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.     

The optimal multi-stage contest

This paper investigates the optimal (effort-maximizing) structure of multi-stage sequential-elimination contests. We allow the contest organizer to design the contest structure using two instruments: contest sequence (the number of stages, and the number of contestants remaining after each stage), and prize allocation. When the contest technology is sufficiently noisy, we find that multi-stage contests elicit more effort than single-stage contests. For concave and moderately convex impact functions, the contest organizer should allocate the entire prize purse to a single final prize, regardless of the contest sequence. Additional stages always increase total effort. Therefore, the optimal contest eliminates one contestant at each stage until the finale when a single winner obtains the entire prize purse. Our results thus rationalize various forms of multi-stage contests that are conducted in the real world.     

Bidding in hierarchies

This paper reconsiders the comparison between hierarchical contests and single-stage contests. A condition is given that characterizes whether and when the aggregate equilibrium payoff of contestants is higher in the single-stage contest, and when the single-stage contest is more likely to award the prize to the contestant who values it most highly. The outcome depends on inter- and intra-group heterogeneity, and is not driven by free-rider incentives.     

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     

Repeated Contests with Asymmetric Information

The same contestants often meet repeatedly in contests. Behavior in a contest potentially provides information with regard to one's type and can therefore influence the behavior of the opponents in later contests. This paper shows that if effort is observable, this can induce a ratchet effect in contests: high ability contestants sometimes put in little effort in an early round in order to make the opponents believe that they are of little ability. The effect reduces overall effort and increases equilibrium utility of the contestants when compared with two unrelated one-shot contests. It does, however, also introduce an allocative inefficiency since sometimes a contestant with a low valuation wins. The model assumes an imperfectly discriminating contest. In an extension I show that, qualitatively, results are similar in a perfectly discriminating contest (all pay auction).     

Using a Simple Contest to Illustrate Mechanism Design

This article describes a simple classroom activity that illustrates how economic theory can be used for mechanism design. The rules for a set of contests are presented; the results typically obtained from these contests illustrate how the prize structure can be manipulated in order to produce a particular outcome. Specifically, this activity is designed to show how changing the prize structure can impact both the contestants’ average efforts and the effort level of the hardest-working contestant. The activity can be run in a 50-minute class, has instructions that fit on a single piece of paper, and, although it can be run in large classes, requires only six students.     

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.     

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.     

A note on modelling tournaments

Existing studies on tournaments utilize different specifications of the production function. Comparison of these results is difficult. This paper shows that the strategic behavior of agents in a tournament can be analyzed more generally using a specification of the production function along the lines of the standard agency model. I also show that in mixed contests, the more able contestant would in equilibrium have a higher probability of winning the contest despite attempts to use effort to compensate for ability by the less able contestant (Proposition 3).The author would like to thank Joseph Stiglitz for suggesting the idea for this paper, Jerry Green for reading an earlier version of the paper, and three anonymous referees for helpful comments and suggestions. The usual disclaimer applies.     

Candidate quality and US Senate elections

We find that candidate quality is a key determinant of US Senate election outcomes. We model the results for the last 10 US Senate election cycles, from 2012 back to 1994, for a total of 345 election contests. For the contests where an incumbent seeks re-election, a quality challenger can significantly diminish the advantage that usually attaches to incumbency. For the open-seat contests, which tend to be more competitive, candidate quality can swing a close election. Governors who seek election to the US Senate receive the largest boost, as indicated by our candidate-quality variables vector. Wave effects and presidential coattail effects are also shown to be contributing factors in certain cases.     

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.     

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.     

INDUCING VARIETY: A THEORY OF INNOVATION CONTESTS

This article analyzes the design of innovation contests when the quality of an innovation depends on the research approach, but the best approach is unknown. Inducing a variety of research approaches generates an option value. We show that suitable contests can induce such variety. The buyer‐optimal contest is a bonus tournament, where suppliers can choose only between a low bid and a high bid. This contest implements the socially optimal variety for a suitable parameter range. Finally, we compare the optimal contest to scoring auctions and fixed‐prize tournaments.     

Hybrid Contests

This paper examines hybrid contests where participants commit two types of resources to improve their probability of winning the prize. The first type is forfeited ex ante, before the prize is allocated, by winners and losers alike, while the second is committed ex ante by all contenders but expended ex post, after the prize is allocated, and only by the contestant that wins the prize. The model yields a number of interesting results. Among them is the finding that, as the number of contestants increases, the ex ante expenditures of individual contestants decrease while the ex post expenditure increases. Even more interesting, the total of the ex ante and ex post expenditures by the contenders in a hybrid contest may decrease with the number of competitors. The study also finds that there is no rent overdissipation, and compares the total expenditures in the contest and “all‐pay” allocation mechanisms.     

Rank-order tournaments and selection

Rank-order tournaments are often presented as devices for aligning incentives in a principal-agent setting. In most of this literature agents are expected to be identical so that the principal is indifferent ex ante as to who wins the contest, implying that the selection properties of the tournament can be ignored. In this paper we consider a tournament which is not necessarily symmetric, and in which agent type is private information. The principal cares about who wins, but the basic tournament will not achieve perfect selection; the lower-type agent may sometimes win. In a two-player tournament we present a simple reward system in which the winner's reward depends upon which (if any) of two “test standards” is passed; conditions are presented under which this system allows the principal to choose the best agent. This system can be extended in a simple manner to rank types in ann-player tournament. We suggest that the theory can be applied to internal labor markets and research contests.     

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.     

On Elimination in Contests: A Perspective on Output Maximization

This paper investigates the desirability of adding a preliminary elimination stage for output maximization in a winner‐take‐all contest framework in which the contestant who achieves the highest (random) output wins. We find that, generally, the desirability of an elimination stage does not monotonically depend on the productivity of the effort; adding a preliminary stage can improve output for both concave and convex production functions. This result contrasts sharply with current insight from effort maximization, which argues that adding a preliminary stage can increase effort supply only if the production function is concave.     

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.     

ENHANCING EFFORT SUPPLY WITH PRIZE‐AUGMENTING ENTRY FEES: THEORY AND EXPERIMENTS

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.     

MECHANISM AND IMPACTS OF EMBEZZLEMENT IN A DUOPOLISTIC MARKET*

We consider the case in which a corruptible manager contemplates to embezzle his/her firm's revenues by overstating the production costs in a duopolistic market. In order to embezzle more, the manager chooses to increase the firm's output. This partially corrects the market failures associated with the oligopoly distortion. Nevertheless, pervasive and large‐scale embezzlement is detrimental and should be addressed, although moderate embezzlement might be socially good. We also consider how the length of tenure affects the manager's behaviours: extending the tenure cannot eradicate embezzlement, and an insufficient rise of the reward rate may only defer embezzlement.