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In this paper, we compare two procedures for allocating a sequence of fixed prizes in multi-stage nested Tullock contests, in which the number of prizes equals the number of contestants. In a winner-leave (loser-leave) procedure, in each stage, the prizes of the stage are allocated to winners (losers) according to their ranks, and prizes in early stages are higher (lower) than those in later stages. Players who obtain prizes leave the contest and the others proceed to the next stage of competition. For both procedures, it is effort-maximizing to allocate one prize in each stage. Provided that the positive prizes in the sequence are homogeneous, the optimally designed loser-leave procedure generates higher total effort if and only if the number of positive prizes is lower than a threshold. If the positive prizes in the sequence are heterogeneous, then the loser-leave procedure may generate higher total effort, even if the number of positive prizes in the sequence is in the high range.

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 investigate optimal favoritism using identity-contingent prizes in a two-player Tullock model. Besides the usual balance effect, prize allocation has an extra efficiency effect: One additional unit of prize tends to induce more effort, if it is used as the winning prize for the stronger player whose marginal cost is lower. We find that a total-effort-maximizing (contest) designer should offer a larger prize to the strong player if and only if the contest is sufficiently noisy. Our results are in contrast to conventional wisdom obtained from contest models with biased winner selection rules, in which leveling the playing field is always preferable in a two-player setting.

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.

In this paper, we study the existence of monotone equilibrium in first price auctions where bidders have a three-dimensional private type, i.e. their private values, degrees of risk aversion and initial wealth. Bidders' utility functions belong to the class of constant relative risk aversion (CRRA) or constant absolute risk aversion (CARA). The bidders' types are independent across bidders, while a bidder's private value, initial wealth and degree of risk aversion are allowed to be correlated. We show that a monotone equilibrium always exists in a general setting allowing for asymmetric bidders. Moreover, with symmetric bidders, a symmetric monotone equilibrium strategy must exist. A bidder's equilibrium strategy increases with bidders' private values and degrees of risk aversion. When bidders have CRRA utility, equilibrium bids decrease with initial wealth; when bidders have CARA utility, equilibrium bids are invariant to initial wealth.

Multiple prizes are usually awarded in contests (e.g., internal promotions, school admissions, sports, etc.) and players exert effort to increase their chances for winning a higher prize. A multi-prize contest model must provide each player's probabilities of winning each prize as functions of all players’ efforts. We find that the key notion of “hypothetical subcontests” in the contest axiomatization literature initiated by Skaperdas (1996) can be appropriately interpreted and precisely defined from a ranking perspective, as well as the relevant axioms of “ Subcontest Consistency ” and “ Independence of Irrelevant Contestants ” (IIC). When there are less than four contestants (exclusive), a set of axioms properly adapted from those in Skaperdas (1996) and Clark and Riis (1998a) are sufficient and necessary for axiomatizing the widely adopted multi-prize nested lottery contest of Clark and Riis (1996a) . When there are more than four contestants (inclusive), including a new axiom of “ Independence of Irrelevant Ranks ” (IIR) is necessary and sufficient.

We study auctions with selective entry and risk averse bidders. Our model accounts for risk averse bidders' endogenous participation decision and thus encompasses the existing entry models. We establish entry and bidding equilibrium in first-price auction and ascending auction mechanisms and show that bidders' entry behavior differs between these two mechanisms with different forms of risk aversion. Our approach provides testable implications of risk aversion in terms of entry behavior. We analyze a timber auction data set and propose a simple test for the form of bidders' risk aversion based on our model implications

We consider a multi-battle team contest in which players from two rival teams form pairwise matches to fight in distinct component battles, which are carried out sequentially or (partially) simultaneously. A team wins if and only if its players win a majority of battles. Each player benefits from his team's win, while he can also receive a private reward for winning his own battle. We find that the outcomes of past battles do not distort the outcomes of future battles. Neither the total expected effort nor the overall outcome of the contest depends on the contest's temporal structure or its feedback policy. (JEL C72, D72, D74, D82)

This paper investigates whether a contest organizer should disclose private information about bidders’ abilities in a multi-prize all-pay auction. Bidders’ abilities are randomly distributed and observed by the contest organizer; the organizer decides whether to disclose this information publicly. We find that concealing the information elicits higher expected total effort, regardless of the distribution of abilities. In addition, we find that the rent-dissipation rate of the contest does not depend on the disclosure policy.

We establish the effort-maximizing rule of allocating heterogeneous prizes when contestants privately observe their effort efficiencies. With ex ante symmetric players and increasing virtual effort efficiency, an all pay auction maximizes the total expected effort.

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.

This paper studies the effort-maximizing design of a complete-information contest with endogenous entry. A fixed pool of homogenous potential players with identical marginal bidding cost must incur an entry cost to enter the contest before they bid for prize(s). The designer can flexibly adjust the impact function of a generalized nested lottery contest and use a fixed budget to fund single or multiple prizes. Applying Dasgupta and Maskin (Rev Econ Stud 53(1):1–26, 1986), we establish the existence of symmetric equilibrium for all contest mechanisms concerned. A uniform upper bound for expected overall bids is identified for any eligible contest, assuming that potential bidders play symmetric equilibria. We show that the upper bound can be achieved through a Tullock contest with a single contingent prize, which adopts compatible bundles of success function and entry fees/subsidies. In particular, we identify the conditions under which the optimum can be achieved by solely setting the right discriminatory power in a Tullock contest with a single fixed prize. Finally, our analysis characterizes the optimal shortlisting rule, which reveals that the contest designer generally should exclude potential bidders to elicit higher bids.

This paper proposes a multi-prize “reverse” nested lottery contest model, which can be viewed as the “mirror image” of the conventional nested lottery contest of Clark and Riis (1996a) . The reverse-lottery contest model determines winners by selecting losers based on contestants’ one-shot effort through a hypothetical sequence of lotteries. We provide a microfoundation for the reverse-lottery contest from a perspective of (simultaneous) noisy performance ranking and establish that the model is underpinned by a unique performance evaluation rule. We further demonstrate that the noisy-ranking model can be interpreted intuitively as a “worst-shot” contest, in which contestants’ performances are evaluated based on their most severe mistakes. The reverse-lottery contest model thus depicts a great variety of widely observed competitive activities of this nature. A handy closed-form solution for a symmetric equilibrium of the reverse-lottery contest is obtained. We show that the winner-take-all principle continues to hold in reverse-lottery contests. Moreover, we find that a reverse-lottery contest elicits more effort than a conventional lottery contest whenever the prizes available to contestants are relatively scarce.

This paper investigates the competitive effect of cross-shareholdings in winner-take-all all-pay auctions with two asymmetric bidders. We show that cross-shareholdings may paradoxically create a “pro-competitive” effect and elicit more effort than a standard contest without cross-ownership. This observation runs in contrast to the anti-competitive effect that cross-shareholdings usually create in standard oligopolistic settings (such as Cournot or Bertrand competitions). Both bidding costs and the sizes of cross-shares affect the resultant total effort non-monotonically. Neither a cross-share nor a higher bidding cost necessarily decreases effort supply. A complete account of equilibrium bidding behaviors is provided and the necessary and sufficient conditions under which cross-shareholdings lead to higher or lower levels of overall effort are identified. However, the pro-competitive effect comes at a loss of efficiency.

In auctions with private information acquisition costs, we completely characterize (socially) efficient and (revenue) optimal two-stage mechanisms, with the first stage being an entry right allocation mechanism and the second stage being a traditional private good provision mechanism. Both efficiency and revenue optimality require that the second-stage selling mechanism be ex post efficient and the number of entry slots be endogenously determined. We show that both efficient and optimal entry can be truthfully implemented in dominant strategies, and can also be implemented via all-pay, though not uniform-price or discriminatory-price, auctions.

This paper studies optimal auction design with asymmetric linear financial externalities among bidders. When the matrix Γ that relates biddersʼ payoffs to their payments is nonsingular, the payment-related component in the design objective must equal a unique linear combination of its counterparts in bidderʼs payoffs. If all multipliers of the linear combination are nonnegative, a modified Myerson procedure is discovered for deriving the optimal design. If any multiplier is negative, an arbitrarily high value can be achieved for design objective by setting proper fixed transfers to bidders. When the matrix Γ is singular, the unbounded optimum result typically prevails. We applied our method to auctions with cross shareholdings and charity auctions for revenue-maximizing and efficient designs.

This paper studies the optimal design of R&D contests. A “sponsor” (e.g. the US Department of Defense or the World Health Organization) wants to improve the quality of the winning products. To do so, it partitions its budget between two schemes: an inducement prize and efficiency-enhancing subsidies to the firms competing in the contest. Prizes and subsidies have different functions, and they provide complementary incentives. In the optimally designed contest, subsidies increase while the prize decreases, if the innovation process is more challenging. Further, sensible conditions are identified under which the optimal contest implements either a “handicapping” scheme (by preferentially subsidizing the “underdog”) or a “national champion” scheme (by favoring the “favorite”). Our analysis yields a number of useful implications and sheds light on an array of R&D incentive schemes, such as the DoD's design competitions and vaccine development incentives.

This article proposes a multi-prize noisy-ranking contest model. Contestants are ranked in descending order based on their perceived performance, which is subject to random perturbation, and they are rewarded based on their ranks. Under plausible conditions, we establish that our noisy performance ranking model is stochastically equivalent to the family of multi-prize lottery contests built upon ratio-form contest success functions. We further establish the equivalence of our model to a contest model that ranks contestants by their best performance out of multiple independent attempts. These results therefore shed light on the micro-foundations of the popularly adopted lottery contest models. The “best-shot ranking rule” reveals a common thread that connects a broad class of seemingly disparate competitive activities (such as rent-seeking contests, patent races, research tournaments), and unifies them through a common performance evaluation mechanism.

We study how a contest organizer who seeks to maximize participant effort should disclose the information on the actual number of contestants in an imperfectly discriminatory contest with stochastic entry. When each potential contestant has a fixed probability of entering the contest, the optimal disclosure policy depends crucially on the properties of the characteristic function H(⋅)=f(⋅)/f′(⋅), where f(⋅) is the impact function. The contest organizer prefers full disclosure (full concealment) if H(⋅) is strictly concave (strictly convex). However, the expected equilibrium effort is independent of the prevailing information disclosure policy if a linear H(⋅) (Tullock Contest) applies.

I. INTRODUCTION In most auction literature, bidders are passively endowed with private information about their valuations. The analysis then focuses on optimal elicitation of private information. On many occasions, bidders may instead have to incur costs to collect this information. (1) Auction design in these cases has to balance between information acquisition and information elicitation, which are interdependent. The performance of an auction depends not only on the bidding equilibrium but also crucially on the information acquisition equilibrium. (2) As a salient feature, auction design with information acquisition costs has been complicated by entry coordination among bidders due to multiple entry equilibria issue. In a symmetric independent private value (IPV) setting of McAfee and McMillan (1987) with fixed information acquisition cost, Levin and Smith (1994) note that the ex ante efficient and revenue-maximizing auction (a second-price auction with no entry fee and no reserve) induces many asymmetric entry equilibria other than the targeted symmetric one. The existence of asymmetric entry equilibria in this setup fundamentally lies in bidders' constant marginal cost of entry. (3) Information acquisition costs could be private information of bidders just as with their private information about values. For example, in the cases of construction procurements or U.S. timber auctions, many aspects of prebid information acquisition and analyzation are private knowledge of bidders. (4) Clearly, when information acquisition costs are private information of bidders, auction design has to additionally take into account information elicitation at the information acquisition stage. This aspect of analysis has yet to be reflected in the literature, while the case with fixed information acquisition costs has been thoroughly studied by Milgrom (1981), McAfee and McMillan (1987), Engelbrecht-Wiggans (1987, 1993), Harstad (1990), Levin and Smith (1994), McAfee, Quan, and Vincent (2002), Ye (2004, 2007), and Cremer, Spiegel, and Zheng (2009) among others. A widely recognized insight of these studies is that ex ante efficiency can be achieved through a second-price auction while setting the reserve at the seller's valuation. If ex ante entry fees can be used to extract all the expected surplus of bidders then there is a congruence between the revenue and total surplus. (5) This article advances this line of research by studying the implications of private acquisition costs on auction design with an emphasis on bidders' coordination at the information acquisition stage. Specifically, we consider the IPV setting of McAfee and McMillan (1987) and Levin and Smith (1994) while allowing the information acquisition costs to be bidders' private information, which follow a continuous distribution. In light of our previous discussion, this article attempts to answer the following questions: (1) How are ex ante efficient and revenue-maximizing auctions affected by this additional dimension of private information? In other words, how does information rent extraction at the entry stage affect the auction designs? (2) Can this private information alleviate (rather than aggravate) the problem of entry coordination? With the dispersion in private information acquisition costs, the multiplicity of entry equilibria arising from the constant marginal cost in Levin and Smith (1994) might be avoidable. The types with higher information acquisition costs must have less incentive to enter, which could reduce the number of entry equilibria. The questions are: Will sufficient dispersion coordinate bidders and induce a unique entry? If yes, how much dispersion is enough? Due to the potential multiplicity of entry equilibria for any given mechanism, it is rather difficult to compare performances across varieties of mechanisms. To overcome this difficulty, we come up with an alternative approach. Our analysis begins with characterizing efficient and revenue-maximizing auctions for any given feasible entry pattern, which can be described through a vector of bidders' entry thresholds of acquisition costs. …

This paper studies the mechanism that a profit-making principal should adopt to provide a discrete public good when the values of the consumers are their private information and their participation is voluntary. The free-riding issue is resolved through threatened nonprovision of the good by the provider. Every bidder is asked to announce his or her virtual value as defined in Myerson (1981). The public good is provided if and only if the sum of the bidders' announced virtual values exceeds the provision cost. When a provision decision results, each bidder pays an amount that is determined by the announcement of other consumers. No one pays when a nonprovision decision results. We find that this mechanism is implementable through an all-pay auction. A restricted profit-maximizing mechanism that implements efficient allocation is also characterized. As in Gradstein (1994), when provision is always efficient, that is, the sum of consumers' values always exceeds the provision cost, efficient allocation is achievable through a profit-maximizer. However, this is not the case when provision is not efficient.

This paper examines the variation in total effort expended by participants when prizes are awarded in a grand contest as opposed to a number of subcontests. When contestants are homogeneous, under a mild and plausible condition (regular contest technology), a grand contest generates more effort than any set of subcontests. When no restrictions are placed on the contest technology, the results further demonstrate an “increasing-return-to-scale” property such that each individual responds to a proportional increase in the number of contestants and the number of each prize by increasing individual effort. Therefore, when a collection of identical subcontests forms a grand contest, the total effort always increases and the grand contest leads to a higher rent-dissipation rate. Our results apply to a wide variety of competitive activities, such as high-profile sports (e.g., diving and gymnastics in the Olympic Games), the internal labor market and the “quota” system for public resource allocation.

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.

This paper studies revenue-maximizing auctions in an independent private value setting where potential bidders have known positive opportunity cost of bidding. The main findings are as follows. Firstly, there is no loss of generality in deriving the revenue-maximizing auctions within the class of threshold-entry mechanisms. Secondly, for any given set of entry thresholds, a second-price sealed-bid auction with properly set reserve prices and entry subsidy is revenue-maximizing. Thirdly, a variety of auctions are revenue-maximizing within the symmetric threshold-entry class. Two of them involve no entry subsidy (fee). Fourthly, we identify sufficient conditions under which it is in the seller’s interest to limit the number of potential bidders even if the revenue-maximizing symmetric threshold-entry auction is adopted. Lastly, the revenue-maximizing auction implements asymmetric entry across symmetric bidders in many cases.

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.

This paper develops a measure of assortative mating on a dichotomous trait that takes into account changes in the trait distribution. Our empirical analysis shows that controlling for these changes is vital in examining the evolution of assortative mating.

This paper derives the effort-maximizing contest rule and the optimal endogenous entry in a context where potential participants bear fixed entry costs. The organizer is allowed to design the contest under a fixed budget with two strategic instruments: he sets the value of the prize purse, and arranges a monetary transfer (entry subsidy or fee) for each participating contestant. In other words, the budget can either be used to subsidize participation or an entry fee can be charged to fund the prize purse. The results show that the optimally designed contest attracts exactly two participating contestants in its unique subgame perfect equilibrium (when there is a positive fixed entry cost) and extracts all the surplus from participating contestants. The study also shows that the direction and amount of the monetary transfer depend on the magnitude of the entry cost: the contest organizer subsidizes entry when contestants bear substantial entry costs, but charges an entry fee to fund the prize purse whenever the entry cost is sufficiently low.

This paper investigates the optimal (effort-maximizing) structure of multi-stage sequential-elimination contests with pooling competition in each stage. We allow the contest organizer to design the contest structure in two arms: contest sequence (the number of stages, and the number of remaining contestants in each stage), and prize allocation. First, we find that the optimality of "winner-take-all" (single final winner, single final prize, no intermediate prizes) is independent of the contest sequence. Second, we show that the more complete the contest sequence is, the more efforts can be induced from the contestants. Therefore, the optimal contest eliminates one contestant at each stage until the finale, while a single winner takes over the entire prize purse. Our results not only rationalize various forms of multi-stage contests conducted in reality, such as the well-known Fox TV show "American Idol", but also shed light on the design of internal organizational hierarchy.