δ

c1(x1) exp

a1(x)δ

+ c2(x2) exp

a2(x)δ

with δ =δλ a positive constant

(iii) if for all i (=1, 2) and x ∈ XSym, we have ri(x) < 1, then the winprobabilities Ψ are such that

Ψi(x)

=

W (Zi)

(1−ri(x))si(ai(x),ri(x))

11−ri(x)

W (Z1)(1−r1(x))s1(a1(x),c1(x))

11−r1(x)

+

W (Z2)

(1−ri(x))si(ai(x),ri(x))

11−r2(x)

where Zi ≡ (1 − ri(x))si(ai(x), ri(x)) exp((ri(x)− 1)K(ci(xi)))with W the Lambert function, si(ai(x), ri(x)) and K(ci(xi)) twofunctions of ai(x), ri(x) and ci(xi), respectively.24

Proof. See Appendix B. �

Property (i) reveals a surprising connection between the stochasticchoice models introduced by Machina (1985) and the problemof the designer of a multi-dimensional contest: the decisionproblem of the planner (i) is based on deterministic (and known)preferences over lotteries, (p1, p2) = p, which can be induced by asample size n, but at a cost that depends (in terms of the monetaryprize) on how much p deviates from p∗. Hence, our equivalenceresult builds on an explicit trade-off between the provision ofincentives to exert x for all contestants and the ensuing cost ofimplementation. Finally, it is worth noting that if r1(x) = r2(x) =

1 and c1(x1) = c2(x2) for all x ∈ Xsym, we obtain the multinomialgeneralized Tullock contest form of CSF.25

Theorem 1 (ii)–(iii) is based upon the following result, which isof interest in its own right.

Proposition 2. Fix a profile x ∈ XSym. Under hypotheses A.4– A.5 ,the P-prize bP : ∆NE

2 (x) → R++ is equivalent to

bP(p) = λ−

i

pi lnpi

c∗i

with λ > 0 and c∗

i ≡ ci(x).

23 Function φ′+(t) denotes the right derivative of φ(t) at point t .

24 These functions are defined in Appendix B.25 This is exactly the form of CSF originally proposed by Hirshleifer (1989).Skaperdas (1996) provided an axiomatic justification for this form.

Page 6: Equivalence of optimal noisy-ranking contests and Tullock contests

Y. Pelosse / Journal of Mathematical Economics 47 (2011) 740–748 745

Proof. See Appendix B. �

In words, this says that the value of increasing the numberof relevant dimensions n is given by the relative-entropy (orinformation-gain) measure suggested by Kullback and Leibler(see Kullback, 1959).26

3.1. Concluding remarks

We have shown that a class of contests assuming Tullock formcontest success functions could be viewed as the reduced-formversion of an optimally designed noisy-ranking contest wherecontestants compete in several dimensions. As this equivalenceresult comes from the standard profit-maximization assumptionfor the contest organizer, this reveals a natural structuralconnection between these two families of contests.

In our model, the underlying stochastic choice of the planneryields a class of Tullock contest success functions. Using thisapproach to derive other popular success functions would be anatural way to proceed. We leave this for future research.

Appendix A

Proof of Lemma 1We show that any profile lying in XSym can be implemented as

an NE by a finite prize b(n | x) for any sample size n. To see this,consider a profile x = (x1, x2) ∈ XSym and two other arbitrary(pure) strategies,xi andxi in (X)K such thatxi ≻ xi ≻ xi. Notethat, in equilibrium, for any i, the least upper bound for the prizeensuring that i does not prefer a strategy xi over a strategy xiwherein he exerts less effort must verify

bi(n | x) = infb ∈ R++ : v(b)1pi(i wins | x \xi; n)

≥ 1ci(x |xi) s.t xi ≻xiwhere 1ci(x | xi) ≡ ci(xi) − ci(xi). The existence of a finite(positive) prize verifying the above inequality follows since ci andpi(i wins | x; n) are strictly increasing with respect to the order≻, and ci(xi) − ci(xi) is bounded above, as ci(xi) is bounded.Moreover, v(·) is strictly increasing in b. Hence, givenn, there existsa finite b(n | x) = max

b1(n), b2(n)

such that player i (=1, 2)

prefers playing xi rather thanxi. This proves that no players wantto put in less effort. Next, consider deviations towards strategiesxi ≻ xi.

By assumption, disutility functions ci are sufficiently convex.This automatically implies that the agent who requires the prizeb(n | x) to exert xi ≻ xi will not switch to axi ≻ xi. For disutilityfunctions that are sufficiently convex, this property will also holdfor the other agent. �

Corollary 1. Suppose that the planner wants to implement the profilex = (x1, x2) in XSym. Then, for any such profile that is optimallyimplemented as an NE of Γ (Πn, b), the planner only needs to findthe sample size n∗ minimizing b(n | x) = max

b1(n), b2(n)

.

Proof. Obvious from Lemma 1. �

Next, we considerxi such that

xi ∈ arg maxxi∈(X)K \{xi}

v−1

c∗

i − ci(xi)1pi(i wins | x \xi; n)

,

26 This representation of the P-prize hinges on three main properties due toHobson (1969) which characterize the relative-entropy measure, up to a positivescalar.

where v−1 is the inverse valuation function27 with xi ≻ xi fori (=1, 2), for any n.28

Proof of Lemma 3

Lemma 3. Under hypotheses A.1 and A.2 , the administrator’soptimal rule n∗ inducing (b(n∗

| x), n∗) is equivalently described bythe choice of a pair (b(n∗

| x), p(n∗| x)) for all x ∈ XSym.

Lemma 3 is an immediate consequence of A.1 and Lemma 2.Athat we shall now state below.

Lemma 2.A. Fix deviation at xi for all i (=1, 2). Under hypothe-ses A.1 and A.2 , there exists a strictly increasing (resp. decreas-ing) function Φ1(· | x) ∈ [0, 1]p1({1,...,K}|x) (resp. Φ2(· | x) ∈

[0, 1]p2({1,...,K}|x)) such that 1p1(1 wins | x∗\ x1; n) = Φ1(· |

x)◦p1(n | x)(resp.1p2(2 wins | x∗)\x2; n) = Φ2(· | x)◦p2(n | x).

Proof. Let ∆i(· | x) : {1, . . . , K} → [0, 1] with ∆i(n | x) =

1pi(i wins | x \ xi; n). By A.1 and A.2, we have p1(· | x) and∆1(· | x) strictly increasing in n while p2(· | x) and ∆2(· | x)are strictly decreasing in n. Thus, for any i = 1, 2, p(· | x)i and∆i(· | x) represent the same order> on {1, . . . , K} and there existsa strictly increasing function Φ i(· | x) with∆i(· | x) = Φ i(· | x) ◦

pi(· | x). �

Each of the next results are identically stated conditional onx ∈ XSym.

Notations. Recall that 1c∗

i := c∗

i − ci(xi) with c∗

i ≡ ci(xi).Similarly, in order to simplify notations, we will systematicallyomit to condition on x. Hence we shall write pi(· | x) ≡ pi(·),∆i(· | x) ≡ ∆i(·), . . . etc. We will also write Φi as the continuousextension of Φ i(· | x) to the convex hull of {pi(1 | x), . . . ,pi(K | x)}.

Proof of Proposition 1. We show that the optimal sample size n∗

is NE-optimal if and only if1pi(i wins | x \xi; n∗) > 0,∀i (=1, 2)and 1c∗1

1c∗2=

1p1(1 wins|x\x1;n∗)

1p2(2 wins|x\x2;n∗).

The first property is obvious because win probabilities areincreasing w.r.t. order ≻.29 For the second condition, let |r| bethe absolute value of r ∈ R. The following well-known algebraicidentity will be useful

maxbI1(p1), bI2(p2)

=

12

bI1(p1)+ bI2(p2)

+

12

bI1(p1)− bI2(p2) .

Observe that bI is minimized when bI1(p1) = bI2(p2). Thus, itfollows from the assumption made on v that

bI1(p1(n∗)) = bI2(p2(n

∗)) ⇔1c∗

1

1c∗

2=1p1(1 wins | x \x1; n∗)

1p2(2 wins | x \x2; n∗). (1)

Define v∗

i (pi) ≡ v(bIi(pi)). Hence, Eq. (1) amounts to v∗

1(p∗

1) =

v∗

2(p∗

2).Existence of a unique optimal NE-probability distribution. Let h(pi) =

1 − pi. To economize on notations, we denote ∆NE2 for any

set ∆NE2 (x). Thus, using Lemma 2.A, we can define Λ(p1) =

1c∗

2Φ1(p1) − 1c∗

1Φ2 ◦ h(p1), with 1c∗

1 defined as in the proofof Proposition 1. Moreover, we know that 1c∗

2Φ1(p1) is strictly

27 This function is well-defined since v is assumed to be strictly monotonicincreasing.28 By assumption, the productivities of agents are independent of n.29 Here, assumption A.1 is used as a shortcut of the proof of Dubey andWu (2001),Theorem 4, p.329.

Page 7: Equivalence of optimal noisy-ranking contests and Tullock contests

746 Y. Pelosse / Journal of Mathematical Economics 47 (2011) 740–748

increasing in p1 while 1c∗

1Φ2 ◦ h(p1) is strictly decreasing in p1.Using A.3, we have Λ(p1) < 0 for all p1 < p1(n∗) ∈ ∆NE

2 andΛ(p1) > 0 for some p1 ≥ p1(n∗) ∈ ∆NE

2 . Hence, the IntermediateValue Theorem implies the existence of a p∗

1 such that Λ(p∗

1) = 0which is equivalent to the optimality condition given in Eq. (1).This completes the proof of existence. Uniqueness follows from thestrict monotonicity ofΛ. �

Appendix B

To prove Proposition 2, we need to state an axiomaticcharacterization of the relative-entropy function (see Kullback,1959) D : ∆k × int(∆k) → R+ suggested by Hobson (1969) forthe special case of k = 2. Then Proposition 2 follows easily.

Let∆k be the k− 1-dimensional unit simplex in Rk and int(∆k)

be the relative interior of∆k in Rk.

Theorem (Hobson, 1969). Let D : ∪k∈{1,2}∆k × int(∆k) → R+ bea function which satisfies the following three axioms.

(i) D((p1, p2), (q1, q2)) is symmetric w.r.t. its arguments;

(ii) D(( 12 ,12 ), (

12 ,

12 )) < D((p1, p2), (

12 ,

12 ));

(iii) D( pipi ,qiqi) = 0,∀i (=1, 2).

Axioms (i)–(iii) hold if and only if D(p, q) is equivalent toλ

∑2i=1 pi ln

piqi

with λ > 0.

Proof of Proposition 2. Using A.4, we can set (q1, q2) = (c∗

1 , c∗

2 )

and definebP : ∪k∈{1,2}∆k × int(∆k) → R+.30 With c∗= (c∗

1 , c∗

2 ),we define the ∨-shaped functionbP(p, c∗) = max{bP1(p1, c∗

1 ),bP2

(p2, c∗

2 )}, andbPi (pi, c∗

i ) = v−1(c∗ipi),∀i (=1, 2). The proof shows

thatbP(p, c∗)meets axioms (i)–(iii) of the above theorem. Axioms(i) and (ii) are easily checked. Axiom (iii) follows by convention, asthere is no prize when there is no contest (there is no contest gamein (iii) since there is only one single agent in this case). This impliesthatbP(p, c∗) = λ

∑i pi ln

pic∗ i

with λ > 0. When c∗ is fixed, wehave bP(p) = λ

∑i pi ln

pic∗ i

. �

Proof of Theorem 1 (i)The proof makes use of several lemmata that we shall show

below.

Lemma 2. Under hypotheses A.1 and A.2 , there exists a strictlyincreasing function φ : R++ → R++ such that bI = φ ◦ bP .

Proof. Let c∗

i define as in the main text in A.4 and 1c∗

i as in theproof of Proposition 1. Define two functions ϕi and ϑi such thatϕi(pi) =

1c∗iψi(pi)

and ϑi(pi) =c∗ipi

for any i (=1, 2). By Lemma 2.A,eachΦi(pi) is increasing in pi,∀i (=1, 2). Hence, thismeans that ϕiand ϑi are two decreasing functions. Moreover, the two compositefunctions v−1

◦ ϕi(pi) and v−1◦ ϑi(pi) are decreasing in pi as

v−1 is an increasing function.31 By definition, we have bIi(pi) =

v−1◦ ϕi(pi) and bPi (pi) = v−1

◦ ϑi(pi). This implies that thereexists a strictly increasing functions φi : R++ → R++ such that,bIi(pi) = φi ◦ bPi (pi) for any i (=1, 2). Since bI and bP are defined

30 Clearly, when k = 1, we have int(∆1) = {1}.31 Recall that v has been assumed strictly increasing.

as the two ∨-shaped functions bI(pi) = maxbI1(p1), bI2(p2)

and

bP(pi) = maxbP1(p1), bP2(p2)

, this implies that bI increases if and

only if bP increases. Hence, we conclude that there exists a strictlyincreasing function φ : R++ → R++ so that bI = φ ◦ bP . �

Recall that∆NE2 denotes any set∆NE

2 (x). We shall now establishLemma 1.B showing that it is without loss of generality towrite theI-prize as,

bI(p) =δbP(p)−g(p), ∀p ∈ ∆NE2

with δ > 0. Let conv(A) be the convex hull of set A. Then, inLemma 1.B it is shown that g(p) can be expressed linearly bytwo functions gi : conv({pi(1), . . . , pi(K)}) → R,∀i (=1, 2) withg(p) = g1(p1)+ g2(p2).

Lemma 1.B. Using Lemma 2, the I-prize, φ(bP(p)), can be writtenwithout loss of generality as

φ(bP(p)) =δbP(p)−g(p), ∀p ∈ ∆NE2 ,

whereg : ∆NE2 → R is a convex function andδ ≥

φ′+(0)

.Proof. The I-prize, bI , is a well-defined function over a convexand compact domain. Moreover, φ is concave by A.5 and strictlyincreasing by Lemma 2. This means φ′(0) ≥ φ′

+(t) ≥ 0 for all

t ≥ 0. Thus, φ(t) = δt − φ(t) meets φ′(t) = δ − φ′(t) ≥ φ′(0).Hence, φ(bp(p)) = δbp(p) − φ(bp(p)) is a convex function on∆NE

2 . �

Having shown that bI(p) = δbp(p) − g(p), we are left tocharacterizeg .Lemma 2.B. For any i (=1, 2), gi(pi) = aip

rii +βi for some constants

ai = 0, βi and ri = 0.

Proof. Part 1. We first show thatg(p) = g1(p1) + g2(p2). First,recall that

bIj(pj) = v−11c∗

j

Φj(pj)

.

Let h(pi) = 1 − pi. Thus, setting Φj|i(pi) := Φj ◦ h(pi), we canrewrite bIj(pj) as

bIj|i(pi) = v−1

1c∗

j

Φj|i(pi)

.

Similarly, we can redefine bPj (pj) as

bPj|i(pi) = v−1 c∗

j

h(pi)

.

Hence we can set

gi(pi) =

δ2max

bPi (pi), bPj|i(pi)

12max

bIi(pi), bIj|i(pi)

, ∀i (=1, 2),

so thatg(p) = g1(p1)+ g2(p2).

Part 2. Let (p1, p2) ∈ ∆NE2 and define

Q =q ∈ R++ : (qp1, 1 − p1q) ∈ ∆NE

2

.

We set li : Q → R++ with l1(q) = q and l2(q) =1−p1q1−p1

. We showthat for each q ∈ Q there exists a continuous function fq(.) suchthat

g1(qp1)+ g2(q′p2) = fq [g1(p1)+ g2(p2)] ,

Page 8: Equivalence of optimal noisy-ranking contests and Tullock contests

Y. Pelosse / Journal of Mathematical Economics 47 (2011) 740–748 747

where q′= l2(q). Thus, for each i we can write

gi(li(q)pi) =

δ2max

bPi (li(q)pi), bPj|i(li(q)pi)

12max

bIi(li(q)pi), bIj|i(li(q)pi)

.

Let bP(pi) ≡ maxbPi (pi), bPj|i(pi)

and bI(pi) ≡ max{bIi(pi), bIj|i

(pi)},∀i (=1, 2). Note that for any q ∈ Q , gi(li(q)pi) increasesas gi(pi) increases. To see this, note that the gi’s are increasingfunctions. Taking the derivative of gi we have g ′

i (pi) =δ2∂bP (pi)∂pi

−12∂φ◦bP (pi)∂bP (pi)

∂bP (pi)∂pi

. Using A.5, we conclude that g ′

i (pi) > 0. Thisimplies that gi(li(q)pi) > gi(li(q)p′

i) ⇔ gi(pi) > gi(p′

i),∀pi > p′

i .Since this is true for any i (=1, 2), it follows that for any q ∈ Q ,there exists an increasing function fq(.) such that

g1(qp1)+ g2(l2(q)p2) = fq [g1(p1)+ g2(p2)] .

Since both gi(pi) and gi(li(q)pi) are continuous in pi32 for any

i (=1, 2), it follows that fq(.) is also a continuous function. Definef (q, .) = fq(.) then

g1(qp1)+ g2(li(q)p2) = f [q, g1(p1)+ g2(p2)] .

For any given q ∈ Q , an infinitesimal increase in q translates intosimultaneous infinitesimal increases in the pi’s (this follows sincel2 is a continuous function). Since gi(li(q)pi) is continuous in allli(q)pi’s, it follows that f (q, .) is also continuous in q.

Part 3. We show that gi(pi) = aipcii + βi,∀i = 1, 2 for some

constants ai and ci = 0 and βi.To do so, we will use different results from Aczél (1966). First,

let

u := g1(p1) and v := g2(p2).

Then p1 = g−11 (u) and p2 = g−1

2 (v). This means we have

g1(qg−11 (u))+ g−1

2 (l2(q)g−12 (v)) = f [q, u + v] .

Part 4. Denotegi := gi ◦ li(q) ◦ g−1i andf := f (q, .). Then

f (u + v) = g1(u)+g2(v). (2)

Eq. (2) is a Pexider equation. The solution of Eq. (2) is (see Aczél,1966, p. 142),

g1(u) = bu + a1, g2(v) = bv + a2,

for some constant b, a1 and a2. Restoring the definitions of gi andf , we obtain

gi(li(q)pi) = b(li(q))gi(pi)+ ai(li(q)), ∀i (=1, 2),

for any q ∈ Q and for some continuous function b(li(q)) andai(li(q)). The above equation is again a Pexider equation with twodistinct solutions for strictly monotonic gi (Aczél, 1966, p. 145)

gi(pi) = ai ln pi + βi for ai = 0and gi(pi) = aipri + βi for ai and ri = 0 withai(li(q)) = βi(1 − li(q)ri), b(li(q)) = li(q)ri ,

∀i (=1, 2) and a1a2ri = 0. Sinceg is convex (see Lemma 1.B), weconclude that the only possible solution is gi(pi) = aipri + βi. �

32 This follows since v is differentiable andΦi is the continuous extension ofΦ i .

Completion of Proof of Theorem 1 (ii)–(iii)Step 1. Uniqueness of p(n∗). The relative entropy function is

continuously differentiable in p on the relative interior of thesimplex. Moreover, it suffices to use Theorem 1 (i) to see thatfor ri ≤ 1, i = 1, 2 it is w.l.o.g to set bI(p) = δD(p, c∗) −∑

i ai(x)prii −

∑i βi for some constants ai(x) and βi. Thus bI is a

(strictly) convex real-valued function with a convex domain. Byadopting the convention that pi(n + 1) > pi > pi(n) implies thatpi ≡ pi(n), there exists a unique interior solution to

minp∈∆NE

2

δD(p, c∗)−

−i

prii ai(x)

. �

Step 2. (Derivations) Denote χ as the Lagrange multiplier and setδ ≡δλ, ai ≡ ai(x), ri ≡ ri(x) (resp. ri = 1 for all i) and qi ≡ c∗

i .Theorem 1 (ii) The necessary conditions for an interior solutionimply that

pi = ln qi expai + χ

δ− 1

∀i (=1, 2).

Using the condition p1 + p2 = 1, yields the result.Theorem 1 (iii) Let W be the inverse of the function f defined byf (w) = w exp(w) where w is any real number. The first ordercondition is ln pi = K(qi) + si(ai, ri)p

ri−1i with K(qi) ≡

1δ[χ −

1] + ln qi − 1 and si(ai, ri) ≡1δriai. This condition can be rewritten

as, (1 − ri)si(ai, ri) exp((ri − 1)K(qi)) = w exp(w) where w exp(w) = (1 − ri)si(ai, ri) exp((ri − 1)K(qi)) and w = (1 − ri)si(ai, ri) exp((ri − 1) ln pi). Hence, W ((1 − ri)si(ai, ri) exp((ci −

1)K(qi))) = (1 − ri)si(ai, ri) exp((ri − 1) ln pi). Solving this lastequation with the condition p1 + p2 = 1, yields the solution. �

The results above show that we have win probabilitiesof the Tullock form for each x ∈ XSym. Hence, we have by construc-tion a map Ψi : XSym → {pi(1 | x), . . . , pi(K | x)} for all i whichcompletes the proof. �

References

Aczél, J., 1966. Lectures on Functional Equations and their Applications. AcademicPress, New York.

Amegashie, J.A., 2002. Committees and rent-seeking effort under probabilisticvoting. Public Choice 112, 345–350.

Baye,M.R., Hoppe, H.C., 2003. The strategic equivalence of rent-seeking, innovation,and patent-race games. Games and Economic Behavior 217–226.

Blavatskyy, P.R., 2010. Contest success function with the possibility of a draw:axiomatization. Journal of Mathematical Economics 46, 267–276.

Che, Y.K., Gale, I., 1998. Caps on political lobbying. American Economic Review 88,643–651.

Clark, D., Konrad, K., 2007. Contests with multi-tasking. Scandinavian Journal ofEconomics 109 (2), 303–319.

Congleton, R.D., 1984. Committees and rent-seeking effort. Journal of PublicEconomics 25, 197–209.

Corchón, L., Dahm, M., 2011. Welfare maximizing contest success functionswhen the planner cannot commit. Journal of Mathematical Economicsdoi:10.1016/j.jmateco.2010.12.018.

Corchón, L., Dahm, M., 2010. Foundations for contest success functions. EconomicTheory 43 (1), 81–98.

Dahm, M., Porteiro, N., 2005. A micro-foundation for non-deterministic contests ofthe logit form.

Dasgupta, A., Nti, K.O., 1998. Designing optimal contests. European Journal ofPolitical Economy 14, 587–603.

Dasgupta, P., Stiglitz, J., 1980. Uncertainty, industrial structure, and the speed ofR&D. Bell Journal of Economics 11, 1–28.

Dixit, A.K., 1987. Strategic behavior in contests. American Economic Review 77,891–898.

Dubey, P., Haimanko, O., 2003. Optimal scrutiny in multi-dimension promotioncontests. Games and Economic Behavior 42 (1), 1–24.

Dubey, P., Wu, C.W., 2001. Competitive prizes: when less scrutiny induces moreeffort. Journal of Mathematical Economics 36, 311–336.

Epstein, G.S., Nitzan, S., 2006. The politics of randomness. Social Choice andWelfare27, 423–433.

Fu, Q., Lu, J., 2008. Unifying contests: from noisy ranking to ratio-form contestsuccess functions. Working Paper. University Library of Munich.

Page 9: Equivalence of optimal noisy-ranking contests and Tullock contests

748 Y. Pelosse / Journal of Mathematical Economics 47 (2011) 740–748

Fullerton, R.L., McAfee, R.P., 1999. Auctioning entry into tournaments. Journal ofPolitical Economy 107, 573–605.

Green, J., Stokey, N., 1983. A comparison of contests and contracts. Journal ofPolitical Economy 91, 349–364.

Grossman, G.M., Helpman, E., 2001. Special Interest Politics. MIT Press, Cambridge.Hirshleifer, J., 1989. Conflict and rent-seeking success functions: Ratio vs. difference

models of relative success. Public Choice 63, 101–112.Hirshleifer, J., Riley, J., 1992. The Analytics of Uncertainty and Information.

Cambridge University Press, New York, NY.Hobson, A., 1969. A new theoremof information theory. Journal of Statistical Physics

1, 383–391.Jia, H., 2008. A stochastic derivation of contest success functions. Public Choice 135,

125–130.Klumpp, T., Polborn, M.K., 2006. Primaries and the newHampshire effect. Journal of

Public Economics 90, 1073–1114.Konrad, K.A., 2009. Strategy and Dynamics in Contests. Oxford University Press.Kullback, S., 1959. Information Theory and Statistics. Wiley, New York.Lazear, E.P., Rosen, S., 1981. Rank order contest as optimum labor contracts. Journal

of Political Economy 89, 841–864.Machina, M., 1985. Stochastic choice functions generated from deterministic

preferences over lotteries. Economic Journal 95, 575–594.

McFadden, D.L., 1974. Conditional logit analysis of qualitative choice behavior.In: Zarembka, P. (Ed.), Frontiers in Econometrics. Academic, New York,pp. 105–142.

Münster, J., 2008. Group contest success functions. Economic Theory 41 (2),345–357.

Nalebuff, B., Stiglitz, J., 1983. Prizes and incentives: toward a general theory ofcompensation and competition. Bell Journal of Economics 14, 21–43.

Nitzan, S., 1994. Modelling rent-seeking contests. European Journal of PoliticalEconomy 10 (1), 41–60.

Rai, B.K., Sarin, R., 2009. Generalized contest success functions. Economic Theory 40(1), 139–149.

Rosen, S., 1986. Prizes and incentives in elimination contests. American EconomicReview 76, 701–715.

Skaperdas, S., 1996. Contest success functions. Economic Theory 7, 283–290.Skaperdas, S., Gan, L., 1995. Risk aversion in contests. Economic Journal 105,

951–962.Tullock, G., 1980. Efficient rent seeking. In: Buchanan, J.M., Tollison, R.D., Tullock, G.

(Eds.), Toward a Theory of the Rent-Seeking Society. Texas A&M UniversityPress, College Station, pp. 97–112.

Weibull, J., Laslier, J.F., 2009. The Concorcet jury theorem and preferenceheterogeneity. Working paper.