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.     

Risk aversion in symmetric and asymmetric contests

We analyze existence, uniqueness and properties of equilibria in incompletely discriminating Tullock contests with logistic contest success functions, when contestants are risk averse. We prove that a Nash equilibrium for such a contest exists, but give an example of a symmetric contest with both symmetric and asymmetric equilibria, showing that risk aversion may lead to multiple equilibria. Symmetric contests have unique symmetric equilibria but additional conditions are necessary for general uniqueness. We also study the effects on incumbents of additional competitors entering the contest under these conditions and examine the effects of risk aversion on rent dissipation in symmetric and asymmetric contests.     

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).     

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.     

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.     

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 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.     

Orchestrating Rent Seeking Contests

Contests have different, sometimes quite complex, organisational structures. In particular, while most of the existing literature focusses on simultaneous contests, multistage contests are also quite frequently encountered. This paper seeks to provide a rationale for the latter by endogenising the choice of a contest structure, which is made by an organiser of a contest interested in maximising the efforts expended by the contenders.     

在线创新竞赛的基本问题与研究展望

在线创新竞赛是组织实施开放式创新的新途径,学术界关于在线创新竞赛的研究日益丰富。然而,现有文献分布于心理学、经济学、管理学、教育学等各个领域,为了梳理现有研究成果并展望未来研究方向,分别从什么是在线创新竞赛、谁在参与在线创新竞赛和如何实施在线创新竞赛3个角度对现有文献进行了述评,厘清了在线创新竞赛的主要特征与类型、参与主体动机及要素设计等基本问题。最后,探讨了开放式创新和“互联网+”背景下,在线创新竞赛的未来研究方向。     

Endogenous entry in contests

We report the results of laboratory experiments on rent-seeking contests with endogenous participation. Theory predicts that (a) contest entry and rent-seeking expenditures increase with the size of the prize and (b) earnings are equalized between the contest and the outside option. While the directional predictions offered in (a) are supported in the data, the level predictions are not. Prediction (b) is not supported in the data: when the prize is large, contest participants earn more than the outside option. When the prize is small, contest participants earn less. Previous studies of gender and contest competition suggest that females should (a) not perform as well in the contest; and (b) enter at a lower rate. We find some support for (a) but not for (b). Women participate in the contest at the same rate as men.     

Optimal contest design: volume and timing of rent seeking in contests

This paper undertakes a comparative analysis of rent-seeking contests in terms of the amount and the timing of effort they elicit from the participants. The optimal contest structure—the one that maximizes the discounted sum of efforts—is found to hinge on the degree of impatience of the contest organizers, the more patient of whom prefer longer contests consisting of pairwise matches among the contestants. Heterogeneity of the contestants' quality also turns out to play an important role in the comparison.     

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.     

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.     

Quantity-cum-quality contests

Quality and quantity are very common features of production processes. People care about these two features and they tend to be connected. I consider a contest in which the quantity and quality of output are rewarded. The output in the quality contest plays a dual role. It counts in the quality contest but it is also converted into quantity-equivalent output to obtain total output in the quantity contest. This latter feature implies that the two contests are interlinked. I find that when the unit cost of producing quality is sufficiently high, then treating quality and quantity as the same has a disincentive effect on the production of quality. In contrast, when the unit cost of producing quality is sufficiently low, treating quality and quantity as the same has no disincentive effect on the production of quality. I also find an equilibrium in which no one exerts effort in the quantity contest. When there is a binding budget constraint on effort, I find that effort in the quantity contest is smaller relative to the unconstrained case but effort in the quality contest may remain unchanged.     

DISCRIMINATION IN A MODEL OF CONTESTS WITH INCOMPLETE INFORMATION ABOUT ABILITY

We study contests with private information and identical contestants, where contestants' efforts and innate abilities generate output of varying qualities. The designer's revenue depends on the quality of the output, and she offers a reward to the contestant achieving the highest quality. We characterize the equilibrium behavior, outcomes, and payoffs for both nondiscriminatory and discriminatory (where the reward is contestant‐dependent) contests. We derive conditions under which the designer obtains a larger payoff when using a discriminatory contest and describe settings, where these conditions are satisfied.     

Marginal Subsidies in Tullock Contests

In a general Tullock contest, we examine a situation where a limited resource can be used to provide marginal subsidies to either player (weak or strong), or to increase the prize directly. We show that to maximize total effort, subsidizing the weak/strong player is preferred when the contest is sufficiently accurate/inaccurate. This result generalizes to n‐player lottery contests. In a lottery contest (Tullock contest with ), we derive the optimal scheme for a full range of resource: when the resource is small, it is optimal to only subsidize the weak player; when it is large, both players should be subsidized simultaneously.     

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.     

Experimental comparison of multi-stage and one-stage contests

This article experimentally studies a two-stage elimination contest and compares its performance with a one-stage contest. Contrary to the theory, the two-stage contest generates higher revenue than the equivalent one-stage contest. There is significant over-dissipation in both stages of the two-stage contest and experience diminishes over-dissipation in the first stage but not in the second stage. Our experiment provides evidence that winning is a component in a subject's utility. A simple behavioral model that accounts for a non-monetary utility of winning can explain significant over-dissipation in both contests. It can also explain why the two-stage contest generates higher revenue than the equivalent one-stage contest.     

Evolutionary equilibrium in Tullock contests: spite and overdissipation

Tullock's analysis of rent-seeking as a contest is reconsidered from an evolutionary point of view. We show that evolutionarily stable behavior in a Tullock contest exists and differs from behavior in Nash equilibrium. Evolutionarily stable behavior in these contests is robust in a strong sense and may entail overdissipation of the contested rent.     

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.