Elimination contests with collusive team players

We consider a standard two-stage elimination (Tullock) contest where multiple (team) players can perfectly and publicly collude with each other throughout. We analyze and compare equilibrium outcomes under various seedings where the collusive players meet or are separated in the group stage. We identify the impact of collusion on the contest organizer and non-collusive players, as well as the organizer's optimal seeding. We find that collusion, while always undermining fairness of the competition, can hurt or benefit the organizer, depending on the discriminatory powers of the two stages. We also discuss issues such as sequential group-stage competitions, comparison between the elimination contest and the corresponding one-shot contest, secret collusion, and large discriminatory powers.     

DISCRIMINATION IN A MODEL OF CONTESTS WITH INCOMPLETE INFORMATION ABOUT ABILITY

We study contests with private information and identical contestants, where contestants' efforts and innate abilities generate output of varying qualities. The designer's revenue depends on the quality of the output, and she offers a reward to the contestant achieving the highest quality. We characterize the equilibrium behavior, outcomes, and payoffs for both nondiscriminatory and discriminatory (where the reward is contestant‐dependent) contests. We derive conditions under which the designer obtains a larger payoff when using a discriminatory contest and describe settings, where these conditions are satisfied.     

Lottery versus all-pay auction contests: A revenue dominance theorem

We allow a contest organizer to bias a contest in a discriminatory way; i.e., she can favor specific contestants by designing the contest rule in order to maximize total equilibrium effort (resp. revenue). The two predominant contest regimes are considered, all-pay auctions and lottery contests. For all-pay auctions the optimal bias is derived in closed form: It implies extreme competitive pressure among active contestants and low endogenous participation rates. Moreover, the exclusion principle advanced by Baye et al. (1993) becomes obsolete in this case. In contrast, the optimally biased lottery induces a higher number of actively participating contestants due to softer competition. Our main result regarding total revenue comparison under the optimal biases reveals that the all-pay auction revenue-dominates the lottery contest for all levels of heterogeneity among contestants. The incentive effect due to a strongly discriminating contest rule (all-pay auction) dominates the participation effect due to a weakly discriminating contest rule (lottery).     

Contests where there is variation in the marginal productivity of effort

Summary. We provide a characterization of participants' behavior in a contest or tournament where the marginal productivity of effort varies across contestants and individual productivity is private information. We then consider the optimal design of such a contest. We first analyze contestant behavior for the usual type of contest, where the highest output wins. Abilities need not be independently distributed. We demonstrate that there is a unique symmetric equilibrium output function, that output is increasing in ability, and that marginal effort is increasing in ability, while effort decreases when the cost of effort increases. Next we consider the case where the highest output need not win, with independently distributed abilities. We analyze the contest designer's decisions in choosing contest rules optimal from her perspective. We show that the output produced, probability of winning, and contest designer's expected revenue are generally increasing in contestants' ability. We examine the relationship between the marginal cost of producing output and marginal utility per dollar of the net award for winning. Received: July 30, 1998; revised version: August 7, 2000     

Asset Prices and Trading Volume in a Beauty Contest

Speculators buy an asset hoping to sell it later to investors with higher private valuations. If agents are uncertain about the distribution of private valuations and about the beliefs of others about this distribution, a beauty contest with an infinite hierarchy of beliefs arises. Under Harsanyi's assumption of a common prior the infinite beliefs hierarchy is readily solved using Bayes' law. This paper shows that common knowledge of the "beliefs formation rule," mapping the private valuation of each gent into his first-order belief, also simplifies the beliefs hierarchy while allowing for disagreement among agents. We analyse the resulting speculation in a stylized asset market. Several statistics, computed only from readily observable quote, return and volume data, are evaluated in terms of their power to discriminate between genuine disagreement and the Harsanyian case. Only statistics that relate volume and volatility, or volume and changes in best offers, have the necessary discriminatory power.     

Efficiency and competition in the long run: The survival of the unfit

In a dynamic contest the current incumbent competes against a randomly assigned entrant in a private value all pay auction each period. We focus on equilibria where the beliefs about the incumbent's type and the employed strategies are stationary. We show that inefficient types survive, even if the entrants arrive very frequently, because the entrant plays more aggressively than the incumbent, allowing a low type entrant to win against a high type incumbent. In an example we show that if the incumbent is challenged more often, then the equilibrium type of the incumbent is higher on average. When the value of the prize is the same for all players (the case studied in the public choice literature), the equilibrium rent of the bidders is fully dissipated as the incumbent is challenged infinitely often. The technical contribution lies in showing the existence of stationary equilibrium in an incomplete information game.     

OVERINVESTMENT IN TEAM SPORTS LEAGUES: A CONTEST THEORY MODEL

This paper applies contest theory to provide an integrated framework of a team sports league and analyzes the competitive interaction between clubs. We show that dissipation of the league revenue arises from 'overinvestment' in playing talent as a direct consequence of the ruinous competitive interaction between clubs. This overinvestment problem increases if the discriminatory power of the contest function increases, revenue-sharing decreases, and the size of an additional exogenous prize increases. We further show that clubs invest more when they play in an open league compared with a closed league. Moreover, the overinvestment problem within open leagues increases with the revenue differential between leagues.     

Multi-activity contests

In many contests, players can influence their chances of winning through multiple activities or “arms”. We develop a model of multi-armed contests and axiomatize its contest success function. We then analyze the outcomes of the multi-armed contest and the effects of allowing or restricting arms. Restricting an arm increases total effort directed to other arms if and only if restricting the arm balances the contest. Restricting an arm tends to reduce rent dissipation because it reduces the discriminatory power of the contest. But it also tends to increase rent dissipation if it balances the contest. Less rent is dissipated if an arm is restricted as long as no player is excessively stronger than the other with that arm. If players are sufficiently symmetric in an arm, both players are better off if that arm is restricted. Nevertheless, players cannot agree to restrict the arm if their costs of using the arm are sufficiently low.     

Hotelling–Downs Competition with Free Entry When Voters Have an Option to Contest

Osborne shows that for almost all distributions of voters’ preferences, a pure strategy Nash equilibrium does not exist in the classical Hotelling–Downs model of electoral competition with free entry. We show that equilibrium is generically possible if in addition one allows voters an option to announce their candidacy to compete side‐by‐side with office‐seeking players. The model studied in this paper renders Osborne and the celebrated citizen‐candidate model à la Osborne and Slivinski as two extreme cases. We characterize the equilibrium set with two central questions: (i) can there be equilibria where only voters contest? and (ii) are equilibria with contesting office‐seeking players possible? We also show that in our general setting, extremists are typically voter‐candidates so that in every two‐party contest, office‐seeking politicians stay out of competition.     

Unfair contests

Real-world contests are often “unfair” in the sense that outperforming all rivals may not be enough to be the winner, because some contestants are favored by the allocation rule, while others are handicapped. This paper analyzes an unfair, two-player discriminatory contest (all-pay auction) with private values. We characterize equilibrium strategies, provide closed form solutions, and illustrate additional strategic issues arising in such unfair contests.     

Quantity-cum-quality contests

Quality and quantity are very common features of production processes. People care about these two features and they tend to be connected. I consider a contest in which the quantity and quality of output are rewarded. The output in the quality contest plays a dual role. It counts in the quality contest but it is also converted into quantity-equivalent output to obtain total output in the quantity contest. This latter feature implies that the two contests are interlinked. I find that when the unit cost of producing quality is sufficiently high, then treating quality and quantity as the same has a disincentive effect on the production of quality. In contrast, when the unit cost of producing quality is sufficiently low, treating quality and quantity as the same has no disincentive effect on the production of quality. I also find an equilibrium in which no one exerts effort in the quantity contest. When there is a binding budget constraint on effort, I find that effort in the quantity contest is smaller relative to the unconstrained case but effort in the quality contest may remain unchanged.     

Auctions with variable supply: Uniform price versus discriminatory

We examine an auction in which the seller determines the supply after observing the bids. We compare the uniform price and the discriminatory auction in a setting of supply uncertainty, where uncertainty is caused by the interplay of two factors: the seller's private information about marginal cost and the seller's incentive to sell the profit-maximizing quantity, given the received bids. In every symmetric mixed strategy equilibrium, bidders submit higher bids in the uniform price auction than in the discriminatory auction. In the two-bidder case, this result extends to the set of rationalizable strategies. As a consequence, we find that the uniform price auction generates a higher expected revenue for the seller and a higher trade volume.     

Information feedback and contest structure in rent-seeking games

We investigate the effects of information feedback in rent-seeking games with two different contest structures. In the share contest a contestant receives a share of the rent equal to her share of rent-seeking expenditures, while in the lottery contest a contestant wins the entire rent with probability equal to her share of rent-seeking expenditures. In share contests average expenditures converge to equilibrium levels when subjects only get feedback about own earnings, and additional feedback about rivals' choices and earnings raises average expenditures. In lottery contests information feedback has an opposite, and even stronger, effect: when subjects only get feedback on own earnings we observe high levels of rent dissipation, usually exceeding the value of the rent, and additional feedback about rivals' choices and earnings has a significant moderating influence on expenditures. In a follow-up treatment we make information feedback endogenous by allowing contestants in a lottery contest to make public or private expenditures. Subjects make the vast majority of expenditures privately and overall excess expenditures are similar to the lottery contest with own feedback.     

Effort maximization in asymmetric contest games with heterogeneous contestants

Contest rules are set up by administrators who frequently have discretionary power in specifying the details of these rules, i.e., they can bias the contest rules toward specific contestants in order to further their prime objective. We derive the optimal bias of the contest rule for a contest administrator, who is interested in maximizing the total efforts expended in the contest. The solution is obtained in closed form for a widely used class of n-person contest games. Setting the optimal bias has important implications: (i) there is never exclusion of strong players, instead there is (endogenously induced) inclusion of weak contestants; (ii) the contest administrator will optimally level the playing field by encouraging weak contestants, but he will not equalize the contestants’ chances unless they are identical; and (iii) at least three contestants will be active in equilibrium of the optimal contest, irrespective of heterogeneity.     

Savings and default

We develop a model in which individuals compete for a fixed pool of prizes by investing effort in a contest. Individuals belong to two separate and identifiable groups. We say that the contest is discriminatory if a lower share of prizes is reserved for one group than for the other. We show that it can be difficult for an observer to detect the presence or absence of discrimination in the contest, as both regimes can be observationally equivalent. In particular, one group’s belief that it is allocated a lower share of prizes than the other group can be consistent with observed data even if no such group quotas actually exist. Conversely, the belief that the contest does not discriminate can be consistent with data when, in fact, discrimination exists. Incorrect beliefs will therefore not be revised, as the contest generates no evidence to the contrary.     

Information in conflicts

We consider two-player contests for a prize of common but uncertain value. For settings where one player knows the value of the prize, while the other only knows its prior distribution, we give conditions for when the uninformed agent is ex ante strictly more likely to win the prize than is the informed agent. In the special case of a lottery contest, equilibrium expenditures are lower under asymmetric information than if either both agents are informed or neither agent is informed.     

When less is more: Rationing and rent dissipation in stochastic contests

This paper shows how to maximize revenue when a contest is noisy. We consider a case where two or more contestants bid for a prize in a stochastic contest where all bidders value the prize equally. We show that, whenever a Tullock contest yields under-dissipation, the auctioneer?s revenue can be increased by optimally fixing the number of tickets. In particular, in a stochastic contest with proportional probabilities, it is possible to obtain (almost) full rent dissipation. We test this hypothesis with a laboratory experiment. The results indicate that, as predicted, revenue is significantly higher in a lottery with rationing than in a standard lottery. On the other hand, an alternative rationing mechanism that does not limit total expenditures fails to increase revenue relative to a standard lottery.     

LOBBYING CONTESTS WITH ENDOGENOUS POLICY PROPOSALS

Lobbyists choose what to lobby for. If they can precommit to certain policy proposals, their choice will have an influence on the behavior of opposing lobbyists. Hence lobbyists have an incentive to moderate their policy proposals in order to reduce the intensity of the lobbying contest. This logic has been explored in a number of recent papers. I reconsider the topic with a perfectly discriminating contest. With endogenous policy proposals, there is a subgame-perfect equilibrium where the proposals of the lobbyists coincide and maximize joint welfare; moreover, this equilibrium is the only one that survives repeated elimination of dominated strategies. Hence there is no rent dissipation at all. A politician trying to maximize lobbying expenditures would prefer an imperfectly discriminating contest.     

Difference-Form Contests and the Robustness of All-Pay Auctions

In much of the existing literature on rent-seeking games, the outcome of the contest is either infinitely sensitive or relatively insensitive to contestants' efforts. The current paper presents a family of contest games that permit characterization of equilibrium for all levels of sensitivity of the outcome to contestants' efforts. Specifically, the outcome of the contest depends on the difference between efforts, which encompass the lottery and the all-pay auction as polar cases. The equilibrium converges to that of the all-pay auction as the probability of winning the prize grows infinitely sensitive to one's effort, and the main qualitative features of equilibrium persist over a large parameter region. Journal of Economic Literature Classification Numbers: D44, D72.     

Population uncertainty in contests

Summary. We study contests where the set of players is a random variable. If it is known for certain that there will be at least one participant, then aggregate contest expenditure in equilibrium is strictly lower in a contest with population uncertainty than in a non-uncertain contest with the same expected number of players. This suggests an explanation of, for example, why empirical studies show rent-seeking expenditures to be much lower than predicted by other theories.Received: 29 June 2003, Revised: 14 December 2004, JEL Classification Numbers: C72, D44, D72, D82, K41. Correspondence to: Karl Wärneryd