Are sequential round‐robin tournaments discriminatory?

I examine sequential round‐robin tournaments with three and four symmetric players. Each player is matched once with each other player and each match is organized as a Tullock contest. A single prize is allocated to the player with the most matches won. I show that, depending on the position of their matches in the sequence of the tournament, the players' ex ante winning probabilities and expected payoffs differ. Compared to tournaments with matches organized as all‐pay auctions, however, discrimination is weaker and partially reversed. Moreover, aggregate effort in tournaments with suitable Tullock contests is larger than in tournaments with all‐pay auctions. The fairness of round‐robin tournaments may be improved by an endogenous sequence of matches or the requirement that players fix their effort ex ante.     

Optimal seedings in elimination tournaments

We study an elimination tournament with heterogenous contestants whose ability is common-knowledge. Each pair-wise match is modeled as an all-pay auction. Equilibrium efforts are in mixed strategies, yielding complex dynamics: endogenous win probabilities in each match depend on other matches’ outcome through the identity of the expected opponent in the next round. The designer seeds competitors according to their ranks. For tournaments with four players we find optimal seedings for three different criteria: (1) maximization of total tournament effort; (2) maximization of the probability of a final among the two top ranked teams; (3) maximization of the win probability for the top player. We also find the seedings ensuring that higher ranked players have a higher winning probability. We compare our predictions with data from NCAA basketball tournaments.     

On the Coalition-Proofness of the Pareto Frontier of the Set of Nash Equilibria

This article shows that the Pareto efficient frontier of the Nash equilibrium set of games with strategic substitutes is coalition-proof under the following conditions: (1) the game has three players, or, alternatively, a player's payoff depends on her own strategy and on the sum (but not on the composition) of other players' strategies; (2) an increase in a player's strategy either raises all other players' payoffs monotonically or reduces them monotonically; and (3) each player's payoff is strictly concave in her own strategy. Under these conditions, the Pareto dominance refinement is equivalent to the coalition-proof Nash equilibrium refinement.Journal of Economic LiteratureClassification Number: C72.     

Doping and cheating in contest-like situations

Individuals who compete in a contest-like situation (for example, in sports, in promotion tournaments, or in an appointment contest) may have an incentive to illegally utilize resources in order to improve their relative positions. We analyze such doping or cheating within a tournament game between two heterogeneous players. Besides the direct and indirect costs of doping, three major effects are identified which determine a player's decision to deviate from a no-doping situation — a cost effect, a likelihood effect and a base-salary effect. Moreover, the influence of heterogeneity and exogenous performance risk on a no-doping outcome, and the influence of ex-ante and ex-post doping tests on the players' investments are discussed.     

On the Competition of Asymmetric Agents

Abstract. Rank-order tournaments are usually implemented in organizations to provide incentives for eliciting employees' effort and/or to identify the agent with the higher ability, for example in promotion tournaments. We close a gap in the literature by experimentally analyzing a ceteris paribus variation of the prize spread – being the major design feature of tournaments – in a symmetric and an asymmetric setting. We find that effort significantly increases with the prize spread as predicted by standard theory. However, only for sufficiently large prize spreads weak players competing against strong players strain themselves all the more and sorting of agents is feasible.     

Belief-Based Equilibria in the Repeated Prisoners' Dilemma with Private Monitoring

We analyze repeated prisoners' dilemma games with imperfect private monitoring and construct mixed trigger strategy equilibria. Such strategies have a simple representation, where a player's action only depends upon her belief that her opponent(s) are continuing to cooperate. When monitoring is almost perfect, the symmetric efficient outcome can be approximated in any prisoners' dilemma game, while every individually rational feasible payoff can be approximated in a class of such games. The efficiency result extends when there are more than two players. It requires that monitoring be sufficiently accurate but does not require very low discounting when a public randomization device is available. Journal of Economic Literature Classification Numbers: C73, D82.     

Interim Bayesian Nash equilibrium on universal type spaces for supermodular games

We prove the existence of a greatest and a least interim Bayesian Nash equilibrium for supermodular games of incomplete information. There are two main differences from the earlier proofs and from general existence results for non-supermodular Bayesian games: (a) we use the interim formulation of a Bayesian game, in which each player's beliefs are part of his or her type rather than being derived from a prior; (b) we use the interim formulation of a Bayesian Nash equilibrium, in which each player and every type (rather than almost every type) chooses a best response to the strategy profile of the other players. There are no restrictions on type spaces and action sets may be any compact metric lattices.     

Contests where there is variation in the marginal productivity of effort

Summary. We provide a characterization of participants' behavior in a contest or tournament where the marginal productivity of effort varies across contestants and individual productivity is private information. We then consider the optimal design of such a contest. We first analyze contestant behavior for the usual type of contest, where the highest output wins. Abilities need not be independently distributed. We demonstrate that there is a unique symmetric equilibrium output function, that output is increasing in ability, and that marginal effort is increasing in ability, while effort decreases when the cost of effort increases. Next we consider the case where the highest output need not win, with independently distributed abilities. We analyze the contest designer's decisions in choosing contest rules optimal from her perspective. We show that the output produced, probability of winning, and contest designer's expected revenue are generally increasing in contestants' ability. We examine the relationship between the marginal cost of producing output and marginal utility per dollar of the net award for winning. Received: July 30, 1998; revised version: August 7, 2000     

Difference-form contest success functions and effort levels in contests

We examine players' equilibrium effort levels in a contest with difference-form contest success functions in which two players compete with each other to win a prize. We show the following. At the pure-strategy Nash equilibrium of a simultaneous-move game, and in the subgame-perfect equilibrium of a sequential-move game, only one of the players expends effort or neither player expends effort. If one player's composite strength is far greater than the other player's, only the player with greater composite strength expends effort whether they move simultaneously or sequentially. If the players' valuations for the prize and their marginal probabilities of winning at (0, 0) are sufficiently small, neither player expends effort whether they move simultaneously or sequentially.     

Group efforts when performance is determined by the “best shot”

We investigate the private provision of a public good whose level is determined by the maximum effort made by a group member. Costs of effort are either commonly known or privately known. For symmetric perfect-information games, any number of players may be active and we characterize the unique (mixed-strategy) equilibrium in which active contributors use the same strategy. Increasing the number of active players leads to stochastically lower individual efforts and level of the public good. When information is private, the symmetric equilibrium is in pure strategies. Increasing the number of players yields a pointwise reduction in the equilibrium contribution strategy but an increase in equilibrium payoffs. Comparative statics with respect to costs and levels of risk aversion are derived. Finally, whether information is public or private, equilibria are inefficient—we provide mechanisms that improve efficiency.     

Contests with rank-order spillovers

This paper presents a unified framework for characterizing symmetric equilibrium in simultaneous move, two-player, rank-order contests with complete information, in which each player??s strategy generates direct or indirect affine ??spillover?? effects that depend on the rank-order of her decision variable. These effects arise in natural interpretations of a number of important economic environments, as well as in classic contests adapted to recent experimental and behavioral models where individuals exhibit inequality aversion or regret. We provide the closed-form solution for the symmetric Nash equilibria of this class of games, and show how it can be used to directly solve for equilibrium behavior in auctions, pricing games, tournaments, R&D races, models of litigation, and a host of other contests.     

代理人能力分布不均情形下首位晋升制与末位淘汰制的比较研究

在基于相对业绩比较的锦标赛激励机制中,首位晋升制与末位淘汰制是两种常见的竞赛激励机制。在代理人能力分布不均情形下,将代理人之间的拆台行为引入传统的锦标赛模型,可以研究代理人能力分布不同时首位晋升制与末位淘汰制下代理人的行为以及对委托人收益的影响。通过数理模型推导可获得以下结论:第一,首位晋升制下代理人的努力水平和拆台水平均比较高,委托人采用首位晋升制更有利,能力分布对委托人最优选择不产生影响。第二,低能力代理人占多数时,高能力代理人努力水平过高,受到的拆台也更多,高能力代理人占多数时,低能力代理人努力水平更高受到的拆台更多,能力分布对代理人的行为产生影响。     

Round‐Robin Tournaments with a Dominant Player

We analyze the subgame perfect equilibrium of the round‐robin tournament with one strong (dominant) and two weak players, and we compare this tournament and the one‐stage contest with respect to the players' expected payoffs, expected total effort, and their probabilities of winning. We find that if the contest designer's goal is to maximize the players' expected total effort, then – if the asymmetry between the players is relatively low – the one‐stage contest should be used. However, if the asymmetry is relatively high, then the round‐robin tournament should be used.     

Fairness and desert in tournaments

We model the behavior of agents who care about receiving what they feel they deserve in a two-player rank-order tournament. Perceived entitlements are sensitive to how hard an agent has worked relative to her rival, and agents are loss averse around their meritocratically determined endogenous reference points. In a fair tournament sufficiently large desert concerns drive identical agents to push their effort levels apart in order to end up closer to their reference points on average. In an unfair tournament, where one agent is advantaged, the equilibrium is symmetric in the absence of desert, but asymmetric in the presence of desert. We find that desert concerns can undermine the standard conclusion that competition for a fixed supply of status is socially wasteful and explain why, when the distribution of output noise is fat-tailed, an employer might use a rank-order incentive scheme.     

Aggregate comparative statics

In aggregative games, each playerʼs payoff depends on her own actions and an aggregate of the actions of all the players. Many common games in industrial organization, political economy, public economics, and macroeconomics can be cast as aggregative games. This paper provides a general and tractable framework for comparative static results in aggregative games. We focus on two classes of games: (1) aggregative games with strategic substitutes and (2) nice aggregative games, where payoff functions are continuous and concave in own strategies. We provide simple sufficient conditions under which positive shocks to individual players increase their own actions and have monotone effects on the aggregate. The results are illustrated with applications to public good provision, contests, Cournot competition and technology choices in oligopoly.     

Welfare effects of payment truncation in piece rate tournaments

This article analyzes the optimal response of a principal to the regulatory proposal which would truncate agents’ bonus payment in a piece rate tournament at zero. In a model with risk-neutral and heterogenous abilities agents, we analyze the principal’s problem of optimal choice of contract parameters under both regular and truncated tournament scenarios. The results show that the principal could significantly mitigate potential welfare losses due to tournament truncation by adjusting the payment scheme. The optimal adaptation to tournament truncation results in a situation where both higher and lower ability players would benefit from the policy while average ability players would lose.     

Markov perfect equilibria in industries with complementarities

Summary.   This paper considers the existence and computation of Markov perfect equilibria in games with a “monotone” structure. Specifically, it provides a constructive proof of the existence of Markov perfect equilibria for a class of games in which a) there is a continuum of players, b) each player has the same per period payoff function and c) these per period payoff functions are supermodular in the player's current and past action and have increasing differences in the player's current action and the entire distribution of actions chosen by other players. The Markov perfect equilibria that are analyzed are symmetric, not in the sense that each player adopts the same action in any period, but rather in the sense that each player uses the same policy function. Since agents are typically distributed across many states they will typically take different actions. The formal environment considered has particular application to models of industries (or economies) in which firms face costs of price adjustment. It is in this context that the results are developed. Received: November 9, 1999; revised version: February 10, 2000     

Robust equilibria under non-common priors

This paper considers the robustness of equilibria to a small amount of incomplete information, where players are allowed to have heterogeneous priors. An equilibrium of a complete information game is robust to incomplete information under non-common priors if for every incomplete information game where each player's prior assigns high probability on the event that the players know at arbitrarily high order that the payoffs are given by the complete information game, there exists a Bayesian Nash equilibrium that generates behavior close to the equilibrium in consideration. It is shown that for generic games, an equilibrium is robust under non-common priors if and only if it is the unique rationalizable action profile. Set-valued concepts are also introduced, and for generic games, a smallest robust set is shown to exist and coincide with the set of a posteriori equilibria.     

A note on modelling tournaments

Existing studies on tournaments utilize different specifications of the production function. Comparison of these results is difficult. This paper shows that the strategic behavior of agents in a tournament can be analyzed more generally using a specification of the production function along the lines of the standard agency model. I also show that in mixed contests, the more able contestant would in equilibrium have a higher probability of winning the contest despite attempts to use effort to compensate for ability by the less able contestant (Proposition 3).The author would like to thank Joseph Stiglitz for suggesting the idea for this paper, Jerry Green for reading an earlier version of the paper, and three anonymous referees for helpful comments and suggestions. The usual disclaimer applies.