Unfair contests

Real-world contests are often “unfair” in the sense that outperforming all rivals may not be enough to be the winner, because some contestants are favored by the allocation rule, while others are handicapped. This paper analyzes an unfair, two-player discriminatory contest (all-pay auction) with private values. We characterize equilibrium strategies, provide closed form solutions, and illustrate additional strategic issues arising in such unfair contests.     

Effort maximization in asymmetric contest games with heterogeneous contestants

Contest rules are set up by administrators who frequently have discretionary power in specifying the details of these rules, i.e., they can bias the contest rules toward specific contestants in order to further their prime objective. We derive the optimal bias of the contest rule for a contest administrator, who is interested in maximizing the total efforts expended in the contest. The solution is obtained in closed form for a widely used class of n-person contest games. Setting the optimal bias has important implications: (i) there is never exclusion of strong players, instead there is (endogenously induced) inclusion of weak contestants; (ii) the contest administrator will optimally level the playing field by encouraging weak contestants, but he will not equalize the contestants’ chances unless they are identical; and (iii) at least three contestants will be active in equilibrium of the optimal contest, irrespective of heterogeneity.     

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.     

Expected revenue of all-pay and first-price sealed-bid auctions with affiliated signals

In the general symmetric auction framework of Milgrom and Weber (1982) it is shown—as a new manifestation of thelinkage principle—that the all-pay sealed-bid auction yieldshigher expected revenue than the standard first-price sealed-bid auction. This raises the question why sealed-bid auctions of the standard first-price variety are observed in practice whereas the all-pay variety is not.     

Repeated Contests with Asymmetric Information

The same contestants often meet repeatedly in contests. Behavior in a contest potentially provides information with regard to one's type and can therefore influence the behavior of the opponents in later contests. This paper shows that if effort is observable, this can induce a ratchet effect in contests: high ability contestants sometimes put in little effort in an early round in order to make the opponents believe that they are of little ability. The effect reduces overall effort and increases equilibrium utility of the contestants when compared with two unrelated one-shot contests. It does, however, also introduce an allocative inefficiency since sometimes a contestant with a low valuation wins. The model assumes an imperfectly discriminating contest. In an extension I show that, qualitatively, results are similar in a perfectly discriminating contest (all pay auction).     

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.     

On the continuous equilibria of affiliated-value,all-pay auctions with private budget constraints

We consider all-pay auctions in the presence of interdependent, affiliated valuations and private budget constraints. For the sealed-bid, all-pay auction we characterize a symmetric equilibrium in continuous strategies for the case of N bidders. Budget constraints encourage more aggressive bidding among participants with large endowments and intermediate valuations. We extend our results to the war of attrition where we show that budget constraints lead to a uniform amplification of equilibrium bids among bidders with sufficient endowments. An example shows that with both interdependent valuations and private budget constraints, a revenue ranking between the two auction formats is generally not possible. Equilibria with discontinuous bidding strategies are discussed.     

Multi-stage sequential all-pay auctions

We study multi-stage sequential all-pay contests (auctions) where heterogeneous contestants are privately informed about a parameter (ability) that affects their cost of effort. We characterize the perfect Bayesian equilibrium of these contests and analyze the effect of the number of contestants and their types on the contestants׳ expected highest effort.     

Auctions with endogenous participation and an uncertain number of bidders: experimental evidence

Attracting bidders to an auction is a key factor in determining revenue. We experimentally investigate entry and bidding behavior in first-price and English clock auctions to determine the revenue implications of entry. Potential bidders observe their value and then decide whether or not to incur a cost to enter. We also vary whether or not bidders are informed regarding the number of entrants prior to placing their bids. Revenue equivalence is predicted in all four environments. We find that, regardless of whether or not bidders are informed, first-price auctions generate more revenue than English clock auctions. Within a given auction format, the effect of informing bidders differs. In first-price auctions, revenue is higher when bidders are informed, while the opposite is true in English clock auctions. The optimal choice for an auction designer who wishes to maximize revenue is a first-price auction with uninformed bidders.     

Total effort, competitive balance and the optimal contest success function

This paper shows that, in the constrained case, the value weighted contest success function (CSF) maximizes not only total effort, but every objective function which is increasing in total effort and competitive balance. In the unconstrained case, a first price all-pay auction with a reserve price is optimal only if the impact of competitive balance on the objective is small. Otherwise, a modified value weighted CSF is optimal.     

Non-monotoniticies and the all-pay auction tie-breaking rule

Discontinuous games, such as auctions, may require special tie-breaking rules to guarantee equilibrium existence. The best results available ensure equilibrium existence only in mixed strategy with endogenously defined tie-breaking rules and communication of private information. We show that an all-pay auction tie-breaking rule is sufficient for the existence of pure strategy equilibrium in a class of auctions. The rule is explicitly defined and does not require communication of private information. We also characterize when special tie-breaking rules are really needed.      

Maximum efforts in contests with asymmetric valuations

Efforts may be reduced when players with different valuations participate in a contest. This paper considers the problem of designing a contest to elicit maximum aggregate effort from players with asymmetric valuations. Optimal designs for different classes of contest technologies are computed and characterized. A value weighted contest is optimal in the concave case. In the unconstrained case, the optimal contest is equivalent to a first price all-pay auction with a reserve price. The optimal design discounts the effort of the high valuation player in order to induce him to compete vigorously.     

The symmetric multiple prize all-pay auction with complete information

This paper extends the analysis of the n-player all-pay auction with complete information to cover the case of m≤n prizes, valued in weakly decreasing order, but symmetrically across players. We provide a complete characterization of the Nash equilibrium distributions for this class of auctions and provide an exact expression for the expected revenue generated.     

Bundle and separate sales in auctions with entry

A seller sells dissimilar objects while taking the auction rule as given. Should the seller sell the objects separately or as a bundle? Generally, when the number of bidders is small bundling helps to stimulate the competition and raise revenue. In this paper, we show that if entry is costly enough the process of endogenous participation generates sufficient competition to make bundling unnecessary in auctions. Thus whether a bundle auction generates a higher or a lower revenue ultimately depends first on the level of the entry costs and then, possibly, on the number of potential bidders.     

Lotteries vs. All‐Pay Auctions in Fair and Biased Contests

The form of contests for a single fixed prize can be determined by a designer who maximizes the contestants' efforts. This article establishes that, under common knowledge of the two asymmetric contestants' prize valuations, a fair Tullock‐type endogenously determined lottery is always superior to an all‐pay‐auction; it yields larger expected efforts (revenues) for the contest designer. If the contest can be unfair (structural discrimination is allowed), then the designer's payoff under the optimal lottery is equal to his expected payoff under the optimal all‐pay auction.     

Optimal seedings in elimination tournaments

We study an elimination tournament with heterogenous contestants whose ability is common-knowledge. Each pair-wise match is modeled as an all-pay auction. Equilibrium efforts are in mixed strategies, yielding complex dynamics: endogenous win probabilities in each match depend on other matches’ outcome through the identity of the expected opponent in the next round. The designer seeds competitors according to their ranks. For tournaments with four players we find optimal seedings for three different criteria: (1) maximization of total tournament effort; (2) maximization of the probability of a final among the two top ranked teams; (3) maximization of the win probability for the top player. We also find the seedings ensuring that higher ranked players have a higher winning probability. We compare our predictions with data from NCAA basketball tournaments.     

Auctions in which losers set the price

We study auctions of a single asset among symmetric bidders with affiliated values. We show that the second-price auction minimizes revenue among all efficient auction mechanisms in which only the winner pays, and the price only depends on the losers' bids. In particular, we show that the kth price auction generates higher revenue than the second-price auction, for all k>2. If rationing is allowed, with shares of the asset rationed among the t highest bidders, then the (t+1)st price auction yields the lowest revenue among all auctions with rationing in which only the winners pay and the unit price only depends on the losers' bids. Finally, we compute bidding functions and revenue of the kth price auction, with and without rationing, for an illustrative example much used in the experimental literature to study first-price, second-price and English auctions.     

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.     

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。     

Bid Functions in Auctions and Fair Division Games: Experimental Evidence

In auctions a seller offers a commodity for sale and collects the revenue. In fair division games the object is collectively owned by the group of bidders who equally share the revenue. We run an experiment in which the participants face four types of allocation games (auctions and fair division game under two price rules, first– versus second–price rule). We collect entire bid functions rather than bids for single values and investigate price and efficiency of the different trading institutions. We find that the first–price auction is more efficient than the second–price auction, whereas economic rationality assuming heterogeneous bidders suggests the opposite. Furthermore, we study the structure of individual bid functions.