An extensive literature in economics and social science addresses contests, in which players compete to outperform each other on some measurable criterion, often referred to as a player's score, or output. Players incur costs that are an increasing function of score, but receive prizes for obtaining higher score than their competitors. In this paper we study finite games that are discretized contests, and the problems of computing exact and approximate Nash equilibria. Our motivation is the worst-case hardness of Nash equilibrium computation, and the resulting interest in important classes of games that admit polynomial-time algorithms. For games that have a tie-breaking rule for players' scores, we present a polynomial-time algorithm for computing an exact equilibrium in the 2-player case, and for multiple players, a characterization of Nash equilibria that shows an interesting parallel between these games and unrestricted 2-player games in normal form. When ties are allowed, via a reduction from these games to a subclass of anonymous games, we give approximation schemes for two special cases: constant-sized set of strategies, and constant number of players.
Szymanski @cite_33 considers the application of the theory of contests to the design of sporting competitions. The allocation of prizes in dynamic sport contests in which players determine their efforts at different stages of the game (e.g., at the beginning of each half in a soccer game) is considered in @cite_28 . There, the focus is on comparing rank-based versus score-based prizes when spectators care about contestants' efforts or about the suspense'' of the game (see also @cite_6 @cite_28 ). @cite_19 study a version of the contest design problem where the prize fund may be chosen by the designer, who wants to maximize the effort elicited, with prizes representing the price paid for effort. A related line of research @cite_7 @cite_5 has addressed from a game-theoretic perspective, the impact of the point scoring system on offensive versus defensive play in the UK Premier League; our concern here is slightly different, being focused on highly competitive versus weakly competitive play.
@cite_33 @cite_7 @cite_28 @cite_6 @cite_19 @cite_5
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This paper reviews the literature on commercial sport through the lens of the economic theory of contests tournaments. It seeks to draw together research on incentives in individualistic sports such as golf and footraces with the research on uncertainty of outcome and competitive balance in team sports such as baseball and soccer. The contest framework is used to analyze issues such as the optimal distribution of prizes, the impact of revenue sharing, salary caps, draft rules, and a variety of other restraints commonly employed in sports leagues. The paper draws heavily on a comparative analysis of contest organization, in particular between North America and Europe. This article addresses effects caused by the transition from a 2-1-0 to a 3-1-0 award system in soccer. The first part of the article discusses consequences of the transition on offensive versus defensive play. This part may be seen as a valuable supplement to work by Brocas and Carrillo (2004) as the choice of a different game theoretic framework provides increased insight into the concept offensive defensive play in soccer. The second and main part of the paper addresses additional effects induced by the award system transition, especially effects on competitive imbalance. It is shown by simple game theory that under a relatively general set of team descriptions, such a transition may affect competitive balance adversely. In the final sections of the paper some empirical examples strengthen the hypothesis on adverse competitive effects. In a dynamic model of sports competition, if spectators care only about contestants' efforts, incentive schemes depending linearly on the final score difference dominate rank order schemes based only on who wins. If spectators also care about suspense, defined as valuing more contestants' efforts when the game is closer, rank order schemes can dominate linear score difference schemes, and this will be the case when the demand for suspense is sufficiently high. Under additional assumptions, we show that the optimal rank order scheme dominates a broad class of incentive schemes. Copyright © The Author(s). Journal compilation © Royal Economic Society 2009. We study a contest with multiple (not necessarily equal) prizes. Contestants have private information about an ability parameter that affects their costs of bidding. The contestant with the highest bid wins the first prize, the contestant with the second-highest bid wins the second prize, and so on until all the prizes are allocated. All contestants incur their respective costs of bidding. The contest's designer maximizes the expected sum of bids. Our main results are: 1) We display bidding equlibria for any number of contestants having linear, convex or concave cost functions, and for any distribution of abilities. 2) If the cost functions are linear or concave, then, no matter what the distribution of abilities is, it is optimal for the designer to allocate the entire prize sum to a single ''first'' prize. 3) We give a necessary and sufficient conditions ensuring that several prizes are optimal if contestants have a convex cost function. We study all-pay contests under incomplete information where the reward is a function of the contestant's type and effort. We analyse the optimal reward for the designer when the reward is either multiplicatively separable or additively separable in effort and type. In the multiplicatively separable environment the optimal reward is always positive while in the additively separable environment it may also be negative. In both environments, depending on the designer's utility, the optimal reward may either increase or decrease in the contestants' effort. Finally, in both environments, the designer's payoff depends only upon the expected value of the effort-dependent rewards and not the number of rewards.