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We study equilibrium player ordering in a dynamic all‐pay contest between two teams. The contest lasts two periods, and each team consists of two players who perform in different periods on behalf of their teams. The team with the higher aggregate output wins the prize, which is a public good to its players. Each team has one stronger player and one weaker player, and the two teams can differ in their values of the prize. The teams maximize their winning odds by strategically assigning their players to different periods. We find that when the intrateam heterogeneity in player ability is not excessive, the teams would allocate their stronger players to the late positions as the “anchormen.” When both the intrateam ability gap and interteam heterogeneity in teams' values become excessively large, the team with high value always places its stronger player in the early position, who will place a large bid to preempt late competition. (JEL C7, D7, D8)

We study collusion in a second-price auction with two bidders in a dynamic environment. One bidder can make a take-it-or-leave-it collusion proposal, which consists of both an offer and a request of bribes, to the opponent. We show that there always exists a robust equilibrium in which the collusion success probability is one. In the equilibrium, for each type of initiator the expected payoff is generally higher than the counterpart in any robust equilibria of the single-option model ( Eso and Schummer (2004) ) and any other separating equilibria in our model.

This article studies cost‐minimizing two‐stage procurement with Research and Development (R&D). The principal wishes to procure a product from an agent. At the first stage, the agent can conduct R&D to discover a more cost‐efficient production technology. First‐stage R&D efficiency and effort and the realized second‐stage production cost are the agent's private information. The optimal two‐stage mechanism is implemented by a menu of single‐stage contracts, each specifying a fixed provision price and remedy paid by a defaulting agent. A higher delivery price is paired with a higher default remedy, and a more efficient type opts for a higher price and higher remedy.

In this paper, we analyze the role of negative prizes in contest design with a fixed budget, risk-neutral contestants, and independent private abilities. The effort-maximizing prize allocation rule features a threshold. When the highest effort is above the threshold, all contestants with lower efforts receive negative prizes. These negative prizes are used to augment the prize to the contestant with the highest effort, which better incentivizes contestants with higher abilities. When no contestant's effort exceeds the threshold, all contestants equally split the initial budget (or a portion of it) to ensure their participation. We find that allowing negative prizes can increase the expected total effort dramatically. In particular, if no bound is imposed on negative prizes, the expected total effort can be arbitrarily close to the highest possible effort inducible when all contestants have the maximum ability with certainty. The above contest is shown to be the optimal mechanism for a more general class of mechanisms.

Multi-battle competitions are ubiquitous in real life. In this paper, we examine the effort-maximizing reward design in sequentially played multi-battle competitions. The organizer has a fixed prize budget, and rewards players contingent on the number of battles they win in a three-battle contest. Battles are played between two opposing players or between selected pairs of players from two opposing teams. A full spectrum of contest technologies in the Tullock family is accommodated. We find that the optimal design is implemented by a contest prize for the grand winner who wins the majority of battles together with uniform battle prizes to battle winners. For competitions between two individuals, the optimal design varies with the discriminatory power of the contest technology. When it is in the low range, winner-take-all is optimal. For the intermediate range, as discriminatory power increases, the optimal prize structure evolves continuously from winner-take-all to a proportional-division rule due to the need to mitigate the growing momentum/discouragement effect. For the high range, a whole span of prize structures extracts full surplus and thus is optimal. In contrast, winner-take-all is optimal for team competitions, regardless of the contest technology, in which the momentum/discouragement effect does not exist.

We study optimal dynamic selling mechanisms in a two-stage model where the buyer can search for a better price at the second stage. When this outside price is public, the optimal selling mechanism takes the form of a fixed first-stage price with price matching in the second stage. In contrast, when the outside price is the buyer's private information, the optimal mechanism is a menu of two contracts: a first-stage sale at a higher price with immediate delivery, or a first-stage sale at a lower price with second-stage delivery. Thus the optimal form of search deterrence depends on the observability of the buyer's outside option.

In this paper, we study information disclosure policies in all-pay auctions with incomplete information. Two symmetric players have either high or low private value. The contest organizer observes players' values ex post, and can commit ex ante to four different symmetric information policies exhaustively: She can fully disclose or conceal the players' types, and she can disclose their types if and only if when both are high or if and only if when both are low. We characterize the unique equilibriums, and completely rank the four policies by various criteria. We find the full concealment policy extracts highest aggregate expected effort, followed by the policy of disclosing types if and only if when both are high. The policy of disclosing types if and only if when both are low induces least aggregate expected effort. Players enjoy highest expected payoffs under the policy of disclosing types if and only if when both are low; the other three policies yield the same expected payoffs to players. In terms of prize allocation efficiency, the full concealment policy is ranked highest and the full disclosure policy is ranked lowest. In between, the other two policies' rankings depend on the probability of high type. (JEL C72, D44, D82, D83, J48).

In this paper, we study effort-maximizing contest design under the “reverse” nested lottery contest model of Fu et al. (2014) — which is the “mirror image” of the conventional nested lottery contest of Clark and Riis (1996) . We show that under the reverse-lottery technology, a single-stage winner-take-all grand contest dominates all other feasible designs when the contest is sufficiently noisy. This result is in dramatic contrast to the conventional wisdom on the optimality of multistage elimination contests that is grounded under the conventional nested lottery contest technology in the literature. In the framework of a noisy-performance ranking model, the conventional and reverse models differ only in the noise on players’ performance. Our study therefore reveals the important role that the noise term plays in modeling imperfectly discriminatory contests.

In this paper, we revisit the two-bidder asymmetric all-pay auction of Amann and Leininger (1996) by allowing interdependent values and correlated signals. Both values and signals are distributed on continuous supports. We provide conditions for the existence and uniqueness of a monotone pure-strategy equilibrium (MPSE), and constructively characterize the MPSE when it exists. We find that given the marginal distributions of the signals, the equilibrium allocation is solely determined by how bidders' values depend on their signals. In particular, the equilibrium allocation does not depend on how bidders' signals are correlated.

A contest organizer (e.g., a government regulatory agency) is endowed with the capacity to provide unlimited homogeneous prizes (e.g., medals) that he can use to incentivize contestants to exert productive effort in an all-pay auction with incomplete information. Each agent, at most, wins one prize. We study the optimal number of prizes the organizer should grant in order to induce maximal expected total effort or expected highest effort from agents. Both are single peaked under mild regularity conditions. When players’ abilities follow a family of beta distributions, expected highest effort maximization requires a smaller set of prizes to be awarded; for both goals, the optimal number of prizes weakly increases when the pool of contestants expands or contestant quality improves.

Contestants often need to incur an opportunity cost to participate in the competition. In this paper, we accommodate costly entry and study the effort-maximizing prize allocation rule in a contest environment of all-pay auction with incomplete information as in Moldovanu and Sela (2001). As equilibrium entry can be stochastic, our analysis allows prize allocation rule to be contingent on the number of entrants. With free entry, Moldovanu and Sela establish the optimality of winner-take-all when effort cost function is linear or concave. Costly entry introduces a new trade-off between eliciting effort from entrants and encouraging entry of contestants, which might demand a more lenient optimal prize allocation rule. Surprisingly, we find that the optimality of winner-take-all is robust to costly entry when cost function is linear or concave. On the other hand, we provide examples to show that the new trade-off due to costly entry does make a difference to the optimal design when effort cost function is convex.

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. Ministry of Education (MOE) We acknowledge the Isaac Manasseh Meyer Fellowship, which funded Hammond’s visit to the National University of Singapore, where part of the work on this article was completed. Bin Liu gratefully acknowledges financial support from the National Natural Science Foundation of China (71703138). Jingfeng Lu gratefully acknowledges the financial support from the Ministry of Education, Singapore (R-122-000-252-115).

Following a Bayesian persuasion approach, we establish that full disclosure (resp. concealment) is the contest organizer’s effort-maximizing policy for disclosing the number of actual contestants if the characteristic function of the imperfectly discriminatory contest technology is strictly concave (resp. convex).

We examine how disclosure policy can be optimally designed to incentivize contestants when their participation is exogenously stochastic. In a generalized Tullock contest setting with two players who are asymmetric in both their values and entry probabilities, we fully characterize the necessary and sufficient conditions under which no disclosure dominates full disclosure. We find that the comparison depends solely on a balance effect exercised by entry probabilities on the expected total effort. The optimal disclosure policy must better balance the competition. These conditions continue to hold when the precision r of Tullock contests is endogenously chosen by the designer.

The common-support assumption of future type distributions is well adopted in the dynamic mechanism design literature. This assumption can become restrictive when the support is bounded. In this paper, we extend Eso and Szentes (2007) by allowing shifting supports and formally show that both their analytical methodology and key insights are robust to this generalization.

In this paper, we study task arrangement in competitions between two teams, each with two players. Two complementary tasks, together with a multiplicative random factor, jointly determine a team’s performance. Each player is responsible for one task on a team, and the team with better performance wins. We show that equilibrium actions do not depend on whether the tasks are conducted simultaneously in a single stage or sequentially, with one task in each stage. We further find that each team should allocate their players according to their comparative advantages in different tasks to maximize the team’s winning chance.

We consider auctions with entry based on a general analytical framework we call the Arbitrarily Selective (AS) model. We characterize symmetric equilibrium in a broad class of standard auctions within this framework, in the process extending the classic revenue equivalence results of Myerson (1981) , Riley and Samuelson (1981) and Levin and Smith (1994) to environments with endogenous and arbitrarily selective entry. We also explore the relationship between revenue maximization and efficiency, showing that a revenue maximizing seller will typically employ both higher-than-efficient reservation prices and higher-than-efficient entry fees.

This paper provides a different approach to establish the uniqueness of equilibrium in Tullock contests between two players with asymmetric valuations, when the discriminatory power r is between 1 and 2 . Our result complements that of Ewerhart (2017) in filling up the remaining gap in the literature on the uniqueness of equilibrium in two-player asymmetric Tullock contests.

Estimating bidders’ risk aversion in auctions is a challeging problem because of identification issues. This paper takes advantage of bidding data from two auction designs to identify nonparametrically the bidders’ utility function within a private value framework. In particular, ascending auction data allow us to recover the latent distribution of private values, while first-price sealed-bid auction data allow us to recover the bidders’ utility function. This leads to a nonparametric estimator. An application to the US Forest Service timber auctions is proposed. Estimated utility functions display concavity, which can be partly captured by constant relative risk aversion.

The reverse nested lottery contest proposed by Fu et al. (2014) is the “mirror image” of the classical nested lottery contest of Clark and Riis (1996a), which has been axiomatized by Lu and Wang (2015). In this paper, we close the gap and provide an axiomatic underpinning for the reverse nested lottery contest by identifying a set of six necessary and sufficient axioms. These axioms proposed specify the properties of contestants’ probabilities of being ranked the lowest among all players or within subgroups, while the axiomatization of the classical nested lottery contest by Lu and Wang (2015) relies on axioms on contestants’ probabilities of being ranked the highest among all players or within subgroups.