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.
We next mention some of the more classical literature on continuous contests. An influential line of work is the literature on rent-seeking problems, initiated by Tullock @cite_34 . These are problems in which players compete to receive favorable treatment from a regulator. Another large body of literature initiated by Lazear and Rosen @cite_27 has focused on contests in labor markets. Lazear and Rosen study the merits of rank-based prizes (as an alternative to paying a piece rate) as a means of incentivizing effort by workers in organizations. These models incorporate a random noise process that affects the selection of the winner. In a Tullock contest, the probability of winning is the amount of effort exerted by a player, divided by the total effort. Besides being a model of artificial competition, games of this kind are a model for competition for status within society @cite_22 .
@cite_27 @cite_34 @cite_22
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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. Incentive mechanisms for crowdsourcing have been extensively studied under the framework of all-pay auctions. Along a distinct line, this paper proposes to use Tullock contests as an alternative tool to design incentive mechanisms for crowdsourcing. We are inspired by the conduciveness of Tullock contests to attracting user entry (yet not necessarily a higher revenue) in other domains. In this paper, we explore a new dimension in optimal Tullock contest design, by superseding the contest prize — which is fixed in conventional Tullock contests — with a prize function that is dependent on the (unknown) winner's contribution, in order to maximize the crowdsourcer's utility. We show that this approach leads to attractive practical advantages: (a) it is well-suited for rapid prototyping in fully distributed web agents and smartphone apps; (b) it overcomes the disincentive to participate caused by players' antagonism to an increasing number of rivals. Furthermore, we optimize conventional, fixed-prize Tullock contests to construct the most superior benchmark to compare against our mechanism. Through extensive evaluations, we show that our mechanism significantly outperforms the optimal benchmark, by over three folds on the crowdsourcer's utility cum profit and up to nine folds on the players' social welfare. In this paper we present and analyze a model in which users select among, and subsequently compete in, a collection of contests offering various rewards. The objective is to capture the essential features of a crowdsourcing system, an environment in which diverse tasks are presented to a large community. We aim to demonstrate the precise relationship between incentives and participation in such systems. We model contests as all-pay auctions with incomplete information; as a consequence of revenue equivalence, our model may also be interpreted more broadly as one in which users select among auctions of heterogeneous goods. We present two regimes in which we find an explicit correspondence in equilibrium between the offered rewards and the users' participation levels. The regimes respectively model situations in which different contests require similar or unrelated skills. Principally, we find that rewards yield logarithmically diminishing returns with respect to participation levels. We compare these results to empirical data from the crowdsourcing site Taskcn.com; we find that as we condition the data on more experienced users, the model more closely conforms to the empirical data. Weighted voting is a classic model of cooperation among agents in decision-making domains. In such games, each player has a weight, and a coalition of players wins the game if its total weight meets or exceeds a given quota. A player's power in such games is usually not directly proportional to his weight, and is measured by a power index, the most prominent among which are the Shapley-Shubik index and the Banzhaf index. In this paper, we investigate by how much a player can change his power, as measured by the Shapley-Shubik index or the Banzhaf index, by means of a false-name manipulation, i.e., splitting his weight among two or more identities. For both indices, we provide upper and lower bounds on the effect of weight-splitting. We then show that checking whether a beneficial split exists is NP-hard, and discuss efficient algorithms for restricted cases of this problem, as well as randomized algorithms for the general case. We also provide an experimental evaluation of these algorithms. Finally, we examine related forms of manipulative behavior, such as annexation, where a player subsumes other players, or merging, where several players unite into one. We characterize the computational complexity of such manipulations and provide limits on their effects. For the Banzhaf index, we describe a new paradox, which we term the Annexation Non-monotonicity Paradox.