Contests with endogenous deadlines

This paper analyzes the problem of a contest designer who chooses a starting time and a deadline of the contest to maximize discounted total effort by the contestants. Each contestant secretly decides how much effort to exert between the starting time and the deadline. At the deadline, the contestant who exerted most effort wins a prize, which consists of the endowment of the designer and collected interest. The contest has a unique Nash equilibrium. In the main model, the designer should announce the contest immediately with a short deadline to promote intense competition. I analyze how the optimal starting time and deadline change for a variable contest prize, different types of asymmetries, a Tullock lottery contest success function, and different goal functions of the designer.     

Allocation of Prizes in Contests with Participation Constraints

We study all‐pay contests with an exogenous minimal effort constraint where a player can participate in a contest only if his effort (output) is equal to or higher than the minimal effort constraint. Contestants are privately informed about a parameter (ability) that affects their cost of effort. The designer decides about the size and number of prizes. We analyze the optimal prize allocation for the contest designer who wishes to maximize either the total effort or the highest effort. It is shown that if the minimal effort constraint is relatively high, the winner‐take‐all contest in which the contestant with the highest effort wins the entire prize sum does not maximize the expected total effort or the expected highest effort. Rather, a random contest in which the entire prize sum is equally allocated to all the participants yields a higher expected total effort as well as a higher expected highest effort.     

Contests with investment

Perfectly discriminating contests (all pay auctions) are widely used as a model of situations where individuals devote resources to win some prize. In reality such contests are often preceded by investments of the contestants into their ability to fight in the contest. This paper studies a two stage game where in the first stage, players can invest to lower their bid cost in a perfectly discriminating contest, which is played in the second stage. Different assumptions on the timing of investment are studied. With simultaneous investments, equilibria in which players play a pure strategy in the investment stage are asymmetric, exhibit incomplete rent dissipation, and expected effort is reduced relative to the game without investment. There also are symmetric mixed strategy equilibria with complete rent dissipation. With sequential investment, the first mover always invests enough to deter the second mover from investing, and enjoys a first mover advantage. I also look at unobservable investments and endogenous timing of investments. Copyright © 2007 John Wiley & Sons, Ltd.     

Two-stage elimination contests with optimal head starts

We study two-stage elimination Tullock contests. In the first stage all the players compete against each other; then some advance to the second stage while the others are removed. The finalists compete against each other in the second stage, and one of them wins the prize. To maximize the expected total effort, the designer can give a head start to the winner of the first stage when he competes against the other finalists in the second stage. We show that the optimal head start, independent of the number of finalists, always increases the players’ expected total effort. We also show how the number of players and finalists affect the value of the optimal head start.     

Innovation contests with temporary and endogenous monopoly rents

In this article, a Tullock contest success function is used to model an innovation contest with endogenous innovation height. We can prove stability for this endogenous prize contest. The winner of the contest gains a monopoly rent, which has two dimensions. In the first dimension the winning firm influences the innovation height. The second dimension is the life span of the temporary monopoly. This life span is determined by the contest designer, who can be asocial planner or the consumers. We find interior solutions in both cases, whereas consumers prefer a monopoly life span below the social optimum. Furthermore, the optimal number of firms in the contest is two.     

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.     

Lobbying contests with alternative instruments

This is a model of a contest where, in order to win, each opponent can use two instruments. The probabilities of winning are explored, as well as the expenditures of the interest groups, and the relative rent-dissipation in both cases where the players have the option to use only one instrument (the standard Tullock contest) and where the players have the option to use two instruments in the contest. We show that the use of two instruments strengthens the player with the higher stake, decreases the relative rent dissipation and it decreases total expenditure if the parties are sufficiently asymmetric. Received: February 23, 2001 / Accepted: March 25, 2002 RID="*" ID="*" We are grateful to two anonymous referees and the editor Kai Konrad, for constructive comments.     

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.     

Deadlines in stochastic contests

We consider a two-player contest model in which breakthroughs arrive according to privately observed Poisson processes. Each player’s process continues as long as she exerts costly effort. The player who collects the most breakthroughs until a predetermined deadline wins a prize.We derive Nash equilibria of the game depending on the deadline. For short deadlines, there is a unique equilibrium in which players use identical cutoff strategies, i.e., they continue until they have a certain number of successes. If the deadline is long enough, the symmetric equilibrium distribution of an all-pay auction is an equilibrium distribution over successes in the contest. Expected efforts may be maximal for a short or intermediate deadline.     

Efficient Contests

In their seminal contribution, Lazear and Rosen (1981) show that wages based upon rank induce the same efficient effort as incentive‐based reward schemes. They also show that this equivalence result is not robust toward heterogeneity in worker ability, as long as ability is private information because it is not possible to structure contests to simultaneously satisfy self‐selection constraints and first‐best incentives. This paper demonstrates that efficiency can be achieved by a simple modification of the prize scheme in a mixed (heterogenous) contest where contestants learn their type after entry. If contestants know their type before entering the contest, rent extraction becomes an issue. Implications for optimal contest design are also explored. Finally, the relationship between effort maximizing contests and profit maximizing contests are discussed.     

Contests over joint production on networks

We consider a network of heterogeneous agents where each edge represents a two‐player contest between the respective nodes. In these bilateral contests, agents compete over an endogenous prize jointly produced using their own contest efforts. We provide a necessary and sufficient condition for the existence of Nash equilibrium and characterize the equilibrium total effort for every agent. Our model has insightful results regarding the network type, that is, depending on whether the network is bipartite or nonbipartite. Finally, considering the sum of all expected utilities as an efficiency notion, we investigate the optimal network structure.     

Optimally biased Tullock contests

This paper examines optimally biased Tullock contests. We consider a multi-player Tullock contest in which players differ in their prize valuations. The designer is allowed to impose identity-dependent treatments – i.e., multiplicative biases – to vary their relative competitiveness. The literature has been limited, because a closed-form solution to the equilibrium is in general unavailable when the number of contestants exceeds two, which nullifies the usual implicit programming approach. We develop an algorithmic technique adapted from the general approach of Fu and Wu (2020) and obtain a closed-form solution to the optimum that addresses a broad array of design objectives. We further analyze a resource allocation problem in a research tournament and adapt Fu and Wu’s (2020) approach to this noncanonical setting. Our analysis paves the way for future research in this vein.     

Winner’s effort maximization in large contests

We investigate the temporal structure that maximizes the winner’s effort in large homogeneous contests. We find that the winner’s effort ranges from a lower bound of 0 to an upper bound of one third of the value of the prize, depending on the temporal structure; the upper (lower) bound is approached with an infinite number of players playing sequentially (simultaneously) in the first periods (period). Nevertheless, when the number of players is large but finite, we show that winner’s effort is maximized when all players play sequentially except in the very last period and that, within the family of such optimal temporal structures, more players play simultaneously in the very last period than sequentially in all other periods. Furthermore, out of all players, the percentage of those playing simultaneously in the very last period goes to 100% as the number of players grows larger and larger.     

Reimbursement as a tool to reduce sabotaging in rent‐seeking contests

We study standard rent‐seeking contests with reimbursement and sabotaging. This study is conducted for a symmetric model with complete information. We show that changing the contest mechanism by applying a form of reimbursement could be an effective tool against sabotaging, in addition to the fact that it increases contest designer revenue. Simple changes such as sufficient reimbursement to winners/losers might completely stop sabotaging efforts in the contest.     

“Reverse” nested lottery contests

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.     

Optimal risk taking in an uneven tournament game with risk averse players

We analyze the optimal choice of risk in a two-stage tournament game between two players that have different concave utility functions. At the first stage, both players simultaneously choose risk. At the second stage, both observe overall risk and simultaneously decide on effort or investment. The results show that those two effects which mainly determine risk taking – an effort effect and a likelihood effect – are strictly interrelated. This finding sharply contrasts with existing results on risk taking in tournament games with symmetric equilibrium efforts where such linkage can never arise. Conditions are derived under which this linkage leads to a reversed likelihood effect so that the favorite (underdog) can increase his winning probability by increasing (decreasing) risk which is impossible in a completely symmetric setting.     

Tullock and Hirshleifer: a meeting of the minds

We introduce the serial contest by building on the desirable properties of two prominent contest games. This family of contest games relies both on relative efforts (as Tullock’s proposal) and on absolute effort differences (as difference-form contests). An additional desirable feature is that the serial contest is homogeneous of degree zero in contestants’ efforts. The family is characterized by a parameter representing how sensitive the outcome is to contestants’ efforts. It encompasses as polar cases the (fair) lottery and the (deterministic) all-pay auction. Equilibria have a close relationship to those of the (deterministic) all-pay auction and important properties of the latter hold for the serial contest, too.      

Risk preference heterogeneity in group contests

We analyze the first model of a group contest with players that are heterogeneous in their risk preferences. In our model, individuals’ preferences are represented by a utility function exhibiting a generalized form of constant absolute risk aversion, allowing us to consider any combination of risk-averse, risk-neutral, and risk-loving players. We begin by proving equilibrium existence and uniqueness under both linear and convex investment costs. Then, we explore how the sorting of a compatible set of players by their risk attitudes into competing groups affects aggregate investment. With linear costs, a balanced sorting (i.e., minimizing the variance in risk attitudes across groups) always produces an aggregate investment level that is at least as high as an unbalanced sorting (i.e., maximizing the variance in risk attitudes across groups). Under convex costs, however, identifying which sorting is optimal is more nuanced and depends on preference and cost parameters.     

Endogenous Group Formation in Contests: Unobservable Sharing Rules

We study contests in which players compete by expending irreversible effort to win a prize, the prize is awarded to one of the players, the winner shares the prize with other players in his group, if any, and each group's sharing rule is unobservable to the other groups and the singletons, if any, when the players expend their effort. The number of groups, their sizes, and the number of singletons are exogenous in the first model, whereas they are endogenous in the second model. We show that group formation occurs if the number of players is four or smaller, but does not occur otherwise. We examine the effect of endogenous group formation on total effort level and the profitability of endogenous group formation. In each of the two models, comparing the outcomes of the case of unobservable sharing rules with those of the case of observable sharing rules, we show that the two cases yield quite different outcomes.     

Incentives and Selection in Promotion Contests: Is It Possible to Kill Two Birds with One Stone?

This paper investigates whether a designer can improve both the incentive provision and the selection performance of a promotion contest by making the competition more (or less) dynamic. A comparison of static (one‐stage) and dynamic (two‐stage) contests reveals that this is not the case. A structural change that improves the performance in one dimension leads to a deterioration in the other dimension. This suggests that modifications of the contest structure are an alternative to strategic handicaps. A key advantage of structural handicaps over participant‐specific ones is that the implementation of the former does not require prior identification of worker types. Copyright © 2014 John Wiley & Sons, Ltd.