All-pay Auction Research Articles

Optimal information exchange in contests

We study optimal exchange of private information in a two-player all-pay auction contest with independent private binary values. A benevolent information center who is informed about the players’ values facilitates the exchange of information by disclosing a signal publicly. The informativeness of the signal determines the monotonicity of the unique symmetric equilibrium and the players’ expected payoff. We characterize the upper bound of players’ expected payoff and the corresponding optimal signals utilizing such a relation between the informativeness and the payoff. When the players are ex ante sufficiently heterogeneous, the optimal signals work through an information-rent channel by inducing allocative efficient contests. When the players are ex ante sufficiently homogeneous, the optimal signals work through an unlevel-playing-field channel by inducing asymmetric contests. In order to guarantee efficient allocation, a regulator can punish any exchange of information when the players are sufficiently homogeneous and impose no restrictions when they are sufficiently heterogeneous.

Award scheme in random trial contests

Innovation contests have been an important tool used in product research and development for companies. In the innovation contest literature, most papers assume the homogenous innovation contest model or the all-pay auction model. In this paper, we consider the random trial contest model and study the optimal award scheme. We show that, in this contest model, risk types of contestants play important roles in the award scheme, and the results are independent of the probability density function of the random shock. These generalize the work in literature. In addition, the risk aversion coefficient will decide the allocation manner in a multiple-winner scheme, i.e., a concave allocation manner or a convex allocation manner is optimal.

Your failure is my opportunity—Effects of elimination in contests

In this paper, we consider a sequence of multi-prize all-pay auctions, where low-performing players are eliminated along the way. We provide sufficient conditions such that equilibrium behavior is unaffected by the prospects of winning prizes in later rounds. When these conditions are not met, we identify a novel externality effect and derive equilibria in a model with two rounds. We compare our results to knockout tournaments in which players are partitioned into subcontests in each round.

Auctioning risk: the all-pay auction under mean-variance preferences

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.

Auctioning risk: The all-pay auction under mean-variance preferences

We analyse the all-pay auction with incomplete information and variance-averse bidders. We characterise the 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 uniform valuations, we show that a more variance-averse type always bids higher than her less variance-averse counterpart. Utilising our analytical bidding functions we discuss all-pay auctions with variance-averse bidders from a designer’s perspective. We extend our basic model to include noisy signals and allow for the possibility of variance-seeking preferences and type-dependent attitudes towards risk. (JEL C7, D7, D81. Keywords: Auctions, Contests, Mean-Variance preferences.)

Perseverance and suspense in tug-of-war

We study a tug-of-war game between two players using the lottery contest success function (CSF) and a quadratic cost (of effort) function. We construct a pure strategy symmetric Markov perfect equilibrium of this game, show that it is unique, and provide closed-form solutions for equilibrium strategies and values. In stark contrast to a model of tug-of-war with an all-pay auction CSF, players exert positive efforts until the very last battle in this equilibrium. We deliver a set of empirically appealing results on effort dynamics.

All-pay auctions with private signals about opponents’ values

We study all-pay auctions where each player observes her private value as well as a noisy private signal about the opponent’s value, following Fang and Morris’s (J Econ Theory 126(1):1–30, 2006) analysis of winner-pay auctions with multidimensional private signals. A unique symmetric monotonic equilibrium exists if the signal is not informative enough. When the signal is sufficiently informative, there exists a symmetric non-monotonic equilibrium in which all types of players randomize in overlapping supports. The revenue is lower than that in the standard independent private value setting. Fixing the signal’s informativeness, the all-pay auction raises lower revenue than the second-price auction, whereas the revenue ranking between the all-pay and the first-price auction is ambiguous.

Auction Design with Ambiguity: Optimality of the First-Price and All-Pay Auctions

We study the optimal auction design problem when bidders' preferences follow the maxmin expected utility model. We suppose that each bidder's set of priors consists of beliefs close to the seller's belief, where closeness is defined by a divergence. For a given allocation rule, we identify a class of optimal transfer candidates, named the win-lose dependent transfers, with the following property: each type of bidder's transfer conditional on winning or losing is independent of the competitor's type report. Our result reduces the infinite-dimensional optimal transfer problem to a two-dimensional optimization problem. By solving the reduced problem, we find that: (i) among efficient mechanisms with no premiums for losers, the first-price auction is optimal; and, (ii) among efficient winner-favored mechanisms where each bidder pays smaller amounts when she wins than loses: the all-pay auction is optimal. Under a simplifying assumption, these two auctions remain optimal under the endogenous allocation rule.

The Cooperative Consequences of Contests

Although contests are recognized theoretically as a highly effective method of motivation, the competitive nature of contests may generate unintended negative effects on social interactions in more general settings beyond contests. Using a laboratory experiment of real effort tasks with treatments varying by compensation schemes (all-pay auction contest, Tullock contest, proportional prize contest, and piece rate payment scheme), we test the relative effect of contest formats on pro-social behavior as measured by choices in social dilemma games. Compared to a ranking of compensation schemes based on the correspondence between effort exerted and reward received, our results provide relatively stronger support for the idea that cooperative behavior after a competition is tied to the potential for obtaining ‘fair’ payoff outcomes within the contest. Our random re-matching experimental design ensures that our findings do not result from subjects’ rivalry towards specific competitors, but rather represents a more fundamental shift in prosocial attitude. The results have managerial consequences for structuring incentives in the workplace when a combination of competition and cooperation is necessary among workers.

On Optimal Favoritism in Asymmetric Competitions

Favoritism towards a relatively weaker competitor is widely adopted as an effective instrument to enhance productive effort provision in asymmetric competitions. In this paper, we investigate the effort-maximizing favoritism rule in asymmetric two-player contests with all-pay auction technology, while accommodating fully flexible (nonlinear) favoritism rules. We assume that the players’ competencies (measured by their values of winning the competition, or marginal effort costs) are public information. We find that at the optimum, the weaker player is extremely favored; however his/her winning chance converges to zero. This finding illustrates that the effort-maximizing extreme favoritism rule perversely decreases winner diversity.

The Gender Difference in Mixed Strategy Nash Equilibrium Play

The mixed strategy Nash equilibrium (MNE) is a well-established concept in theory, but empirical and experimental support has been hampered by the need for a large sample size for across-subject choices and serial correlation for within-subject choices. We overcome these difficulties in testing for MNE play by exploiting a previously developed comparative statics effect for a common value pairwise all-pay auction which, strikingly, predicts that bids increase on perceived opponent’s risk tolerance, but not on the bidder’s own. Men bid as predicted, but women’s bids respond neither to their own nor to the opponent’s risk tolerance. Additionally, while the significance of men’s response to their beliefs about their opponent’s risk tolerance increases with the absence of mistakes on a pre-experiment quiz, women’s lack of response is unaffected by their quiz performance. These results are consistent with a prior finding of a gender difference in MNE play with a large sample of professional tennis players. We contribute by showing that MNE play can be tested with an ordinary sample of subjects in a laboratory setting. We show a gender difference in a factor that is likely crucial in most competitive situations: perceptions of the opponent’s risk tolerance.

Money as a weapon: Financing a winner-take-all competition

We investigate the capital structure of pioneering startup firms, which are frequently credited with opening new markets and niches in the digital era and often face the threat of the potential entry of successful, cash-rich firms from adjacent markets. Our analysis is made in the context of a winner-take-all competition in the form of an all-pay auction for the monopolistic position in a new market. We show that a pioneer's optimal capital structure exhibits widespread diversity and is determined by a tradeoff between entry deterrence and post-entry competition intensification. A pure-equity (a mixture of equity and risky debt) structure is optimal when (1) barriers to entry are small (large), (2) the future prospect of the new market is fairly certain and/or, (3) the new market is likely (unlikely) to create large externalities on the potential entrant's existing business. The post-entry competition is likely to engender large losses to both the winner and the loser.

Strong robustness to incomplete information and the uniqueness of a correlated equilibrium

We define and characterize the notion of strong robustness to incomplete information, whereby a Nash equilibrium in a game $$\mathbf{u}$$ is strongly robust if, given that each player knows that his payoffs are those in $$\mathbf{u}$$ with high probability, all Bayesian–Nash equilibria in the corresponding incomplete-information game are close—in terms of action distribution—to that equilibrium of $$\mathbf{u}$$ . We prove, under some continuity requirements on payoffs, that a Nash equilibrium is strongly robust if and only if it is the unique correlated equilibrium. We then review and extend the conditions that guarantee the existence of a unique correlated equilibrium in games with a continuum of actions. The existence of a strongly robust Nash equilibrium is thereby established for several domains of games, including those that arise in economic environments as diverse as Tullock contests, all-pay auctions, Cournot and Bertrand competitions, network games, patent races, voting problems and location games.

Perfect equilibria in games of incomplete information

This paper extends Selten’s (Int J Game Theory 4:25–55, 1975) notion of perfection to normal-form games of incomplete information and provides conditions on the primitives of a game that ensure the existence of a perfect Bayes–Nash equilibrium. The existence results, which allow for arbitrary (compact, metric) type and/or action spaces and payoff discontinuities, are illustrated in the context of all-pay auctions and Cournot games with incomplete information and cost discontinuities.

Competitive balance when winning breeds winners

In contest settings, heterogeneity between contestants generally leads to low effort provision, and many instruments have been suggested to restore competitive balance. We suggest that heterogeneity may evolve depending upon the outcome of previous contests. Restoring competitive balance in this setting is challenging, and we consider the choice faced by a principal who can distribute a prize fund over two consecutive all-pay auction contests. Contestants are heterogeneous in the sense that one of them has a head start in the first contest. The winner of contest one gains the stage prize and a net head start in contest two; the size of this net head start also varies between players. The head start of the early leader can be increased, neutralized or overturned. The principal aims to maximize total effort. We show how the potential head start that the initial underdog can gain is critical in determining expected efforts. When this parameter is small, the principal can do no better than running a single contest. When the initial underdog can catch up and surpass the first-contest favourite, there is scope to do better. Indeed, it is possible to completely neutralize the initial heterogeneity and capture all of the surplus from the players, inciting a level of expected effort equal to the prize value. Competitive balance is achieved by a judicious division of the prize budget.

Playing the field in all-pay auctions

We provide the first examination of all-pay auctions using continuous-time protocols, allowing subjects to adjust their bid at will, observe payoffs almost instantaneously, and gain more experience through repeated-play than in previous, discrete-time, implementations. Unlike our predecessors—who generally find overbidding—we observe underbidding relative to Nash equilibrium. To test the predictions of evolutionary models, we vary the number of bidders and prizes across treatments. If two bidders compete for a single prize, evolutionary models predict convergence to equilibrium. If three bidders compete for two prizes, evolutionary models predict non-convergent cyclical behavior. Consistent with evolutionary predictions, we observe cyclical behavior in both auctions and greater instability in two-prize auctions. These results suggest that evolutionary models can provide practitioners in the field with additional information about long-run aggregate behavior that is absent from conventional models.

On the Exclusion Principle in Tullock Contests

In an influential paper, Fang (Public Choice 112: 351–371, 2002) asserts that the exclusion principle discovered by Baye et al. (1993) for all-pay auction does not apply to lottery in the case in which an organizer cares about the aggregate effort. Serena (2017) shows that the exclusion principle still applies to lottery with (sufficiently) homogeneous contestants if the organizer only cares about the effort of the winner. In this paper, we consider a Tullock lottery contest in which contestants' types are binary and stochastic and the two types are significantly different. We find that Fang's (2002) result still holds but Serena's (2017) result does not. Precisely, to increase the expected winner's effort, it is strictly beneficial to exclude all but two contestants if and only if each contestant's probability of being the weak type is small. If the probability is sufficiently high, it is optimal to include all contestants.

Bid Caps and Disclosure Policies

We examine the effect of publicly disclosing or concealing bidders' types in an all-pay auction with a common bid cap. We call partial (full) disclosure policy the setup where the contest designer's disclosure policy is (not) contingent on type realization. Despite a bid cap possibly increasing the expected expenditures under a partial disclosure policy and the expected expenditures of low-types under a policy of full disclosure, the bid cap unambiguously decreases the expected total expenditures, regardless of the disclosure policy. For any given bid cap and allowing for full and partial disclosure policies, we find that expenditures are minimized with the policy in which a designer discloses bidders' types only if both players have a low valuation of the prize. The expenditures are maximized with the full concealment policy or with the policy in which a designer discloses the information only if both players have a high valuation. All in all, when the designer can choose both the disclosure policy and the bid cap, revenue is maximized employing full concealment with no bid cap.

Eternal Peace in the Tug-of-War?

The tug-of-war between single players is known to have a non-cooperative Markov-perfect equilibrium in which both players expend zero efforts and neither player drives the Markov process to one of the terminal states. We show that these peaceful outcomes vanish if the single players are replaced by teams with team members permanently assigned to the different Markov states and interacting pairwise in an all-pay auction. The reason for this phenomenon is that the members of the teams can externalize future effort costs while the single players cannot. Our analysis also highlights the impact of the discount factor on the expected trajectory of the tug-of-war, the dynamics of the expected efforts, and the degree of rent dissipation.

Contests with insurance

We study all-pay auctions under incomplete information in which contestants have non-linear effort functions. The designer offers the option of insurance for which a contestant pays a premium to the contest designer. If a contestant does not win he is reimbursed the cost of his effort. We demonstrate that contests with insurance may be profitable for a designer who wishes to maximize his expected revenue as based on the contestants’ expected total effort, the premium of the insured contestants, and their reimbursement.