Efficiency-based measures of inequality

Journal of Mathematical Economics 85 (2019) 60–69

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Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco

Efficiency-based measures of inequality✩ ∗

Costel Andonie a , , Christoph Kuzmics b , Brian W. Rogers c a

Harris School of Public Policy, The University of Chicago, United States of America Department of Economics, University of Graz, Austria c Department of Economics, Washington University in St. Louis, United States of America b

article

info

Article history: Received 5 October 2018 Received in revised form 15 August 2019 Accepted 17 September 2019 Available online 28 September 2019 Keywords: Utilitarianism Inequality Contests

a b s t r a c t How should we make value judgments about wealth inequality? Harsanyi (1953) proposes that an individual evaluate wealth possibilities through expected utility under Rawl’s (1971) ‘‘veil of ignorance’’ assuming wealth levels are assigned uniformly. We propose an alternative notion of how wealth levels are assigned, based on a contest. Inequality can be captured through the equilibrium properties of such a game. We connect these inequality measures to existing measures, demonstrating the conditions under which they satisfy the received key axioms of inequality measures (anonymity, homogeneity and the Pigou–Dalton principle), and how they can evaluate a tradeoff between greater total wealth and greater inequality. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Why might a social planner have a concern for inequality? We take up the following answer. As the positions in society become increasingly unequal, each individual will have increasingly strong preferences to occupy the more desirable positions rather than the less desirable positions. But, as those positions are scarce, incentives are brought into stronger conflict, resulting in wasteful competition. As competition for those positions intensifies, the actions that individuals take therefore result in larger inefficiencies, as individuals optimally compete more aggressively for the better roles. These inefficiencies lower the ex ante expected payoff to all.1 To model this intuition we propose a framework which focuses explicitly on this connection between the inequality in a society and the efficiency of the outcomes it produces. Outcomes are generated through a contest, modeled as a non-cooperative game. Let society be identified with a list of wealth levels (or prizes), x = (x1 , . . . , xn ). The planner evaluates the desirability of x by the expected payoff of the contest’s symmetric equilibrium. That is, the planner prefers society x to society x′ precisely when the expected payoff from the contest is higher under x than x′ . ✩ We thank Marcus Berliant and Alberto Vesperoni, as well as participants at the 48th meeting of the Canadian Economic Association for useful comments. ∗ Corresponding author. E-mail addresses: [email protected] (C. Andonie), [email protected] (C. Kuzmics), [email protected] (B.W. Rogers). 1 We are motivated in part by Frank and Cook (1995), who argue convincingly that many professions/industries, such as the music industry, pay dramatically unequal prizes, leading to very inefficient choices by the players in the industry. https://doi.org/10.1016/j.jmateco.2019.09.002 0304-4068/© 2019 Elsevier B.V. All rights reserved.

We have two main goals with this approach. First, we demonstrate that the equilibrium properties of such a contest can be used to generate what we call efficiency-based measures of the inequality in a given society of roles that are perfectly consistent with the axiomatic literature on inequality measures as in e.g. Fields and Fei (1978), Shorrocks (1980), Foster (1983), Shorrocks (1984), and surveyed in Sen and Foster (1997). In particular we show that the equilibrium payoff in different models of contests leads to different inequality measures with particular contests leading to, among others, the mean log deviation (MLD) measure introduced by Theil (1967) and Rawls’s (1971) maxmin measure, and even the Gini coefficient. Second, we use our efficiency-based measures to discuss the tradeoff between equity and efficiency in a society. In particular, we provide necessary and sufficient conditions (for different models of contests) under which a unilateral increase in one prize (keeping constant all other prizes) is detrimental to the overall efficiency in society.2 The non-cooperative formulation on which we rely naturally addresses this question through the equilibrium payoff. Increasing an individual income in society is preferable, all else equal, but if it is already a large prize it also increases competition for it. The net impact in our framework is nontrivial. In this sense, one can view the equilibrium payoff as playing the role of a social welfare function, as advocated by Atkinson (1970), for the purpose of evaluating efficiency-inequality tradeoffs. 2 In the utilitarian view, an increase in any prize represents an improvement. In the axiomatic approach, the focus is largely on relative inequality, which deliberately abstracts from the effects of changing total wealth. Thus neither approach is designed to evaluate such a tradeoff.

C. Andonie, C. Kuzmics and B.W. Rogers / Journal of Mathematical Economics 85 (2019) 60–69

Before proceeding, we offer a brief remark on how we place our contribution in the history of thought on inequality. The ‘‘veil of ignorance’’ is, of course, a thought experiment with a long tradition, introduced to the economics literature by Harsanyi (1953) and expanded upon by Rawls (1971). Behind the veil, an individual is asked to evaluate a society without any knowledge of her own circumstances. Under Harsanyi’s (1953) formulation, an individual imagines she will be placed into one of society’s roles uniformly at random, whereas for Rawls (1971) the individual thinks as if she will be assigned to the least desirable place in society. Rather than adopting either of these notions of what we might call ‘‘fate’’, we suppose instead that individuals believe they will come to occupy a position in society through the (equilibrium) outcome of a contest. In this sense, our framework is very much in the spirit of Harsanyi (1953), with the inclusion of an explicit non-cooperative model for fate, in which an individual’s preferences are dictated by an expected utility calculation at a symmetric ex ante stage. In this paper we study three different models of a contest, each a class of non-cooperative games. We have looked to the modern theory of inequality to guide us somewhat in our choice of these games and the equilibrium concept that we employ. This literature has produced sharp characterizations of a number of measures, allowing us to understand their fundamental properties, see e.g. Fields and Fei (1978), Shorrocks (1980), Foster (1983), Shorrocks (1984) and Sen and Foster (1997). We connect our work to this approach in part to give some confidence that our approach is compatible with axiomatic considerations. The axiomatic literature has agreed on three key axioms that any measure of relative inequality must satisfy, which we will study vis à vis our efficiency-based measures. The first key axiom is anonymity, i.e., the measure should depend only on the list of possible prizes and not on who gets which prize. In other words the inequality measure should not depend on the names or other characteristics of the individuals. This implies, in the case of efficiency-based measures, a focus on symmetric equilibria, in which each individual chooses the same strategy, so that the equilibrium payoff depends only on the list of prizes and not on the individuals’ names. We would also argue, beyond this technical point, that restricting attention to symmetric equilibria is more in line with the ‘‘veil of ignorance’’ view. The evaluation by a player in our contest is deliberately taken at the ex ante stage, before the individual learns anything about her type that might distinguish her from other individuals. She should, therefore, view the game in exactly the same way as any other individual at the ex ante stage. We therefore require the game to be played in a symmetric way, and so all of our efficiency-based measures of inequality satisfy the anonymity axiom. The second key axiom is homogeneity. We focus throughout on contests in which payoffs are affine functions of the monetary prizes mainly to allow our models to be compatible with homogeneity. Observe that, in this case, if payoffs are all multiplied by a given factor, then neither the symmetries nor the incentives in the game change (see e.g. Alos-Ferrer and Kuzmics (2013, Proposition 4)), and so the equilibria are invariant.3 Accordingly, it is natural to construct an efficiency-based inequality measure that is homogeneous of degree zero, allowing one to focus attention on relative inequality. There is an important auxiliary reason to focus on affine payoffs. The Harsanyi (1953) version of the veil of ignorance argument asks an individual to evaluate, according to her own utility function, a society, assuming that she wins each prize with 3 That is to say, the equilibrium is unchanged by a common scalar multiple of prizes.

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equal probability. If the individual judges less equal societies to be worse, then it must be that her utility function for money exhibits risk-aversion. The affine payoffs in our models allow us to demonstrate that the efficiency-based measures deliver similar judgments about inequality even when all individuals are riskneutral. In our setup, then, it is the nature of how the contest captures inefficiencies of competition, rather than individual preferences, that drives a judgment in favor of a more equal society. This illustrates how the non-cooperative view of inequality we advance here, even though it operates very differently from an axiomatic approach, permits similar judgments. The third of these axioms is the Pigou–Dalton principle, which is perhaps the single most important axiom that makes these measures really measures of inequality. It requires that the measure respond negatively when transferring wealth from a poorer position to a richer position (see Pigou (1920), Dalton (1920)). We provide conditions for each of our contest models characterizing under which the Pigou–Dalton axiom is satisfied. We now describe the three models and the results we obtain for each. The first model is based on an ‘‘allocation game’’ taken from Kuzmics et al. (2014). In an allocation game, each of n players demands one of n prizes. If all demands are unique, then players receive their demanded prizes. Otherwise, if there is any mis-coordination, all players receive zero. An allocation game can be thought of as a discrete version of a Nash demand game, or alternatively as a generalized version of battle of the sexes. Allocation games represent, in a sense, the simplest model of a contest that admits the features we require and, therefore, provides a proof of concept for the approach taken here. In its unique symmetric equilibrium with positive payoffs, if prizes are all equal then the equilibrium mixture is uniform. As prizes become less equal, the equilibrium strategy places higher probability on better prizes, and this decreases the probability of successful coordination, lowering the expected payoff.4 In fact, we show that the ex-ante payoff of this equilibrium is a monotone transformation of the mean log deviation (MLD) measure introduced by Theil (1967).5 The second model of a contest is a class of repeated allocation games, in which we focus on symmetric and stationary strategies. In this model, mis-coordination is followed by subsequent stages in which players again compete, until successful coordination is achieved. Our motivation for studying these games is twofold. The class of repeated allocation games is still very simple and based on the same ideas as the one-shot allocation games, but are parameterized not only by the prizes at stake but also by a common discount factor. This allows us to introduce a class of efficiency-based measures of inequality, one for each discount factor, nesting the MLD measure from the one-shot game. Moreover this class provides sufficient generality to nest Rawls’s (1971) maxmin measure, which occurs in the limit where players become unboundedly patient. Notably, this result obtains for a risk-neutral decision maker performing an (equilibrium) expected utility calculation. Using this model, in Proposition 1 we 4 This game also has asymmetric equilibria in which each person demands a distinct prize. These asymmetric equilibria do not change with the values of the different prizes and, therefore, do not capture the competition element that we want to model with these games. The ex-ante payoffs in these equilibria are different for each person and, even if we randomly assign people to the role they are supposed to play in one of these asymmetric equilibria, the ex ante payoff would simply be proportional to the sum of all prizes and, thus, cannot be thought of as a measure of inequality. 5 The MLD is an entropy-based measure that has an exact axiomatic characterization provided by Shorrocks (1980), in which it is shown that MLD belongs to a class of entropy measures that all satisfy an additive decomposability axiom. Shorrocks (1980) suggests that, among this class, MLD is particularly attractive in that the coefficients for the decomposability axiom correspond to population shares.

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C. Andonie, C. Kuzmics and B.W. Rogers / Journal of Mathematical Economics 85 (2019) 60–69

then show that the equilibrium payoff of the repeated allocation game satisfies Pigou–Dalton for every discount factor (and so the result applies to the one-shot allocation game as well). The third model we study is more in spirit with the recent literature on contests (see e.g. Konrad (2009) for a survey), in that individuals compete for prizes by exerting unproductive, but costly, effort.6 While we studied this third model in an attempt to make our results more general, we stress that neither this model nor those based on the allocation game are intended to be of practical importance. Rather, the models are meant to distill the forces of competition into a tractable framework. These contests are, it turns out, sufficiently general that their associated efficiencybased measures need not satisfy Pigou–Dalton. In Proposition 3, we provide an exact characterization of the condition on the contest’s allocation rule that corresponds to the Pigou–Dalton property. Essentially, the condition requires that an increase in one individual’s effort must lead to a probability distribution over prizes that dominates the original distribution in terms of the likelihood ratio. We then show that certain parameterizations, in particular an n-prize generalization of the Tullock contest, as in Clark and Riis (1996), Clark and Riis (1998) and Fu and Lu (2012), give rise to an equilibrium that ranks inequality identically to the Gini coefficient. Having now described our models, let us turn attention to the equity-efficiency tradeoff. In the one-shot allocation game, we characterize the condition under which increasing the value of a prize leads to an increase in expected payoff, which captures a net positive effect of such a change to society’s roles. Our result shows that such a change is desirable provided that the prize is not too great compared to the average prize. For the repeated allocation game, it turns out that the answer depends non-trivially on the discount factor, such that for more patient players one may reach the opposite conclusion than is reached for the one shot allocation game. In Proposition 2, we characterize when it is that increasing the value of a prize results in a social improvement. Finally, for the allocation contests with effort exertion, we demonstrate a more specialized result, that increasing the value of the highest prize can have either a positive or negative effect on social welfare. For the case of the n-prize Tullock contest, we develop a simple condition on parameters such that this increase leads to a reduction in expected payoff. Our results, especially those from the allocation contests, have interesting connections to a line of work in the contest literature. For example, Clark and Riis (1998) and Schweinzer and Segev (2012) investigate the distributions of prizes in Tullock contests with complete information that maximize players’ effort. Moldovanu and Sela (2001) and Moldovanu et al. (2012) study the same question, but in contests with incomplete information, and where contestants can be punished for low effort. Other papers, e.g. Hopkins and Kornienko (2010), investigate how the distribution of prizes, or distribution of endowments (e.g. skills) affects contestants’ welfare in contests with incomplete information; or Hopkins and Kornienko (2004), Hopkins and Kornienko (2009) investigate analogous questions in contests where contestants obtain an extra ‘‘gratifying’’ payoff from being high-ranked in the distribution of consumption. Finally, and perhaps closest to the idea of our paper, we know of two papers that connect equilibrium properties of contests to known measures of inequality. First, Esteban and Ray (2011) propose a model of conflict among various groups in a society, and show that the‘‘degree of conflict’’, in equilibrium, is linearly connected to some measures of inequality including the Gini coefficient and the Herfindahl– Hirschman index. Second, Vesperoni (2015) studies a number 6 It seems worthwhile to extend the analysis from this section to more general contests, such as those characterized in Siegel (2009) and Siegel (2014).

of particular contests and shows that the players’ effort as a percentage of the sum of prizes is proportional to the Gini index, or a generalized variation of the Gini index, generated by the prize structure. The settings for which this property holds include the pair-swap contest, as developed in Vesperoni (2016), the best-shot or worst-shot contests, and possibly others. The paper proceeds as follows. Section 2 defines a measure of inequality and presents the axioms of anonymity, homogeneity, and the Pigou–Dalton principle. We study the equilibria of allocation games in Section 3, while we explore the generalization to repeated allocation games in Section 4. Section 5 presents the model of a contest operating through the exertion of costly effort. We conclude and offer final remarks in Section 6. An appendix collects proofs omitted from the main text. 2. Measures of inequality Let X = Rn++ = {x ∈ Rn |xi > 0∀i} be the set of all vectors in Rn with strictly positive entries in all coordinates.7 Let an element x of X be an allocation and, thus, X the set of all allocations. A measure of inequality is a mapping M : X → R attaching to each allocation a single number, the inequality of the allocation. The following three properties (or axioms) have been proposed by Fields and Fei (1978) as essential properties that a measure of inequality should satisfy (see also Sen and Foster (1997), and Foster (1983)). Axiom 1 (Anonymity). A measure of inequality M is anonymous if for any two allocations x, x′ ∈ X such that x′ is a permutation of x we have M(x) = M(x′ ). In words, the measure of inequality does not depend on any characteristic of a person other than its ‘‘income’’ or ‘‘assigned value’’ in the allocation. Axiom 2 (Homogeneity). A measure of inequality M is homogenous (of degree zero) if M(x) = M(α x) for any allocation x ∈ X and any real number α > 0. The idea behind this axiom is not that more income to all would not be an improvement, but that we want to isolate welfare effects due to a change in inequality from those due to a change in overall payoffs. The previous two axioms tell us what we want a measure of inequality not to depend on. The next, key, axiom considers a transfer of income from a poorer individual to a richer individual, in which case inequality must increase. Axiom 3 (Pigou–Dalton). A measure of inequality M satisfies the Pigou–Dalton transfer principle if for any two allocations x, x′ ∈ X such that there are distinct indices i and j and a real number ∆ > 0 with xi ≤ xj and x′i = xi − ∆ and x′j = xj + ∆ while xl = x′l for any l ̸ = i, j we have M(x) < M(x′ ). 3. Allocation games 3.1. The game The essential ingredient we require for a contest is that, as a prize becomes more valuable relative to other prizes, each individual will compete harder to win that prize, such that the fiercer competition is inefficient, lowering equilibrium payoffs. 7 Throughout the paper we shall consider n fixed. If one wanted to discuss properties such as replication invariance one would have to extend this setup to ⋃∞ cover the space X = n=2 Rn++ .

C. Andonie, C. Kuzmics and B.W. Rogers / Journal of Mathematical Economics 85 (2019) 60–69

With this intuition in mind, we propose the following formulation, which follows Kuzmics et al. (2014) in calling an allocation game the following game characterized by a vector x ∈ X . There are n players. Each player has the same set of pure strategies (or actions), denoted by A = {1, . . . , n}. The set of action-profiles is denoted by An . Payoffs to all players are zero unless an actionprofile that is a permutation of (1, 2, 3, . . . , n) is played, in which case the player who plays action i receives payoff xi .

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3.5. Efficiency–inequality tradeoff Suppose we increase one prize xi by a small amount, such that inequality is increased. Under what conditions does this actually decrease the expected payoff V (x)? To answer this we simply need to differentiate V (x) with respect to one coordinate, say x1 . One can show that the sign of this derivative is equal to the sign of the following expression:

∑ 3.2. Symmetric equilibrium

xj − (n − 1)x1 .

j

This game has a unique symmetric equilibrium that generates positive expected payoffs.8 This equilibrium is necessarily in mixed strategies with full support over the set of all actions, x placing probability ∑n i x on action i. The proof of all non-trivial j=1 j

formal statements in Section 3 is given in Appendix A. 3.3. Equilibrium payoff and induced measures of inequality The expected payoff in this equilibrium is given by

∏ V (x) = (n − 1)! (∑

j

j

xj

xj

)n−1 .

j

xj

The intuition is that if x1 is already large relative to the other prizes, an increase of x1 decreases the probability of coordination P(x). In this example, the lower probability of coordination more than offsets the increase in ex post payoff from the higher x1 . In other words, one may reject an increase in all prizes, if this is done unequally, in favor of the more equal original prize distribution.

j

xj

4. Repeated allocation games

Let us denote the event in which the realized action profile played is a permutation of (1, 2, . . . , n) the event of coordination. Then the probability of coordination in this symmetric equilibrium is given by

∏ P(x) = n! (∑

)n .

Note that the probability of coordination is also equal to the normalized expected payoff, ∑ i.e. expected payoff V (x) divided by the average prize x¯ = 1n j xj . 3.4. Pigou-Dalton Since expected payoffs are increasing in the probability of coordination, any inequality measure based on the coordination probability must be a decreasing function. Note that any monotonic transformation of P will ‘‘rank’’ different allocations in exactly the same way as the original function P. So let us take the negative natural logarithm. This gives us (x ) ∑ i − log P(x) = − log(n!) + n log(n) − log , x¯ i

which, for fixed n, is a monotone transformation of MLD(x) = −

For the case of n = 2 the partial derivative is thus always positive. This means that with only two prizes there is never a tension between inequality and total payoff. A benevolent planner could thus be viewed as having lexicographic preferences: first the planner seeks to maximize the sum of the prizes, then secondarily she seeks to distribute this total as evenly as possible. When n ≥ 3 this is no longer universally true. For the case of n = 3, for instance, if x1 already exceeds the sum of x2 and x3 , increasing x1 further, even without a corresponding decrease of x2 and x3 , is detrimental to the expected payoff. As a concrete example consider the two allocations (3, 1, 1) and (4, 1, 1) with 6 V (3, 1, 1) = 25 > 29 = V (4, 1, 1).

1∑ n

i

log

(x ) i

x¯

,

which is the so-called mean log deviation, a well-known entropy measure of inequality due to Theil (1967). The mean log deviation is known to satisfy the Pigou–Dalton transfer principle. It also satisfies, by design, the axioms of anonymity and homogeneity. For a full axiomatic characterization see Shorrocks (1980, 1984). Notice, thus, that our derivation demonstrates that an inequality measure founded on the equilibrium of a contest, and evaluated ex ante behind a veil of ignorance, satisfies all of the axioms that characterize the MLD measure, for a fixed n. 8 There are many symmetric equilibria with zero expected payoff to all players. There are also asymmetric equilibria with positive payoffs. Asymmetric equilibria are of no interest to us, as we want to investigate the difficulty of achieving an asymmetric outcome as a function of the inequality in the promised asymmetric allocation.

4.1. The game Consider a particular repeated version of the allocation game. The game is repeated at discrete points in time until coordination is achieved. There is never any feedback to the players except for whether or not coordination was achieved. Players discount their payoffs with a common discount factor δ < 1. As for δ = 0 the analysis is the same as in the one-shot allocation games from Section 3, the analysis in this section is a generalization of that of Section 3. In principle players can condition their play on their private history, but we assume here that they do not. That is, we investigate stationary symmetric strategies for this repeated game. 4.2. Symmetric equilibrium The stationary symmetric equilibrium of the repeated allocation game is characterized by a (mixed) strategy α = (α1 , α2 , . . . , αn ), which each player uses, at each time t ≥ 1. As in the simple allocation game, we are interested in the (symmetric) equilibrium with positive payoffs, so that all αi > 0. Let us use v > 0 to denote the expected continuation payoff of a player in the repeated allocation game, at some point t, given that coordination has not yet been achieved. Because αi > 0 for all i and from stationarity, we must have, for each i = 1, . . . , n,

( ) v = xi (n − 1)!Πj̸=i αj + δv 1 − (n − 1)!Πj̸=i αj So the expected payoff of each action i, which is the average of xi and the future (discounted) payoff δv , must equal v . These equations can be re-written as: (1 − δ )v = (n − 1)!

Πj αj (xi − δv ) αi

(4.1)

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C. Andonie, C. Kuzmics and B.W. Rogers / Journal of Mathematical Economics 85 (2019) 60–69

for each i = 1, . . . , n and they imply δv < mini xi . We can now analyze these equations to obtain the following lemma, which is proved in Appendix A. Lemma 1. The repeated allocation game with allocation x ∈ X has a unique stationary symmetric equilibrium that generates positive expected payoff. This expected payoff, denoted by V δ (x), is the value v that solves the following (implicit) equation.9

⎞n−1

⎛ (1 − δ )v ⎝

∑

xj − nδv ⎠

= (n − 1)!Πj (xj − δv )

(4.2)

j

4.3. Equilibrium payoff and induced measures of inequality As pointed out in the Introduction, a common affine transformation of the allocation x does not change players’ incentives in the allocation game. It also does not change the players’ incentives in the repeated game. Thus, V δ (x), is homogeneous of degree one, i.e., V δ (λx) = λV δ (x) for all λ > 0.10 To generate a measure of inequality from the expected payoff that is net of total payoff and homogeneous of degree zero, we normalize the expected payoff by dividing by the average payoff in the allocation, x¯ . Denote this measure by V∗δ (x) = 1x¯ V δ (x).11 4.4. Pigou-Dalton Proposition 1. The measure of efficiency defined by the Pigou–Dalton axiom. Proof. See Appendix A.

V∗δ

satisfies

■

Naturally, when δ = 0, by simply plugging this into Eq. (4.2), we obtain our measure derived for the static game in Section 3, Πx i.e. V∗0 (x) = P(x) = n! (∑ jx j)n . On the other hand, as δ → 1, by j j

plugging this into Eq. (4.2) and by noting that for all δ < 1 we must have δv < mini xi by Eq. (4.1), we obtain that V∗δ converges to

xmin 1 n

∑

j xj

. This is Rawls’s (1971) maxmin measure of inequality. V∗δ (x)

From these extreme cases, it is immediate that depends non-trivially on δ and that, in fact, the choice of δ ∈ [0, 1] can be taken to parameterize a class of measures. We find it interesting that the inequality measure derived from the repeated allocation game delivers the maxmin measure, even though all players are risk-neutral expected utility maximizers and, in particular, there is no role of ambiguity aversion in the model. The intuition behind this dependence of the equilibrium payoff, as a function of the vector of prizes, on the discount factor is somewhat subtle. Note that an increase in the discount factor, i.e., an increase in the players’ patience, has two effects on 9 For the special case of n = 2 this equation can be solved analytically and ( ) √ yields V δ (x) = 2δ (21−δ ) (x1 + x2 ) − (x1 + x2 )2 − 4δ (2 − δ )x1 x2 . 10 Alternatively, the fact that V δ (x) is homogeneous follows from Eq. (4.2). 11 There are some other measures of interest that one could derive from the equilibrium in this repeated game. One∏is the (constant) per-period probability of coordination, denoted by Q δ (x) = n! j αj . Using Eq. (4.1), we can relate Q δ (x) to the expected payoff: V∗δ (x) =

1 x¯

V δ (x) =

Q δ (x) 1 − δ (1 − Q δ (x))

4.5. Efficiency–inequality tradeoff We turn now to investigating the conditions under which an increase in the highest prize delivers a social improvement. The next result provides an implicit, but exact, characterization. Proposition 2. The non-normalized expected payoff in the repeated allocation game, V δ (x), decreases in the value of a prize xi if and only if

) n−1 ( xi − δ V δ (x) > x¯ − δ V δ (x). n Proof. See Appendix A.

■

Note that this proposition implies that for n = 2, as in the oneshot allocation game, inequality must be a concern that comes lexicographically after a concern for the sum of all prizes.12 In other words, the non-normalized expected payoff can never decrease if the value of a single prize is increased (no matter how unequal the distribution is). Taking n to infinity we also get the same result as for the oneshot allocation game. If xi is greater than the average prize and is increased non-normalized expected payoff decreases. For intermediate n the condition given here is somewhat different than the condition we obtained in the one-shot allocation game. For the one-shot allocation game increasing the value of a 1 prize xi is detrimental to expected payoff if and only if n− xi > x¯ . n

In the repeated game with discount factor δ the condition is n−1 xi > x¯ − 1n δ V δ (x). The first condition implies the latter, but not n vice versa. Thus, an increase in xi may be detrimental to expected payoff in the repeated game but not in the one-shot game. Consider an example with allocations x = (3, 1, 1) and x′ = (3, 3, 1). Which one of these allocations is preferred? In the one 18 54 shot game, we have P(x) = 125 < 343 = P(x′ ). Thus, for sufficiently small δ , we also have V∗δ (x) < V∗δ (x′ ). However,

.

Note that V∗δ (x) is a monotone transformation of Q δ (x), so both measures rank societies in the same way. Of course, to interpret these as measures of inequality one needs to reverse their sign. Another measure∑of potential interest is the δ t −1 δ expected time to coordination, given by T δ (x) = Q (x) = t ≥1 t(1 − Q (x)) 1 . Q δ (x)

the equilibrium payoff for a fixed vector of prizes x. As a first, direct, effect it increases the payoff from any fixed stationary and completely mixed strategy. This is simply because such a strategy has a fixed (geometric) distribution governing how long it takes for coordination to be achieved that involves some delay and the more patient players are the less they care about this delay. But there is also a second, indirect, effect: if players are more patient they are willing to go for higher prizes with a higher probability, as they are less concerned about the risk of miscoordination today. This effect is detrimental to the equilibrium payoff, as it increases the expected delay. It is not intuitively obvious which of these effects dominates. More specifically for the present problem, how does a change in discount factor change the dependence of the equilibrium payoff on the vector of prizes? If we raise the value of one of the higher prizes when players are impatient, then they will all go for this prize with only a slightly higher probability, as there is little future value of not coordinating today. But if players are patient, they do not mind waiting for coordination in the future and therefore raise the probability of going for the higher prize more dramatically. Our result shows that, in equilibrium, the value of increasing any non-minimal prize is completely offset by this behavior.

considering δ → 1, we have that V∗1 (x) =

3 5

>

3 7

= V∗1 (x′ ),

showing that the social preference is reversed for sufficiently high discount factors. 12 For n = 2 the condition above reduces to x min < δv , which can never be true.

C. Andonie, C. Kuzmics and B.W. Rogers / Journal of Mathematical Economics 85 (2019) 60–69

5. Contests In this section we study a contest, which is a more general model of competition. Whereas before players competed by simply stating which prize they would like, in the contest they instead choose an effort. Effort in this model is, as in Spence’s (1973) signaling model, completely unproductive and costly. Intuitively, as prizes become less equal, in the allocation game the equilibrium payoff decreases because players’ miscoordinate with higher probability. In the contest, the equilibrium payoff will decrease because players are induced to exert more effort. Of course, when prizes are all equal players exert no effort. Both models capture an equilibrium cost of competing for less equal prizes. After describing the model in Sections 5.1 and 5.2, we characterize in Section 5.3 exactly when the measures from the allocation contest satisfy the Pigou–Dalton criterion, and so can be interpreted as a measure of inequality. In Section 5.4 we show that a special case of the allocation contest produces a measure that ranks inequality identically to the Gini coefficient. In Section 5.5 we study the tradeoff between greater inequality and greater wealth in the contest model.

(i) both ϕ and c are twice continuously differentiable, (ii) ∑ c is increasing and convex, with c ′ (0) = 0 and c ′ (∞) = ∞, ∂ j (iii) j xj ∂ e ϕi (e, . . . , e) is positive, and it is a decreasing funci

∂ ∂ tion in e, with j xj ∂ ei ϕi (∞, j xj ∂ ei ϕi (0, . . . , 0) = ∞ and .∑ . . , ∞) = 0, ∑ j j ∂2 ∂2 (iv) j xj ∂ (e )2 ϕi (e, . . . , e) and j xj , ∂ e ∂ e ϕi (e, . . . , e) are non-

such a way that the cost is homogeneous ∑ of degree one in We shall write this cost function as j xj c(ei ).

∑

j

xj .13

If we add to all bijections also the null-element to indicate that no player gets anything, then the setup of the allocation contest is flexible enough to accommodate the allocation games of Section 3 as a special case of allocation contests. For this we have to set the effort cost function to zero (thus it is homogeneous of degree one) and make ϕ such that only for any permutation of a prespecified set of n distinct effort levels does it provide a match. The utility function to any player i is given by ui (ei , e−i ) =

∑ j

j

xj ϕi (ei , e−i ) −

∑

xj c(ei ).

j

5.1.1. Further assumptions To obtain a symmetric game we require that ϕ is anonymous, i.e. that the probability that a player gets a certain prize, given that player’s effort choice, depends only on the profile of other players’ effort choices and not on the identity of which of the other players chooses what. This implies in particular that if all players choose the same level of effort, they all have an equal chance of getting any given prize. We also employ the following technical assumptions for convenience, for all i, k ∈ I: 13 While this assumption is perhaps unnatural, we make it in order to best compare our approach to the axiomatic approach. Nothing about the allocation contest requires such an assumption.

∑

i

i

j

k

positive for all e. The interpretation of (iii) is that, at any symmetric effort profile, if a player i increases her effort she increases the expected value of the prize she will get, where this increase is greater for lower levels of effort from the other players. Condition (iv) means that the marginal effect of effort of one player on this player’s expected value of the prize is decreasing, and it is reduced by an increase of another player’s effort. 5.2. Symmetric equilibrium Assuming an interior solution (see below) the first order condition for the optimal choice of effort for player i is given by

j

We will study an allocation contest for a given allocation x ∈ X , which is a game with I = {1, . . . , n} players, with E = R+ the set of available non-negative effort levels for each player, and payoffs given by the element of the allocation the player receives minus a cost of producing effort. For a fixed allocation x ∈ X let B denote the set of all bijections (matchings) from the set of prizes {x1 , . . . , xn } to the set of players I. A class of allocation rules is then characterized by a function, ϕ , from the set of all effort profiles, i.e. Rn+ , times the set of allocations X , to the set of all probability distributions over B. Let ϕij (e1 , . . . , en ) denote the probability, induced by ϕ , that player i receives prize xj for the given effort profile and for a fixed allocation x ∈ X . The cost of effort is assumed to be a function of the sum of ∑ all values in the allocation, j xj , and the chosen level of effort in

j

∑

∑ 5.1. The game

65

xj

∑ ∂ j ϕi (ei , e−i ) − xj c ′ (ei ) = 0. ∂ ei j

In a symmetric equilibrium, in which all players choose the same effort level e∗ we, therefore, must have

) ∑ ∑ ( ∂ j ∗ ∗ xj ϕ (e , . . . , e ) = xj c ′ (e∗ ). ∂ ei i j

j

By assumption, the left hand side decreases continuously in e∗ from ∞ to 0 and the right hand side increases continuously in e∗ from 0 to ∞. So there must be a unique solution. Thus, if there exists a symmetric equilibrium of the allocation contest it must be unique. Our assumptions also imply that the second-order condition (for a maximum) is satisfied at this candidate equilibrium point. We therefore know that at the candidate equilibrium all players use a local best response. The candidate equilibrium effort e∗ is also a global best response if, e.g., the expected benefit ∑ j ∗ ∗ j xj ϕi (ei , e−i ) is concave in ei . In this case, the utility ui (ei , e−i ) ∗ is also concave in ei and therefore e must be a global maximum. However, such an assumption is not necessary for a symmetric equilibrium to exist. For example, the Tullock contest studied in Section 5.4 may have a symmetric equilibrium e∗ even if the utility ui (ei , e∗−i ) is not concave everywhere (see the discussion in footnote 17). 5.3. Pigou-Dalton The expected payoff (to any player) in the interior symmetric equilibrium is given by U(x) =

1∑ n

j

xj −

∑

xj c(e∗ ),

j

which is homogeneous of degree one. If one desires instead a measure that is homogeneous of degree zero, so as to satisfy the homogeneity axiom, one can use the cost of effort as such a measure. Note, however, that this is simply a monotone transformation of equilibrium effort, which we will denote by E(x) and, thus, ranks allocations in exactly the same way. We now provide necessary and sufficient conditions on ϕ under which E(x) satisfies the Pigou–Dalton axiom. Let p and q be two discrete probability distributions with support in {x1 , . . . , xn }. Distribution p dominates distribution q in p(x ) p(x ) terms of the likelihood ratio if q(xi ) < q(xj ) if and only if xi < xj . In i

j

66

C. Andonie, C. Kuzmics and B.W. Rogers / Journal of Mathematical Economics 85 (2019) 60–69

other words, p dominates q if the relative likelihood of obtaining the higher prize compared to the lower prize is higher under p than under q.14 Proposition 3. The measure E(x) (and U(x)) satisfies the Pigou– Dalton axiom if and only if, for all possible symmetric equilibrium effort levels e∗ there is an ϵ¯ > 0 such that for every ϵ ∈ (0, ϵ¯ ) the distribution ϕ1 (e∗ + ϵ, e∗ , . . . , e∗ ) dominates the uniform distribution in terms of the likelihood-ratio.15 Proof. See Appendix A.

■

To understand the result, recall first that at a symmetric equilibrium all players obtain the same distribution over prizes, which implies that ϕ1 (e∗ , . . . , e∗ ) is uniform. The condition expressed in this result requires that, at the interior equilibrium, when a player increases her effort slightly, she obtains a better distribution over prizes, where ‘‘better’’ is in the sense of likelihood ratio dominance. An inequality-increasing Pigou–Dalton transfer causes equilibrium effort to increase, exactly when such an increase in effort leads to unambiguously better outcomes.

From this we can write the player’s expected utility from the contest as u(e, g) =

n ∑

xk ϕ k (e, g) −

k=1

To find a symmetric equilibrium we must identify the condition for which the player’s optimal effort level is e = g. To proceed we will choose, for convenience, c(e) = 12 e2 .16 To derive the first order condition for maximization of u we first compute the derivative of ϕ k (e, g) with respect to e. This is:

⎛

1 ∂ϕ (e, g) = ⎝ βk − ∂e e k

(ei )β (e1

)β

+ (e2 )β + · · · + (en )β

,

(5.1)

for some given β > 0 (see Tullock (1980)). Following Clark and Riis (1996), Clark and Riis (1998) and, more recently, Fu and Lu (2012), we specify an n-prize generalization of the Tullock contest for prizes x1 ≥ x2 ≥ · · · ≥ xn , which works as follows. Players simultaneously choose effort levels once and for all. Then the highest prize x1 is allocated according to probabilities (5.1) with β = β1 . The winner is removed, and a Tullock contest among the remaining n − 1 players is run for the second prize x2 , using β = β2 , with the already given effort levels. The contest proceeds iteratively in this fashion until each prize is allocated to a unique player. The generalized Tullock contest is thus parameterized by the vector β = (β1 , . . . , βn ). As we shall be interested in the symmetric equilibria of this contest, we can simplify notation. We shall look at an arbitrary player and call her level of effort e, while the (identical) effort level of all other players effort will be denoted g. We then have that the probability that this player wins prize xk is given by

( ) k−1 ∏ n−1 eβk g βj ϕ k (e, g) = (k − 1)! k − 1 eβk + (n − k)g βk eβj + (n − j)g βj j=1 =

k−1 (n − 1)! ∏

(n − k)!

−

g βj βj

j=1

e + (n − j)g βj

(n − 1)! (n − k − 1)!

k ∏ j=1

g βj eβj + (n − j)g βj

(

∑n

k=1

the distribution ϕ1 (e∗ − ϵ, e∗ , . . . , e∗ ) is dominated by the uniform distribution in terms of the likelihood-ratio.

j=1

⎞

βj n−j+1

xk βk −

∑k

∑n

xk

n

⎠ ϕ k (e, g).

βj

)

j=1 n−j+1

k=1

.

We now show that through an appropriate choice of the contest parameters β , the equilibrium effort level of the n-prize Tullock contest ranks inequality of the prizes identically to the Gini coefficient. That is, given an appropriate specification of the contest parameters, one set of prizes x is more unequal than another set x′ according to the Gini coefficient, if and only if the equilibrium effort for x is higher than for x′ . Let us denote the Gini coefficient of a vector x by G(x). Proposition 4. There exist parameters β of the generalized Tullock contest such that for any two prize vectors x and x′ , we have G(x) > G(x′ ) if and only if E(x) > E(x′ ). Proof. For a given prize vector x with x1 ≥ · · · ≥ xn we can write the Gini coefficient as (see, e.g., Sen and Foster (1997))

∑n G(x) =

k=1 (n

n

− 2k + 1)xk ∑n . k=1

xk

For any γ > 0, it is straightforward to check that if we set βk = γ (n − k + 1) for each k then (e∗ )2 , and so also E = e∗ , is a monotonic transformation of G. We also need to verify that efforts e∗ , which satisfy the necessary first order conditions, are, in fact, an equilibrium. Consider the following: Claim. If βj ≤ 1 for all j, then concave in e.

∑n

k=1

xk ϕ k (e, g) is increasing, and

Combined with the assumption that c is increasing and convex, the claim implies that u(e, g) is concave in e, which implies that e∗ is a global maximum, so that the local maximum is indeed an equilibrium. We conclude by verifying the claim. We write: n ∑ k=1

xk ϕ k (e, g) =

n ∑ k=1

− 14 See, for instance, Krishna (2010, Appendix B) for this definition. There it is also stated that likelihood ratio dominance implies, among other notions of dominance, first-order stochastic dominance. 15 By the assumed smoothness of ϕ we then also have that for small ϵ < 0

k ∑

It is straightforward to see that ϕ k (e, e) = 1n for all k and all e > 0, i.e. the contest is a uniform lottery when efforts are all identical. Requiring that e = g solves the first order condition, ∑n ∑ ∂ϕ k (e,g) = nk=1 xk c ′ (e), gives rise to equilibrium effort k=1 xk ∂e (e∗ )2 =

ϕi1 (e1 , e2 , . . . , en ) =

xk c(e).

k=1

5.4. Tullock contests and the Gini coefficient We now study a special case of our contest model, which can be interpreted as a multi-prize version of the Tullock contest. In a Tullock contest for a single prize x1 among n players, the probability that any player i wins is given by

n ∑

xk (

k−1 (n − 1)! ∏

(n − k)!

(n − 1)! (n − k − 1)!

g βj βj

j=1 k ∏ j=1

e + (n − j)g βj g βj eβj + (n − j)g βj

)

16 This choice is without loss in that any other cost function that is increasing and convex will generate a monotone transformation of the equilibrium effort we derive below, and so it would produce an identical ordinal ranking.

C. Andonie, C. Kuzmics and B.W. Rogers / Journal of Mathematical Economics 85 (2019) 60–69 n−1 ∑

= x1 −

(xk − xk+1 )

k=1

×

(n − 1)! (n − k − 1)!

k ∏

g βj βj

j=1

e + (n − j)g βj

From this expression, it is clear that when βj ≤ 1 for all j, βj

g β β is positive, decreasing, and convex in e. This implies e j +(n−j)g j β k g j that j=1 βj β is also positive, decreasing, and convex in e. e +(n−j)g j

∏

As xk ≥ xk+1 for each k = 1, . . . , n − 1, the claim is verified. Note, therefore, that through a small enough choice of γ above (γ = 1/n suffices), we exhibit a contest for which the unique symmetric equilibrium effort is a monotonic transformation of the Gini coefficient. ■ We next return to studying the Pigou–Dalton property, for the particular case of the generalized Tullock contest. Using Proposition 3, the measure E(x) satisfies the Pigou–Dalton axiom if and only if, for all possible equilibrium efforts e∗ , there exists a small threshold ϵ > 0, such that for all ϵ ∈ (0, ϵ ) the distribution (for, e.g. player 1) ϕ1 (e∗ + ϵ, e∗ , . . . , e∗ ) dominates the uniform distribution in terms of the likelihood-ratio. More explicitly, the condition is that

ϕ + ϵ, e∗ , . . . , e∗ ) ≤ 1 for all 1 ≤ k ≤ n − 1 and all + ϵ, e∗ , . . . , e∗ ) ϕ ϵ ∈ (0, ϵ ). k+1 ∗ (e 1 k ∗ 1 (e

This condition holds if and only if the function:

ψ k (e, e∗ ) =

ϕ ϕ

,

,

i.e. if and only if ∂ψ k (e=e∗ ,e∗ ) ∂e

,...,e ) is non-increasing at e = e∗ , , . . . , e∗ )

k+1 (e e∗ 1 k ∗ 1 (e e

=

∗

∂ψ k (e=e∗ ,e∗ ) ∂e

β

1 ( k+1 (1 e∗

following lemma.

≤ 0. Because in the Tullock contest − n−1 k ) − βk ), we therefore have the

Lemma 2. The measure E(x) satisfies the Pigou–Dalton axiom in 1 the generalized Tullock contest if and only if βk ≥ βk+1 (1 − n− ) for k all k = 1, 2, . . . , n − 1. The conditions of Lemma 2 require, essentially, that the sensitivity of winning a prize to the effort level are sufficiently greater for higher prizes than for lower prizes. These Pigou–Dalton conditions are satisfied for the specification implying the Gini coefficient, ie. βk = γ (n − k + 1) for all k. They are also satisfied for other common specifications, e.g. βk = β > 0 for all k. 5.5. Efficiency-inequality tradeoff We provide here a simple result that allows us to conclude that the efficiency-inequality tradeoff is nontrivial for the allocation contest model. That is, increasing inequality while increasing the sum of all prizes can lead to an increase or a decrease in the ex-ante equilibrium-payoff in contests, depending on the details of the contest. We then apply this result to the generalized Tullock contest. ∑ ∑ ∗ Differentiating U(x) = 1n j xj − j xj c(e ) with respect to x1 we have, immediately: Lemma 3. The non-normalized expected equilibrium payoff U(x) increases in x1 if and only if 1 n

− c(e∗ ) −

∂ e∗ ′ ∗ ∑ c (e ) xj > 0. ∂ x1 j

67

Let us discuss when the condition of Lemma 3 will hold. In contests for which the equilibrium effort E(x) is an inequality measure (i.e. such that E(x) satisfies the Pigou–Dalton axiom, as characterized by the conditions given in Proposition 3), an increase of the largest prize x1 makes the allocation x more ∗ ‘‘unequal’’. We thus must have ∂∂ xe > 0. Because both c(e∗ ) and 1

c ′ (e∗ ) are monotone in e∗ and because ∂∂ xe > 0, Lemma 3 then 1 suggests that an increase of x1 will decrease utility U(x) when: ∗

(1) the effort e∗ is large; ∗ (2) the marginal effect ∂∂ xe > 0 is large. 1

Both effects (1) are (2) are possible in contests where the probability of winning the largest prize x1 is sensitive to marginal ∂ϕ 1 (ei ,e−i )

changes in effort, i.e., in contests where, all else equal, i ∂ e i is large. For example, in the Tullock contest from Section 5.4, the sensitivity of ϕi1 (ei , e−i ) to effort ei is given by β1 . Therefore, the utility U(x) should decrease if β1 is large. We can indeed verify this intuition directly. In the Tullock contest, the equilibrium payoff is given by U(x) =

n ∑

xk ϕk (e∗ , e∗ ) −

k=1

k=1 n

=

=

1∑ n

n

xk −

k=1

n 1∑

n

n ∑

∑

xk

k=1

1

k=1

1 2

(

∑n

k=1 xk βk −

2

⎛ xk ⎝ 1 −

1 xk (e∗ )2 2

βk +

n

∑n

k 1∑

2

j=1

βj j=1 n−j+1

∑k

k=1

xk

⎞

βj n−j+1

In particular, the coefficient of x1 is 1 − negative if β1 > 2 1 . Therefore, if β1 > 1− n

)

1 2

⎠.

β1 + 2

1− 1n

1 β1 , 2 n

which is then an increase

to x1 decreases expected payoff.17 In this case the cost of the marginal effort exerted in equilibrium outweighs the marginal gain in ex post social wealth. 6. Conclusion We propose a novel methodology within which to consider inequality. Our method operationalizes the assignment of individuals to places in society under a veil of ignorance thought experiment. This assignment is modeled explicitly through a game. The individual, at the ex ante stage, when evaluating a society, rests her judgments upon the equilibrium properties of this game, which we think of as a contest. The essential ingredient of the contest – that which allows it to serve as a medium for inequality judgments – is that, as a prize becomes increasingly valuable relative to the other possible prizes, individuals optimally compete harder to be assigned that prize. The increased competition results in inefficiencies, modeled here either as a possibility of mis-coordination or as an unproductive costly investment, which, in either case, decreases the equilibrium payoff. It is for this reason that one can use properties of the equilibrium to serve as measures of inequality. 17 We showed in Section 5.4 that if β ≤ 1 (and all other β ≤ 1) then 1 j the Tullock contest has a symmetric equilibrium e∗ . When β1 > 1, which is 2 required for U(x) to decrease in x1 because 1 > 2, the contest may or may 1− n

not have a symmetric equilibrium e∗ . Its existence depends on ‘‘how convex’’ the cost function is. As an example, when there are three prizes, and the cost function is c(e) = 12 e2 , a symmetric equilibrium exists if β1 < 24 . Thus, when 5

n = 3 prizes, if β1 ∈ (3, 24 ) then the equilibrium utility decreases in x1 . More 5 generally, the ‘‘more convex’’ the cost function is, the larger the range of β1 for which the contest has a symmetric equilibrium.

68

C. Andonie, C. Kuzmics and B.W. Rogers / Journal of Mathematical Economics 85 (2019) 60–69

From this perspective, we show, perhaps surprisingly, that simple models of contests result in inequality measures that are closely related to measures that have been studied axiomatically, including, in particular, the mean log deviation measure and the Gini coefficient. One can view this either as an independent justification for such measures, or as a way to better understand the properties of the contest model we propose. We hope that these ideas may lead to further research. Obviously, the inequality measure depends crucially on how the contest is modeled. Just as there are many different axioms and measures in the literature, there may be many other models of contests that prove interesting. Our analysis has intentionally focused on very simple contests, which has allowed us to study their properties and connect them to existing measures most easily. One potential area for further work is to expand the contest models to include considerations of individual heterogeneity, partially productive effort, partial miscoordination, and so forth, so as to bring the contests closer in line with salient features of the economy. Appendix A. Proof of statements in Section 3 In a symmetric equilibrium all n players use the same mixed strategy σ = (σ1 , σ2 , . . . , σn∑ ) with σi ≥ 0 (the probability n attached to action i ∈ A) and i=1 σi = 1. Suppose there is an i ∈ A with σi = 0. This implies that no player’s realized action choice is equal to action i, but then, given the definition of an allocation game, all players receive a payoff of zero. Thus, for any such σ the expected payoff if all players use σ is zero. To obtain a positive payoff in a symmetric equilibrium we thus need to have that for all i ∈ A σi > 0. This means that, for σ to constitute a symmetric equilibrium players have to be indifferent among all pure actions. Now consider an arbitrary player, w.l.o.g player 1, and suppose mixed strategy σ is used by all players other than player 1. Player 1’s expected ∏ payoff from using action i ∈ A is then given by xi (n − 1)! j̸=i σj , as only if realized play is a permutation of (1, 2, . . . , n) does player 1 obtain a positive payoff (of in fact xi ). For σ to be a symmetric equilibrium we must ∏ thus have that there is a λ > 0 such that for all i ∈ A, xi j̸=i σj = λ. Multiplying (∏n this)expression with σi on both ∑n sides we obtain that σi = xi σ /λ . The condition that j j=1 i=1 σi = 1 then yields that λ =

∏n

j=1

σj

(∑n

)

i=1

xi and thus, the desired result that the

unique positive payoff symmetric equilibrium σ is such that xi

σi = ∑n

j=1

.

xj

The equilibrium expected payoff is then given by∏ the expected payoff for any action i under σ , which is xi (n − 1)! j̸=i σj . If we x use σi = ∑n i x in this expression we obtain the desired j=1 j

∏ V (x) = (n − 1)! (∑

n!

j

j

xj

xj

j

xj

j

xj

)n .

Finally, differentiating V (x) with respect to x1 yields

∂ V (x) = (n − 1)! ∂ x1

∏

j̸ =1

xj

xj ⎝

is the same as the sign of

⎞n−1

⎛ ∏

∂ V (x) ∂ x1

∑

xj ⎠

⎞n−2

⎛ − (n − 1) ⎝

∑

j

j̸ =1

∏

xj ⎠

j

xj ,

j

which has the same sign as

⎛ ∏

xj ⎝

∑

⎞n−2 ⎡ ⎤ ∑ ⎣ xj ⎠ xj − (n − 1)x1 ⎦ ,

j

j̸ =1

j

and hence has the same sign as

∑

xj − (n − 1)x1 .

j

Appendix B. Proof of Lemma 1 Summing∑ Eqs. (4.1) for all i = 1, 2, . . . , n, we obtain (1 −δ )v = (n − 1)!Πj αj ( j xj − nδv ). Using Πj αj from this equation in each of x −δv

the Eqs. (4.1), we have: αi = (∑ xi −nδv ) . Now, given this expression j for αi ’s as a function of v , we can use it in Eq. (4.1) to obtain (4.2). As we already noted, we have δv < xmin , where xmin = minj xj . To show that there is a unique v that solves Eq. (4.2) let us define

⎛ g(v ) = ln (1 − δ) + ln v + (n − 1) ln ⎝

⎞ ∑

xj − nδv ⎠

j

− ln ((n − 1)!) −

∑

ln xj − δv ,

(

)

j

which is obtained from (4.2) by taking logs. Thus we must show there exists a unique zero of g(v ) on v ∈ [0, 1δ xmin ]. It is easy to check that g(0) = −∞ and g( 1δ xmin ) = +∞, so there exists a solution because g() is continuous. To show uniqueness we demonstrate monotonicity of g. We have dg(v ) dv

=

1

v

−

(n − 1)δ

∑ 1 n

j (xj

− δv ) ∑ 1

+δ

∑

1

j

xj − δv

,

1 which is positive as ( n j xj − δv ) j xj −δv ≥ n, which in turn follows from an application of the Cauchy–Schwarz inequality √ ∑ ∑ ∑ 2 2 2 (a ) (b ) ≥ ( a b ) where a = x − δv and bj = j j j j j j j j √1 .

∑

xj −δv

Appendix C. Proof of Proposition 1 Take an allocation x with xi < xk , for some i ̸ = k. We will prove that V δ (x) increases∑if some ∆ > 0 is transferred from xk to xi (thus, not changing j xj ). Consider function g(·) from the proof to Lemma 1. We observe that the transfer ∆ decreases g due to the concavity of the last term in x. Because g increases in v , we then have that V δ (x), which is the unique v that satisfies g(v ) = 0, must increase. Appendix D. Proof of Proposition 2

)n−1 .

The equilibrium probability of coordination is then given by ∏ n ∑nxi , we obtain the desired i=1 σi and, again using σi = j=1 xj ∏

P(x) = n! (∑

The sign of

(∑

j

xj

)n−1

− (n − 1) (∑ )2(n−1) j

xj

(∑

j

xj

)n−2 ∏

j

xj

.

To see this take logs on both sides of Eq. (4.2). Fixing v = V δ (x) take derivatives of the left-hand side and the right hand side with respect to xi . For the left hand side this is ∑ nx−−1nδ V . For the right j j

1 hand side this is x −δv . Thus an increase in xi (not changing v ) i increases the left-hand side ∑ more than the right hand side if and 1 only if n− (xi − δv) > 1n j xj − δv . In this case the value of the n

function g(v ) (defined in the proof of Lemma 1) increases as xi increases, while v is kept fixed. As g is an increasing function and as we need g(v ) = 0 in equilibrium we must have that V δ (x) decreases as a response to an increase in xi under the stated condition.

C. Andonie, C. Kuzmics and B.W. Rogers / Journal of Mathematical Economics 85 (2019) 60–69

Appendix E. Proof of Proposition 3

69

dominates the uniform distribution in terms of the likelihood ratio, etc.

In this proposition, we assume that the probability functions

{ϕ1l (.)}l=1..n depend only on the ranking of prizes, but not on the values of prizes. An example is that of Tullock contests, as in Section 5.4. Consider allocations x and x′ and ∆ > 0 such that x′i = xi − ∆ and x′j = xj + ∆ and x′l = xl if l ̸ = i, j. Assume first that ∆ is small, so the ranking of prizes is unchanged, and therefore the probability functions {ϕ1l (.)}l=1..n remain unchanged when going from x to x′ . At e∗ = E(x) we must have of player 1 ( that the marginal utility ) in her effort choice is

∑

l

xl

∂ ϕ l (e∗ , . . . , e∗ ) ∂ e1 1

− c ′ (e∗ ) = 0. We

shall now investigate under what conditions player 1’s marginal utility, given allocation x′ , at the same effort level e∗ is positive. This would mean that E(x′ ) > E(x). Let, for any l, al = ∂ ϕ l (e∗ , . . . , e∗ ) − c ′ (e∗ ). Player 1’s marginal utility, given allo∂ e1 1 ∑ ′ ∗ cation x′ and given ∑effort level e is then∑given by l xl al which can be written as l xl al − ai ∆ + aj ∆. As l xl al = 0 the marginal utility is simply −ai ∆ + aj ∆ and, thus, positive if and only if aj > ai , i.e.,

∂ i ∗ ∂ j ∗ ϕ (e , . . . , e∗ ) > ϕ (e , . . . , e∗ ). ∂ e1 1 ∂ e1 1 As ϕ is assumed to be smooth, the last inequality is equivalent to:

ϕ1j (e∗ + ϵ, e∗ , . . . , e∗ ) − ϕ1j (e∗ , e∗ , . . . , e∗ ) ϵ ϕ1i (e∗ + ϵ, e∗ , . . . , e∗ ) − ϕ1i (e∗ , e∗ , . . . , e∗ ) > ϵ for ϵ small. Because ϕ1k (e∗ , e∗ , . . . , e∗ ) = small, this is the same as:

1 n

for all k, for ϵ > 0 and

ϕ1j (e∗ + ϵ, e∗ , . . . , e∗ ) ϕ i (e∗ + ϵ, e∗ , . . . , e∗ ) > 1 . 1/n 1/n Thus, if xj ≥ xi , Pigou–Dalton requires E(x′ ) > E(x) which is equivalent to ϕ1 (e∗ + ϵ, e∗ , . . . , e∗ ) dominating the uniform distribution in terms of the likelihood ratio, and conversely for xj < xi . Suppose now that ∆ is ‘‘large’’ so the ranking of prizes is changed. Let us then decompose the transfer ∑ ∆ into a finite sequence of sub-transfers, say {∆k }k with k ∆k = ∆, with the property that each sub-transfer ∆k may change the ranking of prizes only at its endpoints, where it can make two prizes identical. We observe that if two (or more) prizes are equal in an allocation, then the equilibrium effort e∗ will not depend on the order the probabilities ϕ1l are associated with the prizes that are identical. This implies that the equilibrium effort e∗ will continuously vary with the transfers ∆k ’s and will not jump when probability functions {ϕ1l (.)}l=1..n change as a result of a change in the ranking of prizes. Therefore, we can proceed by applying sequentially the argument from above. For example, if xj > xi , we can apply the argument sequentially for each sub-transfer ∆k to obtain that E(x′ ) > E(x) if and only if ϕ1 (e∗ + ϵ, e∗ , . . . , e∗ )

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