Contests with size effects through costs

Eggert and Kolmar [Eggert, W., Kolmar, M., in press. Contests with size effects. European Journal of Political Economy.] examine the effects of group size on individual and aggregate payoffs in contests where prizes increase in the number of players. In a reconsideration of their paper, this note examines contests in which the unit cost of effort decreases in the number of players. In contests for fixed prizes, an increase in the number of players decreases individual and aggregate payoffs. In contests in which players face a trade-off between appropriative and productive investments, an increase in the number of players decreases individual payoffs while increasing aggregate payoffs.     

Effort and Performance in Public Policy Contests

Government intervention often gives rise to contests in which the possible “prizes” are determined by the status quo and some new public policy proposal. In this paper we study a general class of such two‐player public policy contests and examine the effect of a change in the proposed policy, a change that may affect the payoffs of the two contestants, on their effort and performance. Our results extend the existing comparative statics studies that focus, in symmetric contests, on the effect of a change in the value of the prize or, in asymmetric contests, on the effect of one contestant's valuation of the prize. Our results hinge on a fundamental equation that specifies the equilibrium relationship between the strategic own‐stake effect and the strategic rival's‐stake effect. This fundamental equation clarifies the role of the three possible types of ability and stake asymmetry in determining the effect of payoff variations on the efforts and performance of the contestants.     

POLITICALLY CONTESTABLE RENTS AND TRANSFERS*

A rent or transfer is politically contestable when policy decisions are subject to influence by potential beneficiaries and losers. This paper studies contestablility of rents and transfers when contenders place different valuations on the politically allocated prize. Asymmetric valuation inhibits participation by low-valuation contenders. The model explains the phenomena of small numbers of active participants in contests to exercise political influence and low lobbying and other influence-seeking outlays relative to the value of politically allocated prizes.     

Overbidding and overspreading in rent-seeking experiments: Cost structure and prize allocation rules

We study experimentally the effects of cost structure and prize allocation rules on the performance of rent-seeking contests. Most previous studies use a lottery prize rule and linear cost, and find both overbidding relative to the Nash equilibrium prediction and significant variation of efforts, which we term ‘overspreading.’ We investigate the effects of allocating the prize by a lottery versus sharing it proportionally, and of convex versus linear costs of effort, while holding fixed the Nash equilibrium prediction for effort. We find the share rule results in average effort closer to the Nash prediction, and lower variation of effort. Combining the share rule with a convex cost function further enhances these results. We can explain a significant amount of non-equilibrium behavior by features of the experimental design. These results contribute towards design guidelines for contests based on behavioral principles that take into account implementation features of a contest.     

Rent seeking and rent dissipation: A neutrality result

We consider rent seeking contests between at least two agents who might value the prize differently. We capture a wide range of institutional aspects of contests by analyzing a class of contest success functions fulfilling several properties. The main properties are anonymity and a condition on the elasticity of a rent seeker's win probability with respect to her effort. We show the existence of a mixed-strategy equilibrium and establish equilibrium payoffs. In this equilibrium complete rent dissipation holds. Our results imply a partial robustness result for the all-pay auction.     

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.     

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.     

Entry into winner-take-all and proportional-prize contests: An experimental study

This experiment compares the performance of two contest designs: a standard winner-take-all tournament with a single fixed prize, and a novel proportional-payment design in which that same prize is divided among contestants by their share of total achievement. We find that proportional prizes elicit more entry and more total achievement than the winner-take-all tournament. The proportional-prize contest performs better by limiting the degree to which heterogeneity among contestants discourages weaker entrants, without altering the performance of stronger entrants. These findings could inform the design of contests for technological and other improvements, which are widely used by governments and philanthropic donors to elicit more effort on targeted economic and technological development activities.     

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。     

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.     

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.     

Are sequential round‐robin tournaments discriminatory?

I examine sequential round‐robin tournaments with three and four symmetric players. Each player is matched once with each other player and each match is organized as a Tullock contest. A single prize is allocated to the player with the most matches won. I show that, depending on the position of their matches in the sequence of the tournament, the players' ex ante winning probabilities and expected payoffs differ. Compared to tournaments with matches organized as all‐pay auctions, however, discrimination is weaker and partially reversed. Moreover, aggregate effort in tournaments with suitable Tullock contests is larger than in tournaments with all‐pay auctions. The fairness of round‐robin tournaments may be improved by an endogenous sequence of matches or the requirement that players fix their effort ex ante.     

Repeated contests: A general parameterization

Contests often involve players vying for the same prize year after year. This paper characterizes equilibrium effort, both individual and aggregate, in a general parameterization of such repeated contests.     

PUBLIC GOODS,RENT DISSIPATION,AND CANDIDATE COMPETITION*

The early literature on rent seeking presumed that total expenditures in the rent-seeking process would equal the value of a contested prize. Subsequent analysis has shown, however, that a number of circumstances give rise to underdissipation of the contested rent—imperfectly discriminating contests, risk-aversion, or where individuals' valuations of the prize are not identical. This paper shows how underdissipation is associated with the public-good character of contested rents. In particular, total rent dissipation is demonstrated to be less than the average stake of the individual contenders. In addition to assessing rent dissipation in contests for public-good allocations, the paper also endogenizes the value of the prize by placing the rent-seeking contest in a setting of candidate competition.     

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.     

Tournaments of Weakly Heterogeneous Players

We consider three prominent tournament formats—contests, binary elimination tournaments, and round-robin tournaments—in the case when players are heterogeneous in their abilities but the heterogeneity is, in a well-defined sense, weak. Using linear approximation, we characterize equilibrium strategies and payoffs in the three tournament games and compare them to the benchmark symmetric case of identical players. We describe small deviations from the symmetric equilibria by elasticities of a player's equilibrium effort with respect to her own ability and the abilities of her rivals. Our results only require general symmetry and smoothness assumptions but not specific functional forms for the probabilities of winning in tournaments. We show that, in equilibrium: (i) a player's effort and payoff depend on her rivals' abilities in a model-independent way, either through the average ability of the field (for static games), or through the properly discounted average ability of the field (for dynamic elimination tournaments); (ii) players respond stronger to changes in their own relative abilities than to changes in their rivals' relative abilities; (iii) aggregate effort (dissipated rent) does not change compared to the benchmark case; (iv) it is not possible to manipulate aggregate effort through seeding in binary elimination tournaments, although optimal seeding schemes for other purposes can be identified; and (v) balanced seeding and a uniform distribution of relative abilities cancel out the dependence of a player's effort on her rivals' abilities in binary elimination tournaments.     

Sequential two-prize contests

We study a two-stage all-pay auction with two identical prizes. In each stage, the players compete for one prize. Each player may win either one or two prizes. We analyze the unique subgame-perfect equilibrium of our model with two players where each player??s marginal values for the prizes are decreasing, constant, or increasing. We also analyze an equilibrium of the model with more than two players where each player??s marginal values for the prizes are nonincreasing.     

Governing Interest Groups and Rent Dissipation

In a contest of group‐specific public goods we consider the effect that managing an interest group has on the rent dissipation and the total expected payoffs of the contest. While in the first group there is a central planner determining its members’ expenditure in the contest, in the second group there are two different possibilities: either all the members are governed by a central planner or they are not. We consider both types of contests: an all‐pay auction and a Logit contest success function. We show that while governing an interest group decreases free‐riding, it may as well decrease the rent dissipation; at the same time the expected payoffs of the groups may also decrease.     

Collective rent seeking and division of labor

The article analyzes division of labor within multiple groups engaged in collective rent seeking through time. Each agent seeks membership in one of two subgroups within each group, conveniently labelled a leader-subgroup and a follower-subgroup. The within-group generated payoffs are used as input in the between-group competition for the other groups' payoffs and an external rent. Within-group egalitarian allocation, but not relative-effort allocation, alleviates leadership struggle if the rent is large or the group is moderately more decisive than the other groups. The group employing a more egalitarian sharing rule than the other groups alleviates its leadership struggle more successfully. The Folk theorem is used to show the conditions under which leadership struggle gets intensified or alleviated.     

Collective rent seeking and division of labor

The article analyzes division of labor within multiple groups engaged in collective rent seeking through time. Each agent seeks membership in one of two subgroups within each group, conveniently labelled a leader-subgroup and a follower-subgroup. The within-group generated payoffs are used as input in the between-group competition for the other groups' payoffs and an external rent. Within-group egalitarian allocation, but not relative-effort allocation, alleviates leadership struggle if the rent is large or the group is moderately more decisive than the other groups. The group employing a more egalitarian sharing rule than the other groups alleviates its leadership struggle more successfully. The Folk theorem is used to show the conditions under which leadership struggle gets intensified or alleviated.