Two-player Contest Research Articles

Ambiguous persuasion in contests

We study optimal information disclosure via an ambiguous persuasion approach in a two-player contest. The designer can precommit to an ambiguous device to influence the uninformed contestant's belief about his opponent's private type. We fully characterize the optimal ambiguous information structures when players are maxmin expected utility (MMEU) maximizers. Depending on the prior, it is optimal to either induce ambiguity or fully conceal information. We provide a necessary and sufficient condition under which an effort-maximizing organizer can benefit strictly more from using ambiguous persuasion than from using the optimal Bayesian device.

Contests with delegation and initial probabilities of winning

ABSTRACT We study a two-player contest in which each player is endowed with an initial probability of winning the prize and hires a delegate who expends effort on each player’s behalf. We show that (i) the players’ equilibrium delegation contracts are monotonic in the initial probabilities of winning and the impact parameter; (ii) the delegates’ equilibrium effort levels are monotonic in the impact parameter but not in the initial probabilities of winning; and (iii) a contest designer may attain various objectives of contest design by choosing a proper value of the impact parameter.

Information disclosure in sequential (and simultaneous) contests

We study a two-player contest with one-sided incomplete information: One player’s valuation is publicly known and the other’s is his private information. The designer chooses the players’ move sequence (between simultaneous and sequential contests) and the information disclosure policy (between full and no disclosure). We find that if the uninformed player’s valuation is low, a simultaneous (resp. strong-lead sequential) contest with full disclosure is optimal when the valuation difference is large (resp. small); if the uninformed player’s valuation is high, a strong-lead sequential contest with full (resp. no) disclosure is optimal when the valuation difference is large (resp. small).

Competitive balance: Information disclosure and discrimination in an asymmetric contest

We study a design problem for an effort-maximizing principal in a two-player contest with two dimensions of asymmetry. Players have different skill levels and an information gap exists, as only one player knows the skill difference. The principal has two policy instruments to redress the lack of competitive balance due to asymmetry; she can commit to an information-disclosing mechanism, and she can discriminate one of the players by biasing his effort. We characterize the optimal level of discrimination to maximize aggregate effort, showing how this inextricably determines the choice of information disclosure. Applications are found in newcomer-incumbent situations in an internal labor market, sales-force management, and research contests.

Competitive Balance: Information Disclosure and Discrimination in an Asymmetric Contest

We study a design problem for an effort-maximizing principal in a two-player contest with two dimensions of asymmetry. Players have different skill levels and an information gap exists, as only one player knows the skill difference. The principal has two policy instruments to redress the lack of competitive balance due to asymmetry; she can commit to an information-revealing mechanism, and she can discriminate one of the players by biasing his effort. We characterize the optimal level of discrimination to maximize aggregate effort, showing how this is in turn inextricably linked to the choice of information revelation. Applications are found in newcomer-incumbent situations in an internal labor market, sales-force management, and research contests.

Contests with a constrained choice set of effort

We consider a symmetric two-player contest, in which the choice set of effort is constrained. We apply a fundamental property of the payoff function to show that, under standard assumptions, there exists a unique Nash equilibrium in pure strategies. It is shown that all equilibria are near the unconstrained equilibrium. Perhaps surprisingly, this is not the case when players have different prize evaluations.

Competitive Balance: Information Disclosure and Discrimination in an Asymmetric Contest

We study a design problem for an effort-maximizing principal in a two-player contest with two dimensions of asymmetry. Players have different skill levels and an information gap exists, as only one player knows the skill difference. The principal has two policy instruments to redress the lack of competitive balance due to asymmetry; she can commit to an information-disclosing mechanism, and she can discriminate one of the players by biasing his effort. We characterize the optimal level of discrimination to maximize aggregate effort, showing how this inextricably determines the choice of information disclosure. Applications are found in newcomer-incumbent situations in an internal labor market, sales-force management, and research contests.

How Selfish Contestants Use Endogenous Emotions to Increase Subjective Utilities

We study a two-player contest in which each egoistic player can choose to release his emotion information to the rival. Each player selects his emotion-parameter value to maximize his material payoffs and his effort level to maximize his subjective utility. There are different equilibria depending on the difference between the abilities of the players. The favorite reveals his envious emotion and the underdog his altruistic emotion in the equilibrium if the favorite’s ability is moderately higher than that of the underdog. Our results suggest that the classic result of the favorite-as-follower does not occur in the equilibrium of the full game.

Asymmetric endogenous prize contests

We consider a two-player contest in which one contestant has a headstart advantage, but both can exert further effort. We allow the prize to depend on total performance in the contest and consider the respective cases in which efforts are productive and destructive of prize value. When the contest success function takes a logit form, and marginal cost is increasing in effort, we show that a Nash equilibrium exists and is unique both in productive and destructive endogenous prize contests. In equilibrium, the underdog expends more resources to win the prize, but still his probability of winning remains below that of the favorite. In a productive contest, the underdog behaves more aggressively and wins the prize more often in comparison to a fixed-value contest. Thus, the degree of competitive balance—defined as the level of uncertainty of the outcome—depends upon the (fixed or endogenous) prize nature of the contest.

All-pay contests with performance spillovers

This paper generalizes the results of Siegel (2009, 2010) to accommodate performance spillovers, with which a player’s performance in a contest may affect the performance cost of another player. More precisely, we show that, if for any player, the spillovers from other players’ performance enter his cost in an additively separable form, then an all-pay contest has a unique Nash equilibrium. Moreover, we construct the equilibrium payoffs and strategies. Both the equilibrium uniqueness and construction are generalized to multiplicatively separable spillovers in a two-player contest.

Rent-seeking and competitive preferences

In this experiment, I examine the extent to which competitive social preferences can explain over-bidding in rent-seeking contests. The Human treatment is a standard two-player contest. In the Robot treatment, a single player bids against a computerized player, eliminating potential social preference motives. The results show no difference in bids between treatments at the aggregate level. Further analysis shows evidence of heterogeneous treatment effects between impulsive and reflective subjects. Moreover, impulsive subjects are more likely than reflective subjects to deviate qualitatively from the shape of the theoretical best response function.

Information Disclosure in Contests: A Bayesian Persuasion Approach

We examine optimal information disclosure through Bayesian persuasion in a two-player contest. One contestant's valuation is commonly known and the other's is his private information. The contest organiser can precommit to a signal to in uence the uninformed contestant's belief about the informed contestant. We show that to search for the optimal signal when the informed contestant's valuation follows a binary distribution, it is without loss of generality to compare no disclosure with full disclosure; otherwise, such a restriction causes loss of generality. We propose a simple method to compute the optimal signal, which yields explicit solutions in some situations.

Gambling in contests with random initial law

This paper studies a variant of the contest model introduced in Seel and Strack [J. Econom. Theory 148 (2013) 2033-2048]. In the Seel-Strack contest, each agent or contestant privately observes a Brownian motion, absorbed at zero, and chooses when to stop it. The winner of the contest is the agent who stops at the highest value. The model assumes that all the processes start from a common value $x_0>0$ and the symmetric Nash equilibrium is for each agent to utilise a stopping rule which yields a randomised value for the stopped process. In the two-player contest, this randomised value has a uniform distribution on $[0,2x_0]$. In this paper, we consider a variant of the problem whereby the starting values of the Brownian motions are independent, nonnegative random variables that have a common law $\mu$. We consider a two-player contest and prove the existence and uniqueness of a symmetric Nash equilibrium for the problem. The solution is that each agent should aim for the target law $\nu$, where $\nu$ is greater than or equal to $\mu$ in convex order; $\nu$ has an atom at zero of the same size as any atom of $\mu$ at zero, and otherwise is atom free; on $(0,\infty)$ $\nu$ has a decreasing density; and the density of $\nu$ only decreases at points where the convex order constraint is binding.

Gambling in Contests with Random Initial Law

This paper studies a variant of the contest model introduced by Seel and Strack. In the Seel-Strack contest, each agent or contestant privately observes a Brownian motion, absorbed at zero, and chooses when to stop it. The winner of the contest is the contestant who stops at the highest value. The model assumes that all the processes start from a common value $x_0>0$ and the symmetric Nash equilibrium is for each agent to utilise a stopping rule which yields a randomised value for the stopped process. In the two-player contest this randomised value should have a uniform distribution on $[0,2x_0]$.In this paper we consider a variant of the problem whereby the starting values of the Brownian motions are independent, non-negative, random variables that have a common law $\mu$. We consider a two-player contest and prove the existence and uniqueness of a symmetric Nash equilibrium for the problem. The solution is that each agent should aim for the target law $\nu$, where $\nu$ is greater than or equal to $\mu$ in convex order; $\nu$ has an atom at zero of the same size as any atom of $\mu$ at zero, and otherwise is atom free; on $(0,\infty)$ $\nu$ has a decreasing density; and the density of $\nu$ only decreases at points where the convex order constraint is binding.

Deadlines in stochastic contests

We consider a two-player contest model in which breakthroughs arrive according to privately observed Poisson processes. Each player’s process continues as long as she exerts costly effort. The player who collects the most breakthroughs until a predetermined deadline wins a prize.We derive Nash equilibria of the game depending on the deadline. For short deadlines, there is a unique equilibrium in which players use identical cutoff strategies, i.e., they continue until they have a certain number of successes. If the deadline is long enough, the symmetric equilibrium distribution of an all-pay auction is an equilibrium distribution over successes in the contest. Expected efforts may be maximal for a short or intermediate deadline.

Endogenous timing in general rent-seeking and conflict models

We examine simultaneous versus sequential choice of effort in a two-player contest. The timing of moves, determined in a preplay stage prior to the contest subgame, as well as the value of the prize is allowed to be endogenous. Contrary to endogenous timing models with an exogenously fixed prize the present paper finds the following: (1) Players may decide to choose their effort simultaneously in the subgame perfect equilibrium (SPE) of the extended game. (2) The SPE does not need to be unique, in particular, there is no unique SPE with sequential moves if the direct costs of effort are zero. (3) Symmetry among players does not rule out incentives for precommitment to effort locally away from the Cournot–Nash level. (4) Finally, there may be no correlation between win probability and strategic incentives in our framework; a finding most central in the analysis of fixed-prize contests.

Uncertainty in Conflicts

This paper theoretically assesses the role that uncertainty plays in the intensity of conflicts. The standard two-player rent-seeking contest model (Tullock, 1980) is extended to allow for privately known subjective values of the prize. The conflict is modeled as a Bayesian game on which each player’s valuation is drawn independently from arbitrary distributions. We find sufficient conditions for when first-order and second-order stochastic refinements in the distributions cause predictable movements in the conflict’s dissipation. We focus on arbitrary contest success functions and arbitrary independent distributions for each player, allowing us to extend our analysis beyond the case of symmetric equilibria.

The Equivalence of Contests

We use a Tullock-type contest model to show that intuitively and structurally different contests can be strategically and revenue equivalent to each other. We consider a two-player contest, where outcome-contingent payoffs are linear functions of prizes, own effort, and the effort of the rival. We identify strategically equivalent contests that generate the same family of best response functions and, as a result, the same revenue. However, two strategically equivalent contests may yield different equilibrium payoffs. Finally, we discuss possible contest design applications and avenues for future theoretical and empirical research.

Optimal contest design, closeness and the contest success function

This paper considers the optimal design of an asymmetric two-player contest when the designer’s payoff depends not only on performance of the contestants, but also on the closeness of the contest measured by the difference in winning probabilities. In contrast to previous studies, the impact of closeness on the optimal prize depends on the shape of the contest success function. Furthermore, including closeness in the designer’s objective may induce the designer to uniformly increase the contestants’ effort costs. A similar result is obtained in case the designer may handicap the stronger contestant, but an even contest is never optimal.

Optimal Contest Design When the Designer's Payoff Depends on Competitive Balance

This paper investigates optimal contest design when the designer's payoff is increasing in competitive balance between contestants. A two-player contest with asymmetric effort costs (asymmetric abilities) is considered. Competitive balance is measured by the difference in winning probabilities of the contestants. In contrast to previous studies, the impact of competitive balance on the optimal prize is not unique, but depends on the shape of the contest success function. Furthermore, it is shown that including competitive balance in the designer's objective may induce the designer to increase the contestants' effort costs, for example, by tightening the rules of the contest.