Multiple equilibria in Tullock contests

We find the sufficient conditions for the existence of multiple equilibria in Tullock-type contests, and show that asymmetric equilibria arise even under symmetric prize and cost structures. We then present existing contests where multiple equilibria exist under reasonably weak conditions.     

The theory of contests: A unified model and review of the literature

This paper is a brief review of the literature on contests, with focus on rent-seeking. A fairly general contest model is presented. We show that the Tullock contest model and the first-prize sealed-bid auction model are obtained as special cases. Some important modifications of the basic model are reviewed: hierarchical rent-seeking, rent-seeking under risk aversion, insecure rents, sabotage in rent-seeking contests, contest design, commitment and endogenous order of moves, and dynamic rent-seeking.     

Contests with rank-order spillovers

This paper presents a unified framework for characterizing symmetric equilibrium in simultaneous move, two-player, rank-order contests with complete information, in which each player??s strategy generates direct or indirect affine ??spillover?? effects that depend on the rank-order of her decision variable. These effects arise in natural interpretations of a number of important economic environments, as well as in classic contests adapted to recent experimental and behavioral models where individuals exhibit inequality aversion or regret. We provide the closed-form solution for the symmetric Nash equilibria of this class of games, and show how it can be used to directly solve for equilibrium behavior in auctions, pricing games, tournaments, R&D races, models of litigation, and a host of other contests.     

Equilibria in rent-seeking contests with homogeneous success functions

Summary. We characterize pure-strategy Nash equilibria for symmetric rent-seeking contests in which the contest success function is homogeneous of degree zero. The equilibrium strategies have a simple form. We give sufficient conditions for existence of an equilibrium.Received: 11 February 2003, Revised: 27 August 2004, JEL Classification Numbers: D72. Correspondence to: Andrew J. YatesWe thank Douglas Nelson, Jac Heckelman, and two anonymous referees for helpful comments.     

总体努力水平约束下的Tullock竞赛与最优约束机制设计:总体约束,个体约束还是无努力水平约束?

竞赛中参赛者往往面临各种参与约束,本文研究总体努力水平约束下的Tullock竞赛以及竞赛组织者如何最大化总努力水平。我们刻画了总体约束下的Tullock竞赛均衡,并将均衡努力水平与无约束及对称个体约束下的结果相比较。为了诱致出最高的总努力水平,最优总体约束水平和对称个体约束水平都应设定在比对奖品估值较高参与者的估值稍微低一点的水平,而最优非对称个体约束则要把估值较高参与者面临的约束设在比其估值稍微低一点的水平。     

An experimental study of bidding in contests of incomplete information

In many contest situations, such as R&D competition and rent seeking, participants’ costs are private information. We report the results of an experimental study of bidding in contests under different information and symmetry conditions about players’ costs of effort. The theory predicts qualitatively different comparative statics between bids under complete and incomplete information in contests of two and more than two players. We use a 2×3 experimental design, (n=2, n=4)×(symmetric complete information, asymmetric complete information, incomplete information), to test the theoretical predictions. We find the comparative statics of bids across the information and symmetry conditions, and the qualitative differences in comparative statics across group sizes, in partial agreement with the theory.     

The effects of risk preferences in mixed-strategy equilibria of games

We consider the effects of risk preferences in mixed-strategy equilibria of 2×2 games, provided such equilibria exist. We identify sufficient conditions under which the expected payoff in the mixed equilibrium increases or decreases with the degree of risk aversion. We find that (at least moderate degrees of) risk aversion will frequently be beneficial in mixed equilibria.     

Asymmetric contests with general technologies

Summary. We investigate the pure-strategy Nash equilibria of asymmetric, winner-take-all, imperfectly discriminating contests, focussing on existence, uniqueness and rent dissipation. When the contest success function is determined by a production function with decreasing returns for each contestant, there is a unique pure-strategy equilibrium. If marginal product is also bounded, limiting total expenditure is equal to the value of the prize in large contests even if contestants differ. Partial dissipation occurs only when infinite marginal products are permitted. Our analysis relies heavily on the use of share functions and we discuss their theory and application.Received: 28 May 2003, Revised: 26 April 2004, JEL Classification Numbers: C72, D72. Correspondence to: Richard CornesMuch of the research in this paper was undertaken while the first author was a Visiting Scholar at the Centre for Economic Studies, University of Munich. The support of the Centre is gratefully acknowledged.     

Uncertain contest success function

In this article, contestants play with a certain probability in Contest A and with the complementary probability in Contest B. This situation is called contest uncertainty. In both contests, effort is additively distorted by a contest noise parameter which affects the sensitivity of the contest success function (CSF). In Contest A (B), this parameter is linearly added to (subtracted from) effort. We analyze the interaction of contest uncertainty and contest noise on contestant behavior and profit. For symmetric contestants, contest noise has an ambiguous effect on effort and profit. We show that more contest uncertainty can imply greater effort. Furthermore, an introduction of an infinitesimal degree of contest uncertainty can have a large impact on effort and profit. Based on the analysis, this article presents the contest organizer's incentive to manipulate the degree of uncertainty in the contest. For profit or effort maximization, the contest organizer should always eliminate any uncertainty. If contestants are asymmetric, more contest noise increases effort as well as competitive balance if both Contests A and B have the same probability of occurrence.     

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.     

Effort and performance in group contests

We study contest performance by focusing on the per capita payoffs of the contestants in an important class of asymmetric two-group contests. The group with the higher valuation has a greater chance of winning the contest whenever the rent-seeking technology is increasing and concave. We note that equilibrium efforts and payoffs per capita are uniquely defined and can be subjected to a comparative statics analysis. We show that payoff per capita for a group increases with its own valuation but decreases with the valuation of the opposing group. Per capita payoff for a group increases with its population.     

Equilibria in second price auctions with participation costs

We investigate equilibria of sealed-bid second price auctions with bidder participation costs in the independent private values environment. We focus on equilibria in cutoff strategies (participate and bid the valuation iff it is greater than the cutoff), since if a bidder finds it optimal to participate, she cannot do better than bidding her valuation. When bidders are symmetric, concavity (strict convexity) of the cumulative distribution function from which the valuations are drawn is a sufficient condition for uniqueness (multiplicity) within this class. We also study a special case with asymmetric bidders and show that concavity/convexity plays a similar role.     

The best-shot all-pay (group) auction with complete information

We analyze an all-pay group contest in which individual members’ efforts are aggregated via the best-shot technology and the prize is a public good for the winning group. The interplay of within-group free-riding and across-group competition allows for a wide variety of equilibria, according to how well groups overcome internal free-riding. In contrast with the existing literature, we derive equilibria of a symmetric model in which multiple agents per group are active. Our findings differ qualitatively from those of the individualistic all-pay auction: rents are not necessarily dissipated in equilibrium, total expected efforts vary across equilibria, and participation is expected to be greater. Moreover, equilibria with greater symmetry of behavior within a group are shown to have more “wasted” effort but also greater payoffs as overall efforts are lower. In contrast to standard economic intuition, free-riding can be beneficial for players as it reduces competition among groups. Examples of asymmetric group contests are also studied.     

HETEROGENEOUS CONTESTS AND LESS INFORMATIVE SIGNALS*

We consider rank‐order contests with heterogeneous agents in which the principal is restricted to using a fair contest (or a symmetric contest), focusing on the optimal accuracy of output signals. As opposed to the absolute performance evaluation, we show that it is optimal for the principal to deliberately make the signals noisier according to the degree of heterogeneity. Some economic interpretations of controlling noise are discussed.     

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.     

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.     

Collective contests with externalities

This paper examines collective contests associated with externalities. The collective contest is modelled as a two-stage game in which intra-group sharing rules and individual outlays are determined sequentially. Depending upon the restrictions on the intra-group sharing rules and the extent of externalities, we identify three kinds of Nash equilibria, and compare them with the outcome of the contest between individuals. This paper also proposes a real rent-dissipation rate as a measure of social waste when externalities are present. The externalities are shown to have significant effects on the relationship between the number of players and the real rent-dissipation rate.     

Free riders and the optimal prize in public‐good funding lotteries

We prove the existence and uniqueness of an equilibrium in a game where players, whose preferences exhibit constant absolute risk aversion or constant relative risk aversion, contribute to a public good via lottery‐ticket purchases. Contrasting models with risk neutrality, we show that an equilibrium with a strictly positive amount of the public good may not exist without a sufficient number of participants who are not too risk‐averse. We show that players who are more risk‐averse purchase fewer lottery tickets and are more likely to free ride in equilibrium. In fact, it is possible for free riders to place a larger value on the public good than do those who contribute. In a symmetric equilibrium, we show that an upper bound exists for the amount of the public good, even though there are infinitely many participants. Furthermore, we derive a lottery prize that maximizes the amount of the public good in a symmetric equilibrium and find that such a prize always results in an overprovision of the public good.     

On Norms and Coordination Games: A Rent–Seeking Approach

The aim of this paper is to extend the rent–seeking literature to the equilibrium selection problem in competitive coordination games, i.e., games in which more than one equilibrium exists, and individuals' preferences are opposed. We analyze alternative correlated equilibria: contractual agreements and legally enforced equilibria. The latter are to be understood as the outcome of rents–seeking contests in which players invest resources in order to set a norm. The contest is analyzed in its basic two–person setting and later generalized to the two–populations case. There we show that the outcomes depend on the relative payoff structure of the game, the technological properties of the contest, and the population distribution. Finally, the efficiency analysis focuses not only on the extent of the rent dissipation, but also on the comparative analysis of the inefficiencies that arise in the market (not coordinated) equilibrium.     

Price vs quantity in a duopoly supergame with Nash punishments

We examine the endogenous choice between price and quantity behaviour in a duopoly supergame with product differentiation. We find that (i) if cartel profits are evenly split between firms, then only symmetric equilibria s obtained; (i) if instead the additional profits available through collusion are split according to the Nash bargaining solution, there are parameter regions where all subgame perfect equilibria are asymmetric, with firms colluding in price–quantity supergames.