Two-player Contests Research Articles

Optimal tie-breaking rules

We consider two-player contests with the possibility of ties and study the effect of different tie-breaking rules on effort. For ratio-form and difference-form contests that admit pure-strategy Nash equilibrium, we find that the effort of both players is monotone decreasing in the probability that ties are broken in favor of the stronger player. Thus, the effort-maximizing tie-breaking rule commits to breaking ties in favor of the weaker agent. With symmetric agents, we find that the equilibrium is generally symmetric and independent of the tie-breaking rule. We also study the design of random tie-breaking rules that are ex-ante fair and identify sufficient conditions under which breaking ties before the contest actually leads to greater expected effort than the more commonly observed practice of breaking ties after the contest.

Simple equilibria in general contests

We show how symmetric equilibria emerge in general two-player contests in which skill and effort are combined to produce output according to a general production technology and players have skills drawn from different distributions. The model includes the Tullock (1980) and Lazear and Rosen (1981) models as special cases. Our paper provides intuition regarding how the contest components interact to determine the incentive to exert effort and sheds new light on classic comparative statics results. In particular, we show that more heterogeneity can increase equilibrium effort.

Equilibrium existence in two-player contests without absolute continuity of information

Equilibrium existence in two-player contests without absolute continuity of information

On Optimal Favoritism in Asymmetric Competitions

Favoritism towards a relatively weaker competitor is widely adopted as an effective instrument to enhance productive effort provision in asymmetric competitions. In this paper, we investigate the effort-maximizing favoritism rule in asymmetric two-player contests with all-pay auction technology, while accommodating fully flexible (nonlinear) favoritism rules. We assume that the players’ competencies (measured by their values of winning the competition, or marginal effort costs) are public information. We find that at the optimum, the weaker player is extremely favored; however his/her winning chance converges to zero. This finding illustrates that the effort-maximizing extreme favoritism rule perversely decreases winner diversity.

Unobservable Contract and Endogenous Timing in Legal Contests

It is possible that a plaintiff uses information about delegation contract as a means of responding strategic interactions between his lawyer and the defendant. We induce this possibility within the class of two-player contests with delegation. In detail, we have the following main results. (i) If the defendant’s total hourly fee rate is not large, the plaintiff who becomes the underdog wants to hide information on delegation contract. (ii) If the defendant’s total hourly fee rate is large, the plaintiff wants to release the information so that the defendant is the leader and the plaintiff’s lawyer is the follower, while enjoying being the favorite. In addition, we show that if the plaintiff can choose whether to disclose the information, the total expenditure will be reduced.

A General Framework for Studying Contests

We show how symmetric equilibria emerge in general two-player contests in which skill and effort are combined to produce output according to a general production technology and players have skills drawn from different distributions. We also show how contests with heterogeneous production technologies, cost functions and prizes can be analyzed in a surprisingly simple manner using a transformed contest that has a symmetric equilibrium. Our paper provides intuition regarding how the contest components interact to determine the incentive to exert effort, sheds new light on classic comparative statics results, and discusses the implications for the optimal composition of teams.

Information in Tullock contests

In Tullock contests in which the common value of the prize is uncertain and the elasticity of the marginal cost of effort is increasing (decreasing), the effect of changes of players’ information on the equilibrium efforts and payoffs is unambiguous: if information is symmetric, then expected effort decreases (increases) as players become better informed; in two-player contests, the expected effort of a player with information advantage is less (greater) than that of his opponent. Sharper results arise when the cost of effort is linear: Under symmetric information, expected effort and payoff are invariant to changes in the players’ information. In two-player contests, both players exert the same expected effort regardless of their information, although expected effort is smaller when one player has information advantage than when both players have the same information. Interestingly, the expected payoff of a player with information advantage is larger than that of his opponent, even though he wins the prize less frequently.

Catch-Up: A Rule That Makes Service Sports More Competitive

Service sports include two-player contests such as volleyball, badminton, and squash. We analyze four rules, including the Standard Rule (SR), in which a player continues to serve until he or she loses. The Catch-Up Rule (CR) gives the serve to the player who has lost the previous point—as opposed to the player who won the previous point, as under SR. We also consider two Trailing Rules that make the server the player who trails in total score. Surprisingly, compared with SR, only CR gives the players the same probability of winning a game while increasing its expected length, thereby making it more competitive and exciting to watch. Unlike one of the Trailing Rules, CR is strategy-proof. By contrast, the rules of tennis fix who serves and when; its tiebreaker, however, keeps play competitive by being fair—not favoring either the player who serves first or who serves second.

Contests with a Non-Convex Strategy Space

We characterize the Nash equilibria of a class of two-player contests with a non-convex strategy space under the usual concavity assumptions. The analysis sheds light on behavior in international conflicts. For instance, it may explain why some attempts to resolve international conflicts have been successful while others have not.

Payoff Shares in Two-Player Contests

In imperfectly discriminating contests with symmetric valuations, equilibrium payoffs are positive shares of the value of the prize. In contrast to a bargaining situation, players’ shares sum to less than one because a residual share of the value is lost due to rent dissipation. In this paper, we consider contests with two players and investigate the relationship between these equilibrium shares and the parameters of a class of asymmetric Tullock contest success functions. Our main finding is that any players’ shares that sum up to less than one can arise as the unique outcome of a pure-strategy Nash equilibrium for appropriate parameters.

Payoff Shares in Two-Player Contests

In contest models with symmetric valuations, equilibrium payoffs are positive shares of the value of the prize. In contrast to a bargaining situation, these shares sum to less than one because a share of the value is lost due to rent-dissipation. We ask: can every such division into payoff shares arise as the outcome of the unique pure-strategy Nash equilibrium of a simple asymmetric contest in which contestants differ in the effectiveness of their efforts? For two-player contests the answer is shown to be positive.

Contests with Bilateral Delegation: Unobservable Contracts

We study two-player contests in which, in order to win a prize, each player hires a delegate to expend effort on her behalf; neither party's delegation contract is revealed to the rival party when the delegates choose their effort levels. We obtain first the outcomes of this unobservablecontracts case. Next, we perform comparative statics of these outcomes with respect to the higher-valuation player's valuation for the prize. Finally, we compare the outcomes of the unobservable-contracts case with those of the observable-contracts case. We find, among other things, that the unobservability of delegation contracts narrows the gap between the delegates' equilibrium contingent compensation.

Intention-based fairness preferences in two-player contests

We analyze equilibria of two-player contests where players have intention-based preferences. We find that players invest more effort compared to the case with selfish preferences and are even willing to exert effort when the monetary value of the prize converges to zero. As a consequence, overdissipation occurs if the value of the prize is sufficiently small.

Intention-Based Fairness Preferences in Two-Player Contests

We analyze equilibria of two-player contests where players have intention-based preferences. We find that players invest more effort compared to the case with selfish preferences and are even willing to exert effort when the monetary value of the prize converges to zero. As a consequence, overdissipation occurs if the value of the prize is sufficiently small.

Two-Player Lopsided Contests under Different Timing Assumptions

I study contests in which two asymmetric players compete with each other by expending irreversible efforts to win a prize. I consider three types of games, which are distinguished by their different timing assumptions: the simultaneous-move game, the sequential-move game, and the game with endogenous timing. I compare the outcomes obtained under the different timing assumptions.

On Delegation in Contests and the Survival of Payoff Maximizing Behavior

In two-player contests, optimal delegation involves giving the agent incentives to maximize the principal's payoff while in contests with more than two players incentives will be different from the principal's payoff maximization. These results are related to the evolutionary stability of payoff-maximizing preferences in the model of indirect evolution in general symmetric games, which depends on the slope of the reaction function being zero at equilibrium. Further examples of the relationship are also discussed.

Evolutionarily stable preferences in contests

We define an indirect evolutionary approach formally and apply it to (Tullock) contests. While it is known that the direct evolutionary approach in the form of finite population ESS yields more aggressive behavior than in Nash equilibrium, it is now shown that the indirect evolutionary approach yields the same more aggressive behavior, too. This holds for any population size N, if the evolution of preferences is determined by behavior in two-player contests. The evolutionarily stable preferences (ESP) of the indirect approach turn out to be negatively interdependent, thereby “rationalizing” the more aggressive behavior.

Evolutionarily Stable Preferences in Contests

We define an indirect evolutionary approach formally and apply it to (Tullock) contests. While it is known (Leininger, 2003) that the direct evolutionary approach in the form of finite population ESS (Schaffer, 1988) yields more aggressive behavior than in Nash equilibrium, it is now shown that the indirect evolutionary approach yields the same more aggressive behavior, too. This holds for any population size N, if evolution of preferences is determined by behavior in two-player contests. The evolutionarily stable preferences (ESP) of the indirect approach turn out to be negatively interdependent, thereby rationalizing the more aggressive behavior.

The strategic advantage of interdependent preferences in rent-seeking contests

We study rent-seeking contests, where the set of players contains two groups of players – one with independent preferences and the other with (negatively) interdependent preferences. It turns out that the latter experience a strategic advantage in general two-player contests and in n-player-contests with non-increasing marginal efficiency. For general n-player contests with increasing marginal efficiency, the strategic advantage prevails provIDed convexity of contest technologies is sufficiently weak. For strongly convex contest technologies, other types of equilibria exist, including one where indivIDualists receive strictly higher pay-off.

Equilibria and comparative statics in two-player contests

We analyze two-player contests in which each player exerts effort to influence his probability of winning the prize. Assuming the contest success function is homogeneous of degree zero, we characterize Nash equilibria, provide conditions for existence of a pure-strategy equilibrium, and present comparative statics results. For comparative statics, we consider the dependence of equilibrium efforts, success probabilities, and expected payoffs on players' values for the prize. We also analyze the effects on the comparative statics of relaxing the homogeneity assumption. These findings extend previous results based on more restricted forms of the contest success function.