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We consider contests with many, possibly heterogeneous, players and prizes, and show that the equilibrium outcomes of such contests are approximated by the outcomes of mechanisms that implement the assortative allocation in an environment with a single agent that has a continuum of possible types. This makes it possible to easily approximate the equilibria of contests whose exact equilibrium characterization is complicated, as well as the equilibria of contests for which there is no existing equilibrium characterization.

This paper provides a general study of a contest modeled as a multiplayer incomplete-information, all-pay auction with sequential entry. The contest consists of multiple periods. Players arrive and exert efforts sequentially to compete for a prize. They observe the efforts made by their earlier opponents, but not those of their contemporaneous or future rivals. We establish the existence and uniqueness of a symmetric perfect Bayesian equilibrium (PBE) and fully characterize the equilibrium. Based on the equilibrium result, we show that a later mover always secures a larger ex ante expected payoff. Further, we endogenize the timing of moves and show that all players choose to move in the last period in the unique equilibrium that survives iterated elimination of strictly dominated strategies (IESDS).

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.

Many economic, political and social environments can be described as contests in which agents exert costly effort while competing over the distribution of a scarce resource. These environments have been studied using Tullock contests, all-pay auctions and rank-order tournaments. This survey provides a comprehensive review of experimental research on these three canonical contests. First, we review studies investigating the basic structure of contests, including the number of players and prizes, spillovers and externalities, heterogeneity, risk and incomplete information. Second, we discuss dynamic contests and multi-battle contests. Then we review studies examining sabotage, feedback, bias, collusion, alliances, group contests and gender, as well as field experiments. Finally, we discuss applications of contests and suggest directions for future research.

To explore the effects of different incentives on crowdsourcing participation and submission quality, we conduct a randomized field experiment on Taskcn, a large Chinese crowdsourcing site using mechanisms with features of an all-pay auction. In our study, we systematically vary the size of the reward as well as the presence of a soft reserve, or early high-quality submission. We find that a higher reward induces significantly more submissions and submissions of higher quality. In comparison, we find that high-quality users are significantly less likely to enter tasks where a high-quality solution has already been submitted, resulting in lower overall quality in subsequent submissions in such soft reserve treatments. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2013.1845 . This paper was accepted by Uri Gneezy, behavioral economics.

We study all-pay auctions with noise both theoretically and experimentally. First we show theoretically that all-pay auctions with noise (or “luck”) can be surprisingly like standard all-pay auctions with complete information in several important respects. Equilibria in both formats feature the same expected expenditures and payoffs. One difference, however, is that without noise predictions are in mixed strategies, while with noise predictions are in pure strategies. We next report the results of an experiment on our model and find qualitative support for the predicted equivalence of expected expenditures. Our experimental results on the noisy version of the model are particularly interesting, as subjects’ behavior appears to be more closely in line with the model’s theoretical equilibrium predictions than the treatment without noise in terms of reduced variance.

We study optimal information design in static contests where contestants do not know their values of winning. The designer aims at maximizing the total expected effort. Before the contest begins, she commits to the information technology that includes (1) a signal distribution conditional on each values profile (state) and (2) the type of signal disclosure to contestants—public, private or none at all. Upon observing the signal, contestants simultaneously choose effort that maximizes their expected payoff in an all-pay auction game. We find that the optimal information technology involves private signals, which are slightly positively correlated and never reveal the true state precisely if the contestants’ values of winning are different. In settings where public disclosure must be used, the optimal signal distribution generates symmetric beliefs about the values profile, so that, for example, a complete information concealment is optimal, while public and precise disclosure of each state is not.

In this paper, we generalize the General Lotto game (budget constraints satisfied in expectation) and the Colonel Blotto game (budget constraints hold with probability one) to allow for battlefield valuations that are heterogeneous across battlefields and asymmetric across players and for the players to have asymmetric resource constraints. We completely characterize Nash equilibrium in the generalized version of the General Lotto game and find that there exist sets of nonpathological parameter configurations of positive Lebesgue measure with multiple payoff nonequivalent equilibria. Across equilibria each player achieves a higher payoff when he more aggressively attacks battlefields in which he has lower relative valuations. Hence, the best defense is a good offense. We, then, show how this characterization can be applied to identify equilibria in the Colonel Blotto version of the game.

We analyse the all-pay auction with incomplete information and variance-averse bidders. We characterise the unique symmetric equilibrium for general distributions of valuations and any number of bidders. Variance aversion is a sufficient assumption to predict that high-valuation bidders increase their bids relative to the risk-neutral case while low types decrease their bid. Considering an asymmetric two-player environment with uniformly distributed valuations, we show that a variance-averse player always bids higher than her risk-neutral opponent with the same valuation. Utilising our analytically derived bidding functions we discuss all-pay auctions with variance-averse bidders from an auction designer's perspective. We briefly consider possible extensions of our model, including noisy signals, type-dependent attitudes towards risk, and variance-seeking preferences.

We study the two-player, complete information all-pay auction in which a tie ensues if neither player outbids the other by more than a given amount. In the event of a tie, each player receives an identical fraction of the winning prize. Players thus engage in two margins of competition: losing versus tying, and tying versus winning. Two pertinent parameters are the margin required for victory and the value of tying relative to winning. We fully characterize the set of Nash equilibria for the entire parameter space. For much of the parameter space, there is a unique Nash equilibrium which is also symmetric. Equilibria typically involve randomizing over multiple disjoint intervals, so that in essence players randomize between attempting to tie and attempting to win. In equilibrium, expected bids and payoffs are non-monotonic in both the margin required for victory and the relative value of tying.