Difference-Form Contests and the Robustness of All-Pay Auctions

In much of the existing literature on rent-seeking games, the outcome of the contest is either infinitely sensitive or relatively insensitive to contestants' efforts. The current paper presents a family of contest games that permit characterization of equilibrium for all levels of sensitivity of the outcome to contestants' efforts. Specifically, the outcome of the contest depends on the difference between efforts, which encompass the lottery and the all-pay auction as polar cases. The equilibrium converges to that of the all-pay auction as the probability of winning the prize grows infinitely sensitive to one's effort, and the main qualitative features of equilibrium persist over a large parameter region. Journal of Economic Literature Classification Numbers: D44, D72.     

ENHANCING EFFORT SUPPLY WITH PRIZE‐AUGMENTING ENTRY FEES: THEORY AND EXPERIMENTS

Entry fees are widely observed in contests. We study the effect of a prize‐augmenting entry fee on expected total effort in an all‐pay auction setting where the contestants' abilities are private information. An entry fee reduces equilibrium entry but can enhance the entrants' effort supply. Our theoretical model demonstrates that the optimal entry fee is strictly positive and finite. In a laboratory experiment, we empirically test the effect of entry fees on effort supply. Our results provide strong support for the notion that a principal can elicit higher effort using an appropriately set entry fee to augment the prize purse.     

DISCRIMINATION IN A MODEL OF CONTESTS WITH INCOMPLETE INFORMATION ABOUT ABILITY

We study contests with private information and identical contestants, where contestants' efforts and innate abilities generate output of varying qualities. The designer's revenue depends on the quality of the output, and she offers a reward to the contestant achieving the highest quality. We characterize the equilibrium behavior, outcomes, and payoffs for both nondiscriminatory and discriminatory (where the reward is contestant‐dependent) contests. We derive conditions under which the designer obtains a larger payoff when using a discriminatory contest and describe settings, where these conditions are satisfied.     

Designing an optimal contest

The paper brings a mechanism design perspective to the study of contests. We consider the problem of selecting a contest success function when the contest designer may also value the prize. We show that any equilibrium outcome that can be achieved by a concave increasing contest success function can be replicated by a linear contest success function. An expected utility maximizing designer should employ a linear homogeneous contest success function. We explicitly derive the optimal contest for a risk-neutral designer and present comparative statics results. Tullock's contest is optimal only when the designer's valuation for the prize is low.     

Lottery versus all-pay auction contests: A revenue dominance theorem

We allow a contest organizer to bias a contest in a discriminatory way; i.e., she can favor specific contestants by designing the contest rule in order to maximize total equilibrium effort (resp. revenue). The two predominant contest regimes are considered, all-pay auctions and lottery contests. For all-pay auctions the optimal bias is derived in closed form: It implies extreme competitive pressure among active contestants and low endogenous participation rates. Moreover, the exclusion principle advanced by Baye et al. (1993) becomes obsolete in this case. In contrast, the optimally biased lottery induces a higher number of actively participating contestants due to softer competition. Our main result regarding total revenue comparison under the optimal biases reveals that the all-pay auction revenue-dominates the lottery contest for all levels of heterogeneity among contestants. The incentive effect due to a strongly discriminating contest rule (all-pay auction) dominates the participation effect due to a weakly discriminating contest rule (lottery).     

The Important Thing Is Not (Always) Winning but Taking Part: Funding Public Goods with Contests

This paper considers a public good game with heterogeneous endowments and incomplete information affected by extreme free riding. I overcome this problem through the implementation of a deterministic contest in which several prizes may be awarded. I identify a monotone equilibrium, in which the contribution is strictly increasing in the endowment. I prove that it is optimal for the social planner to set the last prize equal to zero, but otherwise total expected contribution is invariant to the prize structure. Finally, I show that private provision via a contest Pareto‐dominates public provision and is higher than the total contribution raised through a lottery.     

Dynamic Difference‐Form Contests

This study presents a multistage contest designed by an organizer who chooses the (indivisible) winner prize together with the ranking scheme. When the organizer's objective is to maximize efforts, the optimal ranking scheme that selects the prize recipient coincides with a version of the piecewise linear difference‐form success function postulated by Che and Gale (2000). Even if the organizer's objective is not necessarily to maximize agents' effort, a noisy ranking of the difference‐form remains optimal.     

Marginal Subsidies in Tullock Contests

In a general Tullock contest, we examine a situation where a limited resource can be used to provide marginal subsidies to either player (weak or strong), or to increase the prize directly. We show that to maximize total effort, subsidizing the weak/strong player is preferred when the contest is sufficiently accurate/inaccurate. This result generalizes to n‐player lottery contests. In a lottery contest (Tullock contest with ), we derive the optimal scheme for a full range of resource: when the resource is small, it is optimal to only subsidize the weak player; when it is large, both players should be subsidized simultaneously.     

When less is more: Rationing and rent dissipation in stochastic contests

This paper shows how to maximize revenue when a contest is noisy. We consider a case where two or more contestants bid for a prize in a stochastic contest where all bidders value the prize equally. We show that, whenever a Tullock contest yields under-dissipation, the auctioneer?s revenue can be increased by optimally fixing the number of tickets. In particular, in a stochastic contest with proportional probabilities, it is possible to obtain (almost) full rent dissipation. We test this hypothesis with a laboratory experiment. The results indicate that, as predicted, revenue is significantly higher in a lottery with rationing than in a standard lottery. On the other hand, an alternative rationing mechanism that does not limit total expenditures fails to increase revenue relative to a standard lottery.     

Are sequential round‐robin tournaments discriminatory?

I examine sequential round‐robin tournaments with three and four symmetric players. Each player is matched once with each other player and each match is organized as a Tullock contest. A single prize is allocated to the player with the most matches won. I show that, depending on the position of their matches in the sequence of the tournament, the players' ex ante winning probabilities and expected payoffs differ. Compared to tournaments with matches organized as all‐pay auctions, however, discrimination is weaker and partially reversed. Moreover, aggregate effort in tournaments with suitable Tullock contests is larger than in tournaments with all‐pay auctions. The fairness of round‐robin tournaments may be improved by an endogenous sequence of matches or the requirement that players fix their effort ex ante.     

Governing Interest Groups and Rent Dissipation

In a contest of group‐specific public goods we consider the effect that managing an interest group has on the rent dissipation and the total expected payoffs of the contest. While in the first group there is a central planner determining its members’ expenditure in the contest, in the second group there are two different possibilities: either all the members are governed by a central planner or they are not. We consider both types of contests: an all‐pay auction and a Logit contest success function. We show that while governing an interest group decreases free‐riding, it may as well decrease the rent dissipation; at the same time the expected payoffs of the groups may also decrease.     

Explaining Overbidding in First Price Auctions Using Controlled Lotteries

In this paper, we study the behavior of individuals when facing two different, but incentive-wise identical, institutions. We pair the first price auction with an equivalent lottery. Once a subject is assigned a value for the auctioned object, the first price auction can be modeled as a lottery in which the individual faces a given probability of winning a certain payoff. This set up allows us to explore to what extent the misperception of the probability of winning in the auction is responsible for bidders in a first price auction to bidding above the risk neutral Nash equilibrium prediction. The first result we obtain is that individuals, even though facing the same choice over probability/payoff pairs, behave differently depending on the type of choice they are called to make. When facing an auction, subjects with high values tend to bid significantly above the bid they choose in the corresponding lottery environment. We further find that in both the lottery and the auction environments, subjects tend to bid in excess of the bid predicted by the risk neutral model, at least for intermediate range values. Finally, we find that the difference between the lottery behavior and the auction behavior is substantially, but not totally, eliminated by showing the subjects the probability of winning the auction.     

The irrelevant-menu affect on valuation

Three experiments are designed to test if the level of irrelevant prizes in the menu has a positive (assimilation) or negative (contrast) effect on the perceived valuation of target objects. Familiar field prizes and binary lotteries over such prizes are placed within “more-expensive” and “less-expensive” menus. Subjects fill-in a sequence of binary choice problems to reveal their preference between given cash and a designated prize from the menu. Between-subject comparisons reveal that the prize-level in the menu positively affects perceived valuations in spite of procedural attempts to rule out menu-dependent preferences and prohibit experimenter bias. The effect also shows within-subject in auction experiments: the price that subjects are willing to pay for given monetary lotteries significantly increases with the average payoff in the irrelevant-menu. The bias finally manifests even when subjects are led to choose the target lottery, independently, from the underlying menu.     

Consuming your way to efficiency: Public goods provision through non-distortionary tax lotteries

We revisit the classical result that taxation of private consumption is distortionary and therefore precludes the efficient provision of public goods. We introduce a nonlinear consumption tax which we call a ‘tax lottery’. Under this scheme, an ad-valorem consumption tax is supplemented with a lottery in which consumers can win cash prizes. The winning probabilities in this lottery depend on all consumers' private good consumption decisions. We show that for a given ad-valorem tax, an appropriately designed lottery can implement an efficient allocation in pure-strategy Nash equilibrium. The lottery component corrects the distortion in private consumption due to the ad-valorem tax, while the resulting tax revenue is sufficient to efficiently provide the public good and pay out the lottery prize.     

The best-shot all-pay (group) auction with complete information

We analyze an all-pay group contest in which individual members’ efforts are aggregated via the best-shot technology and the prize is a public good for the winning group. The interplay of within-group free-riding and across-group competition allows for a wide variety of equilibria, according to how well groups overcome internal free-riding. In contrast with the existing literature, we derive equilibria of a symmetric model in which multiple agents per group are active. Our findings differ qualitatively from those of the individualistic all-pay auction: rents are not necessarily dissipated in equilibrium, total expected efforts vary across equilibria, and participation is expected to be greater. Moreover, equilibria with greater symmetry of behavior within a group are shown to have more “wasted” effort but also greater payoffs as overall efforts are lower. In contrast to standard economic intuition, free-riding can be beneficial for players as it reduces competition among groups. Examples of asymmetric group contests are also studied.     

Overbidding and overspreading in rent-seeking experiments: Cost structure and prize allocation rules

We study experimentally the effects of cost structure and prize allocation rules on the performance of rent-seeking contests. Most previous studies use a lottery prize rule and linear cost, and find both overbidding relative to the Nash equilibrium prediction and significant variation of efforts, which we term ‘overspreading.’ We investigate the effects of allocating the prize by a lottery versus sharing it proportionally, and of convex versus linear costs of effort, while holding fixed the Nash equilibrium prediction for effort. We find the share rule results in average effort closer to the Nash prediction, and lower variation of effort. Combining the share rule with a convex cost function further enhances these results. We can explain a significant amount of non-equilibrium behavior by features of the experimental design. These results contribute towards design guidelines for contests based on behavioral principles that take into account implementation features of a contest.     

Information in conflicts

We consider two-player contests for a prize of common but uncertain value. For settings where one player knows the value of the prize, while the other only knows its prior distribution, we give conditions for when the uninformed agent is ex ante strictly more likely to win the prize than is the informed agent. In the special case of a lottery contest, equilibrium expenditures are lower under asymmetric information than if either both agents are informed or neither agent is informed.     

On the Existence and Uniqueness of Pure-Strategy Nash Equilibrium in Asymmetric Rent-Seeking Contests

There can be three types of heterogeneity among players in a rent‐seeking contest. First, effectiveness of player's effort on the winning probabilities may differ among players. Secondly, players may evaluate the rent or prize of the rent‐seeking contest differently. Thirdly, players may face different financial constraints. This article proves under standard assumptions in the literature that there exists a unique pure‐strategy Nash equilibrium in a general asymmetric rent‐seeking contest with these three types of heterogeneity among players.     

竞赛中的最优奖励:一个拍卖分析框架

本文在全支付(all-pay)拍卖的框架下,研究了一类非对称竞赛活动的最优奖励问题。我们假设竞赛组织者未知参赛者的成本信息,进而可供选择的奖励方式有两种,其一是固定奖励额;其二是线性奖励,最终奖励额由参赛者的投资内生决定。我们证明:如果投资不能完全披露成本信息,固定奖励优于线性奖励;如果投资完全披露成本信息,线性奖励可能优于固定奖励,同时我们得出了实现投资完全披露信息的一个必要条件。我们的结论不依赖于参赛者的对称性假设,不依赖于参赛者具体的成本分布函数形式,也不依赖于竞赛组织者试图最大化参赛者最大努力或最大化所有参赛者努力总和。     

Charitable asymmetric bidders

Recent papers show that the all‐pay auction is better at raising money for charity than the first‐price auction with symmetric bidders under incomplete information. Yet, this result is lost with sufficiently asymmetric bidders under complete information. In this paper, we consider a framework on charity auctions with asymmetric bidders under some incomplete information. We find that the all‐pay auction still raises more money than the first‐price auction. Thus, the all‐pay auction should be seriously considered when one wants to organize a charity auction.