On the Competition of Asymmetric Agents

Abstract. Rank-order tournaments are usually implemented in organizations to provide incentives for eliciting employees' effort and/or to identify the agent with the higher ability, for example in promotion tournaments. We close a gap in the literature by experimentally analyzing a ceteris paribus variation of the prize spread – being the major design feature of tournaments – in a symmetric and an asymmetric setting. We find that effort significantly increases with the prize spread as predicted by standard theory. However, only for sufficiently large prize spreads weak players competing against strong players strain themselves all the more and sorting of agents is feasible.     

Tournaments with Prize‐setting Agents*

In some tournaments, it is the contestants themselves who determine reward allocation. Union members bargain over wage distribution, and some firms allow self‐managed teams to freely determine internal resource allocation, incentive structure, and division of labor. We analyze, and test experimentally, a tournament where heterogeneous agents determine the spread between winner prize and loser prize. We investigate the relationship between prize spread, uncertainty, heterogeneity, and effort. We find that a large prize spread is associated with a low degree of uncertainty and a high degree of heterogeneity, and that heterogeneity triggers effort. By and large, our real‐effort experiment supports the theoretical predictions.     

Proposal power and majority rule in multilateral bargaining with costly recognition

This paper studies a sequential bargaining model in which agents expend efforts to be the proposer. In equilibrium, agents’ effort choices are influenced by the prize and cost effects. The (endogenous) prize is the difference between the residual surplus an agent obtains when he is the proposer and the payment he expects to receive when he is not. Main results include: (1) under the unanimity voting rule, two agents with equal marginal costs propose with equal probabilities, regardless of their time preferences; (2) under a nonunanimity rule, however, the more patient agent proposes with a greater probability; (3) while, under the unanimity rule, the social cost decreases in group heterogeneity, it can increase under a nonunanimity rule; and (4) when agents are identical, the unanimity rule is socially optimal.     

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.     

Asymmetric technologies in contests

I consider a contest between two risk-neutral players over a common-value prize, in which one player has a linear cost-of-effort function and the other a strictly convex cost-of-effort function f. I show that if the value of the prize is above (below) a certain threshold level, then the equilibrium aggregate effort in this contest is larger (smaller) than in a contest in which both players are characterized by the strictly convex cost-of-effort function f, and smaller (larger) than in a contest in which both players are characterized by a linear cost-of-effort function. Therefore, in contrast to the general result in the literature, asymmetry in contests can increase competition.     

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.     

Irrational diversification in multiple decision problems

The paper deals with multiple decision problems, which are similar to the task of guessing the color outcomes of five independent spinnings of a roulette wheel, 60% of whose slots are red and 40% white. Each correct guess yields a prize of $1. The guess of 5 Reds clearly first order stochastic dominates any other strategy. In contrast, subjects diversify their choices when facing a multiple decision problem in which the choice is between lotteries with clear objective probabilities. The diversification is stronger when the subjects face uncertainty without objective probabilities and weaker when the choice problem involves real life actions.     

Contests with Alternative Public‐Good Prizes

I study contests in which a society of players compete, by expending irreversible effort, over which one of alternative prizes should be awarded to them by the decision maker. The prizes are public goods and/or public bads for the players. The players choose their effort levels simultaneously and independently. I define each player's valuation spread as the difference between his valuations for the two public‐good/public‐bad prizes. I establish that the players' equilibrium effort levels depend solely on their valuation spreads, and that the players never expend positive effort for both prizes in equilibrium. Further, I establish that in equilibrium only players with the widest positive valuation spread and players with the widest negative valuation spread expend positive effort. Finally, I establish that the equilibrium effort level expended for each prize and the equilibrium total effort level are determined only by the widest positive valuation spread and the widest negative valuation spread.     

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.     

ARBITRATION,INDIVISIBLE ASSETS AND THE POSSIBILITY OF MALICE

Many real‐life contexts involve allocating an indivisible prize between two claimants. One (or both) claimants may derive ‘malice utility’ from depriving the other claimant. I study an arbitrator who does not know if malice is involved in a particular dispute, but wishes the prize to be allocated to the party with the higher intrinsic valuation for it and discuss two simple mechanisms that achieve his purpose regardless of whether malice is actually present. I examine several extensions.     

Tournaments with gaps

A standard tournament contract specifies only tournament prizes. If agents’ performance is measured on a cardinal scale, the principal can complement the tournament contract by a gap which defines the minimum distance by which the best performing agent must beat the second best to receive the winner prize. We analyze a tournament with two risk averse agents. Under unlimited liability, the principal strictly benefits from a gap by partially insuring the agents and thereby reducing labor costs. If the agents are protected by limited liability, the principal sticks to the standard tournament.     

Fairness in the Mail and Opportunism in the Internet: A Newspaper Experiment on Ultimatum Bargaining

Abstract. On 11 May 2001, readers of the Berliner Zeitung were invited to participate in an ultimatum bargaining experiment played in the strategy vector mode: each participant chooses not only how much (s)he demands of the DM1,000 pie but also which of the nine possible offers of DM100, 200, …, 900 (s)he would accept or reject. In addition, participants were asked to predict the most frequent type of behavior. Three randomly selected proposer–responder pairs were rewarded according to the rules of ultimatum bargaining and three randomly chosen participants of those who predicted the most frequent type of behavior received a prize of DM500. Decisions could be submitted by mail, fax or via the internet. Behavior is described, statistically analyzed and compared to the usual laboratory ultimatum bargaining results.     

Contests with limited resources

Many interesting phenomena (electoral competition, R&D races, lobbying) are instances of multiple simultaneous contests with unconditional commitment of limited resources. Specifically, the following game is analyzed. Two players compete in a number of simultaneous contests. The players have limited resources (budgets) and must decide how to allocate these to the different contests. In each contest the player who expends more resources than his adversary wins a corresponding prize. Mixed-strategy equilibria are characterized in the case of identical values and budgets and the connections with the classical Blotto game are analyzed.     

竞赛中的最优奖励:一个拍卖分析框架

本文在全支付(all-pay)拍卖的框架下,研究了一类非对称竞赛活动的最优奖励问题。我们假设竞赛组织者未知参赛者的成本信息,进而可供选择的奖励方式有两种,其一是固定奖励额;其二是线性奖励,最终奖励额由参赛者的投资内生决定。我们证明:如果投资不能完全披露成本信息,固定奖励优于线性奖励;如果投资完全披露成本信息,线性奖励可能优于固定奖励,同时我们得出了实现投资完全披露信息的一个必要条件。我们的结论不依赖于参赛者的对称性假设,不依赖于参赛者具体的成本分布函数形式,也不依赖于竞赛组织者试图最大化参赛者最大努力或最大化所有参赛者努力总和。     

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.     

Influence and the discretionary allocation of several prizes

When a decision process involves discretion, it may also be open to influence by various actors. If a single prize is at issue, then this influence can be modelled in the context of a rent-seeking game. However, the basic rent-seeking model needs to be extended in order to examine the amount of influence exerted when more than one prize is at stake. In this paper, we investigate the effect of several factors which may affect rent-seeking in this multi-prize framework but are not accounted for in single-prize models; examples here are the number of prizes, the division of the prize mass and the distribution mechanism which is employed.     

Allocation of Prizes in Contests with Size Effects

We examine the allocation of prizes in contests in which the number of contenders affects the prizes and costs. We assume that there are two groups of contenders. The government allocates a prize to the two groups, and the contenders in each group respectively compete for the prize. Examining the prize allocation in such contests, we obtain the following results. The aggregate effort increases in the prize share of the larger group. In contests with size effects through costs, the aggregate resource expended in the contests and the aggregate payoffs are independent of group size distribution if the prize is allocated in proportion to group size. The integration of contests with size effects through prizes can yield higher aggregate effort and payoffs than the decentralized contests.     

Dynamic Difference‐Form Contests

This study presents a multistage contest designed by an organizer who chooses the (indivisible) winner prize together with the ranking scheme. When the organizer's objective is to maximize efforts, the optimal ranking scheme that selects the prize recipient coincides with a version of the piecewise linear difference‐form success function postulated by Che and Gale (2000). Even if the organizer's objective is not necessarily to maximize agents' effort, a noisy ranking of the difference‐form remains optimal.     

Complementarity in contests

To facilitate the study of contests in general equilibrium, we examine winner-take-all contests in which the prize is complementary to the effort of the contestants, as inputs are in production functions or final goods in utility functions. We focus on the effects of technological factors and endowments on the effort and the welfare of the contestants. Most of our findings differ considerably from the standard model of contests in which prize and effort are independent. In particular, we find a critical role for the elasticity of substitution between prize and effort. For example, under low elasticities of substitution, a higher prize reduces the effort exerted by the contestants.     

On the existence of expected multi-utility representations

We prove the following facts related to the expected multi-utility representation of an affine preorder: If the prize space is not compact and if the lottery set consists of all probabilities on the prize space, standard independence and continuity axioms do not guarantee the existence of a (continuous) representation. If the prize space is σ-compact and lotteries have compact support, a representation exists. When the preorder in question is bounded, this result extends to the set of lotteries that consists of all probabilities on the prize space. For the case of monetary lotteries, the boundedness assumption in this last result can be dropped, provided that the preference relation at hand is monotone and risk-averse.