First-price All-pay Auctions Research Articles

Round-robin tournaments with limited resources

We propose a theoretical model of a round-robin tournament with limited resources motivated by the fact that in real-world sport round-robin tournaments, participating teams are sometimes forced to distribute their effort over multiple matches. We assume that participating teams have a limited amount of effort that must be distributed between all matches of the tournament. We model the outcome of each match as a first-price all-pay auction. The prizes are awarded according to the ranking at the end of the tournament, based on the number of wins. The tournament with three teams is solved. In the case of identical budgets and low first prize, we find two types of equilibria—‘effort-saving’ and ‘burning-out’—both leading to unequal payoffs; when the first prize is large, we find multiple families of equilibria with both equal and non-equal payoffs. In the case of non-identical budgets, we show that there exist equilibria where a team with a lower budget gets a higher payoff than a team with a larger budget.

Loss aversion and sunk cost sensitivity in all-pay auctions for charity: Theory and experiments

This paper theoretically and experimentally studies the role of two behavioral biases in all-pay auctions for charity. The theory is developed to predict the effect of loss aversion in the first-price all-pay auction and sunk cost sensitivity in the war of attrition. Using controlled laboratory experiments, auction treatments are designed to test for the presence of these biases. In support of the theory, the results indicate that revenues in incremental bidding mechanisms such as the war of attrition rely on bidders who are sunk cost sensitive. It is shown that this behavioral response can be curbed significantly with a commitment device. Likewise, the results of the experimental first-price all-pay auctions find evidence of loss aversion, which reduces bidders’ average bid. These findings help explain the inconsistencies in revenues from previous all-pay auction studies and indicate a mechanism preference based on the distribution of these behavioral characteristics.

The Effects of Loss Aversion and Sunk Cost Sensitivity in All-Pay Auctions for Charity

All-pay auctions have demonstrated an extraordinary ability at raising money for charity. One mechanism in particular is the war of attrition, which frequently generates revenue well beyond what is theoretically predicted with rational bidders. However, what motivates the behavioral response in bidders remains unclear. By imposing charity auction incentives in the laboratory, this paper uses controlled experiments to consider the effects of loss aversion and sunk cost sensitivity on bidders’ willingness to contribute. The results indicate that revenues in incremental bidding mechanisms, such as the war of attrition, rely heavily on bidders who are sunk cost sensitive. It is shown this behavioral response can be easily curbed with a commitment device which drastically lowers contributions below theoretical predictions. A separate behavioral response due to loss aversion is found in the sealed-bid first-price all-pay auction, which reduces bidders’ willingness to contribute. These findings help explain the inconsistencies in revenues from previous all-pay auction studies and indicate a mechanism preference based on the distribution of these behavioral characteristics.

All-Pay Auctions with a Buy-Price Option

This paper studies all-pay auctions for a product in which there is a buy-price option for bidders to guarantee purchases at a seller-specified price. We analyze symmetric pure-strategy equilibria in the first- and second-price all-pay auctions with the buy-price option. Under these equilibria, the buy-price option will affect bidders' bidding behavior: while it has no impact on low-value bidders, it makes middle-value bidders bid more aggressively and high-value bidders less aggressively. Under the optimal buy-price, the first- and second-price all-pay auctions maintain the same expected profit while the buy-price in the second-price is higher than those in the first-price all-pay auction. Compared to the uniform posted-price selling mechanism, all-pay auctions with the buy-price cannot attain more expected profit under the fixed number of consumers. However, if there is an endogenous entry process, they can attract more consumers and then reach more profit in expectation.

The all-pay auction with complete information and identity-dependent externalities

We derive a necessary and sufficient condition for the existence of equilibria with only two active players in the all-pay auction with complete information and identity-dependent externalities. This condition shows that the generic equilibrium of the standard all-pay auction is robust to the introduction of “small” identity-dependent externalities. In general, however, the presence of identity-dependent externalities invalidates well-established qualitative results concerning the set of equilibria of the first-price all-pay auction with complete information. With identity-dependent externalities, equilibria are generally not payoff equivalent, and identical players may earn different payoffs in equilibrium. These observations show that Siegel’s (Econometrica 77(1), 71–92, 2009) results characterizing the set of equilibrium payoffs in all-pay contests, including the all-pay auction as a special case, do not extend to environments with identity-dependent externalities. We further compare the all-pay auction with identity-dependent externalities to the first-price winner-pay auction with identity-dependent externalities. We demonstrate that the equilibrium payoffs of the all-pay auction and the “undominated strategy equilibrium” payoffs of the winner-pay auction (Funk in Int J Game Theory 25(1), 51–64, 1996) cannot be ranked unambiguously in the presence of identity-dependent externalities by providing examples of environments where equilibrium payoffs in the all-pay auction dominate those of the undominated strategy equilibria in the winner-pay auction and vice versa.

The All-Pay Auction with Complete Information and Identity-Dependent Externalities

We derive a necessary and sufficient condition for the existence of equilibria with only two active players in the all-pay auction with complete information and identity-dependent externalities. This condition shows that the generic equilibrium of the standard all-pay auction is robust to the introduction of small identity-dependent externalities. In general, however, the presence of identity-dependent externalities invalidates well-established qualitative results concerning the set of equilibria of the first-price all-pay auction with complete information. With identity-dependent externalities equilibria are generally not payoff equivalent, and identical players may earn different payoffs in equilibrium. These observations show that Siegel’s (2009) results characterizing the set of equilibrium payoffs in all-pay contests, including the all-pay auction as a special case, do not extend to environments with identity-dependent externalities. We further compare the all-pay auction with identity-dependent externalities to the first-price winner-pay auction with identity-dependent externalities. We demonstrate that the equilibrium payoffs of the all-pay auction and winner-pay auction cannot be ranked unambiguously in the presence of identity-dependent externalities by providing examples of environments where equilibrium payoffs in the all-pay auction dominate those in the winner-pay auction and vice versa.

Wars of attrition and all-pay auctions with stochastic competition

We extend the war of attrition and all-pay auction analysis of Krishna and Morgan (1997) to a stochastic competition setting. We determine the existence of equilibrium bidding strategies and discuss the potential shape of these strategies. Results for the war of attrition contrast with the characterization of the bidding equilibrium strategies in the first-price all-pay auction as well as the winner-pay auctions. Furthermore we investigate the expected revenue comparisons among the war of attrition, the all-pay auction and the winner-pay auctions and discuss the linkage principle as well. Our findings are applicable to future works on contests and charity auctions.

Note on a Two-Player All-Pay Auction with Asymmetrical Bidders and Incomplete Information

The present paper analyzes a general class of first-price all-pay auctions where two players have different bidding technologies and one bidder has a head start advantage over his/her opponent. Prize values are common knowledge and may be different between the players. Equilibria are characterized for the complete information setting and for the case where there is incomplete asymmetrical information because the handicapped player is uncertain about the fact that the advantage is actually awarded.

Mixed-strategy play in single-stage first-price all-pay auctions with symmetric players

We consider sealed-bid, first-price, all-pay auctions with complete information, discrete strategy space, budget constraint, and symmetric players, and then construct the equilibrium solution in mixed strategies for both fixed and variable prizes. The equilibrium solution yields predictions concerning the effects of group size and value of the prize that are tested experimentally. The experimental results support equilibrium play on the aggregate but not individual level. Adaptive learning on the individual level accounts for the major trends in the aggregate expenditures over time.

Asymmetric All-Pay Auctions with Incomplete Information: The Two-Player Case

We prove existence and uniqueness of (Bayesian) equilibrium for a class of generally asymmetric all-pay auctions with incomplete information. Due to its importance in applications some prominence is given to the first-price all-pay auction, for which a detailed characterization of equilibrium and an approximation to equilibrium of its well-studied complete information version are supplied. Furthermore, we relate our uniqueness result to the well-known multiplicity of equilibria in the “war of attrition” (second-price all-pay auction), which emerges as a “limit” point of the class of two-player auction games considered.Journalof Economic LiteratureClassification Numbers: D44, C62, C72.