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.     

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.     

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.     

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.     

Contests with Multi-tasking*

The standard contest model in which participants compete in a single dimension is well understood and documented. Multi‐dimension extensions are possible but are liable to increase the complexity of the contest structure, mitigating one of its main advantages: simplicity. In this paper we propose an extension in which competition ensues in several dimensions, and a competitor that wins a certain number of these is awarded a prize. The amount of information needed to run the contest is hence limited to the number of dimensions won by each player. We look at the design of this contest from the point of view of maximising effort in the contest (per dimension and totally), and show that there will be a tendency to run small contests with few dimensions. The standard Tullock model and its results are encompassed by our framework.     

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.     

Designing an optimal contest

The paper brings a mechanism design perspective to the study of contests. We consider the problem of selecting a contest success function when the contest designer may also value the prize. We show that any equilibrium outcome that can be achieved by a concave increasing contest success function can be replicated by a linear contest success function. An expected utility maximizing designer should employ a linear homogeneous contest success function. We explicitly derive the optimal contest for a risk-neutral designer and present comparative statics results. Tullock's contest is optimal only when the designer's valuation for the prize is low.     

INTENSITY OF COMPETITION, ENTRY AND ENTRY DETERRENCE IN RENT SEEKING CONTESTS

In rent seeking contests agents compete over a given amount of resource a prize. To increase the likelihood of winning the prize, the agents expend efforts. This paper addresses the issue of entry to and entry deterrence from rent seeking contest. The main purpose of the analysis is to provide an explanation for the empirical fact that the number of participants in contests is usually small. I also obtain results on the relationship between the intensity of competition in a contest and the amount of the contenders'expenditures.     

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.     

Lotteries vs. All‐Pay Auctions in Fair and Biased Contests

The form of contests for a single fixed prize can be determined by a designer who maximizes the contestants' efforts. This article establishes that, under common knowledge of the two asymmetric contestants' prize valuations, a fair Tullock‐type endogenously determined lottery is always superior to an all‐pay‐auction; it yields larger expected efforts (revenues) for the contest designer. If the contest can be unfair (structural discrimination is allowed), then the designer's payoff under the optimal lottery is equal to his expected payoff under the optimal all‐pay auction.     

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.     

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.     

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.     

Dynamic Multi‐Activity Contests*

We develop a model of dynamic multi‐activity contests. Players simultaneously choose efforts in long‐run activities, observe each other's efforts in these activities, and then simultaneously choose efforts in short‐run activities. A player's long‐run and short‐run efforts complement each other in determining the player's probability of winning. We compare the outcomes of this two‐stage model to those of the corresponding model in which players choose efforts in all activities simultaneously. Interestingly, effort expenditures are always lower in the sequential multi‐activity contest than in the simultaneous multi‐activity contest. The implications of this result for the organization of military, litigation, innovation, academic, and sporting contests are highlighted.     

Selection and incentives in contests: evidence from horse racing

The designer of internal labour market promotion contests must balance the need to select the best candidate with the need to provide incentives for all candidates. We use an extensive data set from horse racing – where there is abundant variation in contest design features – to analyse if there are particular features that help to achieve these two objectives. We find that contests with higher prize money and fewer participants are the most successful at achieving the dual remit of selection and incentives.     

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.     

Multi-player contests with asymmetric information

We consider imperfectly discriminating, common-value, all-pay auctions (or contests) in which some players know the value of the prize, others do not. We show that if the prize is always of positive value, then all players are active in equilibrium. If the prize is of value zero with positive probability, then there is some threshold number of informed players such that if there are less, all uninformed players are active, and otherwise all uninformed players are inactive.     

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.     

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.     

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.