CONTESTS WITH DRAWS: AXIOMATIZATION AND EQUILIBRIUM

We introduce and axiomatize a class of single‐winner contest success functions that embody the possibility of a draw. We then analyze the game of contest that our success functions induce, having different prizes delivered in the occurrence of a win and a draw. We identify conditions for the existence and uniqueness of a symmetric interior Nash equilibrium and show that equilibrium efforts and equilibrium rent dissipation can be larger than in a Tullock contest (with no possibility of a draw) due to increased competition even if the draw‐prize is null. These results suggest that a contest designer may profit from introducing the possibility of a draw. Finally, we show that this approach naturally extends to multiprize contests with multiple draws across different subsets of the set of players. (JEL C72, D72, D74)     

OPTIMAL CONTESTS

Firms frequently use contests to compensate their employees: an employee's pay depends on the ranking of his output compared to that of others, rather than on the absolute level of his output. This paper analyzes the design of a contest which maximizes the contestants' expected aggregate output, and describes two settings which yield opposite results. In one, prizes should be equal except for that given to the contestant with the lowest output. In the other setting, only the contestant with the highest output should obtain a meaningful prize.     

Two-Player Asymmetric Contests with Ratio-Form Contest Success Functions

I examine players' equilibrium effort levels in two-player asymmetric contests with ratio-form contest success functions. I first characterize the Nash equilibrium of the simultaneous-move game. I show that the equilibrium effort ratio is equal to the valuation ratio, and that the prize dissipation ratios for the players are the same. I also show that the prize dissipation ratio for each player is less than or equal to the minimum of the players' probabilities of winning at the Nash equilibrium and thus never exceeds a half. Then I examine how the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels respond when the players' valuations for the prize or their abilities change.     

PRIZE SCARCITY AND OVERDISSIPATION IN ALL‐PAY AUCTIONS

We compare bidding behavior in complete information all‐pay auction experiments that vary in the prizes and number of players. We confirm the observation from prior single‐prize experiments that there is overbidding relative to equilibrium predictions. Our primary results are that increasing the number of prizes and players proportionally does not reduce overbidding but increasing the number of prizes with a fixed number of players eliminates overbidding. We conclude that the overbidding phenomenon is related to the scarcity of the prize. We provide new theoretical results on the multi‐prize logit equilibrium, and our experimental results are qualitatively consistent with logit equilibrium predictions. (JEL D72, D91, C91, D44)     

INFORMATION AND ENDOGENOUS DELEGATION IN A RENT‐SEEKING CONTEST

We offer a new explanation for the occurrence of delegation in rent‐seeking contests. We consider a two‐player contest for a prize of common value. The players only know that the prize is high or low, with given probabilities. Each player can hire a delegate to act on his behalf. After a delegate is hired, she privately observes the true value of the prize. We derive the conditions under which, respectively, no player, only one player, or both players delegate in equilibrium. (JEL D7)     

CARROTS AND STICKS: PRIZES AND PUNISHMENTS IN CONTESTS

We study optimal contest design in situations where the designer can reward high performance agents with positive prizes and punish low performance agents with negative prizes. We link the optimal prize structure to the curvature of distribution of abilities in the population. In particular, we identify conditions under which, even if punishment is costly, punishing the bottom is more effective than rewarding the top in eliciting effort input. If punishment is costless, we study the optimal number of punishments in the contest. (JEL D44, D82, J31, J41)     

A DESIGNER'S CHOICE BETWEEN SINGLE‐PRIZE AND PARALLEL TOURNAMENTS

In this article, I theoretically and experimentally compare a designer's profits from two tournament designs. The first design is a standard winner‐take‐all tournament with a single prize. The second design features two winner‐take‐all (parallel) tournaments with different prizes where individuals choose which tournament to enter before competing. I develop a simple model that illustrates how the relative performances of these designs change as contestants' abilities differ. The theoretical model shows that the designer's profit is higher (lower) in the parallel tournament when contestants' abilities differ greatly (are similar). I complement these findings with experimental evidence. The experiments show that the parallel tournament is more profitable under high heterogeneity, whereas under low heterogeneity, the designer is better off with the single‐prize tournament. Furthermore, high‐ability agents under‐participate and low‐ability agents over‐participate in the high‐prize tournament relative to the theoretical prediction. (JEL C72, D82, J33, M51, M52)     

PROPERTY RIGHTS AND LOSS AVERSION IN CONTESTS

We analyze the effects of property rights and the resulting loss aversion on contest outcomes. We study three situations: in “gain” two players start with no prize and make sunk bids to win a prize; in “loss” both the players start with prizes and whoever loses the contest loses their prize; and in “mixed” only one player starts with a prize that stays with him if he wins, but is transferred to the rival otherwise. Since the differences among the treatments arise only from framing, the expected utility and the standard loss aversion models predict no difference in bids across treatments. We introduce a loss aversion model in which the property rights are made salient, and as a result the reference point varies across treatments. This model predicts average bids in descending order in the loss, the mixed, and the gain treatment; and higher bids by the player with property rights in the mixed treatment. The results from a laboratory experiment broadly support these predictions. There is no significant difference in bids in the loss (gain) treatment and bids by property rights holder (nonholder) in the mixed treatment. A model incorporating both loss aversion and social preferences explains this result. (JEL C91, C72, D23, D74)     

RISK TAKING IN CONTESTS AND THE ROLE OF CARROTS AND STICKS

We study contests in which contestants choose both work effort and the variance of output (risk). Winner-take-all contests generate incentives for contestants to engage in costly risk taking, which is inefficient if the contest organizer values the aggregate output of all contestants. The addition of a penalty for ranking last (retaining a prize for ranking first) enables the organizer to independently control contestants' incentives to exert productive effort and to increase output variance. In this way, the organizer can eliminate risk-seeking behavior in settings where it is wasteful, but also control risk seeking when it is desirable, such as in research tournaments. ( JEL J33, C72)     

Sequential contests with synergy and budget constraints

We study a sequential Tullock contest with two stages and two identical prizes. The players compete for one prize in each stage and each player may win either one or two prizes. The players have either decreasing or increasing marginal values for the prizes, which are commonly known, and there is a constraint on the total effort that each player can exert in both stages. We analyze the players’ allocations of efforts along both stages when the budget constraints (effort constraints) are either restrictive, nonrestrictive or partially restrictive. In particular, we show that when the players are either symmetric or asymmetric and the budget constraints are restrictive, independent of the players’ values for the prizes, each player allocates his effort equally along both stages of the contest.     

CHAOTIC DYNAMICS IN CONTESTS

Under a myopic best‐reply dynamic, efforts in repeated contests may exhibit chaotic behavior. This may help explain, for example, why experimental data often show nonconvergence to one‐shot equilibrium efforts. (JEL C61, C72, D72, D74, D83)     

OPTIMAL PRICE OF ENTRY INTO A COMPETITION

A continuum of contestants are choosing whether to enter a competition. Each contestant has a type, and of those who enter, the ones with highest types receive prizes. A profit‐maximizing firm controls entry, and charges a price for it. I show that an increase in the value of each prize leads the firm to raise the price while keeping the intensity of entry fixed. Conversely, when the mass of prizes increases, the firm initially keeps the price constant while allowing entry to increase; and later—raises the price. (JEL C72, D82, D83)     

Prize Proliferation1

Recent decades have been characterized by growing numbers of awards, prizes, and honors distributed by a wide range of social worlds—prize proliferation. This article examines how claims for—and counterclaims against—establishing prizes lead to more awards being given. Unlike most constructionist analyses that explore how centralized claims‐making campaigns construct social problems, this study of prize proliferation illustrates how localized, independent social worlds can become arenas for claims, and how such largely unrelated efforts can shape broad social trends.     

Abstinence and Relapse Rates Following a College Campus-Based Quit & Win Contest

Abstract

Objective: To conduct and evaluate Quit & Win contests at 2 2-year college and 2 4-year university campuses. Participants: During Spring semester, 2006, undergraduates (N = 588) interested in quitting smoking signed up for a Quit & Win 30-day cessation contest for a chance to win a lottery prize. Methods: Participants (N = 588) completed a baseline survey, provided a urine sample to verify smoking status before joining the contest, and completed a follow-up survey at contest end to assess abstinence. Participants reporting continuous 30-day abstinence were surveyed again 2 weeks post contest to assess relapse. Results: Participants smoked an average of 9.8 ± 6.7 cigarettes/day on 26.7 ± 5.7 days/month. Among participants completing a follow-up survey (74%), 72.1% reported abstinence during the entire contest period (Intent-to-Treat Analysis = 53.2%). 55.3% of those abstinent at the end of contest had resumed smoking 2 weeks post contest. Conclusions: Campus Quit & Win contests appear feasible, acceptable, and effective at facilitating short-term abstinence. Further research is needed to identify strategies to prevent postcontest relapse.     

CONTESTS WITH STOCHASTIC ABILITIES

We consider the properties of perfectly discriminating contests in which players' abilities are stochastic, but become common knowledge before efforts are expended. Players whose expected ability is lower than that of their rivals may still earn a positive expected payoff from participating in the contest, which may explain why they participate. We also show that an increase in the dispersion of a player's own ability generally benefits this player. It may benefit or harm his rival but cannot benefit the rival more than it benefits himself. We also explore the role of stochastic ability for sequential contests with the same opponent (multibattle contests) and with varying opponents (elimination tournaments) and show that it reduces the strong discouragement effects and holdup problems that may otherwise emerge in such games. High own ability dispersion selects such players into the contest and favors them in elimination contests. ( JEL D72, D74)     

CAPS IN SEQUENTIAL CONTESTS

We study a sequential two‐stage all‐pay auction with two identical prizes. In each stage, the players compete for one prize and each player can win either one or two prizes. The designer may impose a cap on the players' bids in each of the stages. We analyze the equilibrium in this sequential all‐pay auction with bid caps and show that capping the players' bids is profitable for a designer who wishes to maximize the players' expected total bid. (JEL D44, D82, J31, J41)     

SIMULTANEOUS DECISION‐MAKING IN COMPETITIVE AND COOPERATIVE ENVIRONMENTS

We experimentally investigate simultaneous decision‐making in two contrasting environments: one that encourages competition (lottery contest) and one that encourages cooperation (public good game). We find that simultaneous participation in the public good game affects behavior in the contest, decreasing sub‐optimal overbidding. Contributions to the public good are not affected by participation in the contest. The direction of behavioral spillover is explained by differences in strategic uncertainty and path‐dependence across games. Our design allows us to compare preferences for cooperation and competition. We find that in early periods, there is a negative correlation between decisions in competitive and in cooperative environments. (JEL C72, C91)     

IF YOU DON'T LIKE THE OUTCOME,CHANGE THE CONTEST

This article analyzes contests when contestants derive heterogeneous benefits from winning and investments are seen as costs to the contest designer. An example of this is sports leagues maximizing league profits. For sports leagues, optimal contests depend upon the heterogeneity of market sizes and the impact of team investment on winning. Sports leagues have mechanisms to change the structure of contests, implying they have some control of the expected payoff of teams, the value of the athletes, and the balance of the league. Data from Major League Baseball and the National Basketball Association are used to empirically estimate the model. (JEL C72, L10, L83)     

THE OPTIMAL ALLOCATION OF PRIZES IN TOURNAMENTS OF HETEROGENEOUS AGENTS

Tournaments are widely used in organizations, explicitly or implicitly, to reward the best‐performing employees, for example, by promotion or bonuses, and/or to penalize the worst‐performing employees, for example, by demotion, withholding bonuses, or unfavorable job assignments. These incentive schemes can be interpreted as various prize allocations based on the employees' relative performance. While the optimal prize allocation in tournaments of symmetric agents is relatively well understood, little is known about the impact of the allocation of prizes on the effectiveness of tournament incentive schemes for heterogeneous agents. We show that while multiple prize allocation rules are equivalent when agents are symmetric in their ability, the equivalence is broken in the presence of heterogeneity. Under a wide range of conditions, loser‐prize tournaments, that is, tournaments that award a low prize to relatively few bottom performers, are optimal for the firm. The reason is that low‐ability agents are discouraged less in such tournaments, as compared to winner‐prize tournaments awarding a high prize to few top performers, and hence can be compensated less to meet their participation constraints. (JEL M52, J33, J24)     

Equity and effectiveness of optimal taxation in contests under an all-pay auction

The means of contest design may include differential taxation of the prize. This paper considers all-pay auctions under complete information and establishes that, given a revenue-maximizing contest designer who faces a balanced-budget constraint, the optimal taxation scheme corresponding to an all-pay auction is appealing in two senses. First, it ensures extreme equitable final prize valuations. Second, it is effective; it yields total contestants’ efforts that are larger than those obtained under almost any pure-strategy equilibrium of a Tullock-type lottery contest.