On the Existence and Uniqueness of Pure-Strategy Nash Equilibrium in Asymmetric Rent-Seeking Contests

There can be three types of heterogeneity among players in a rent‐seeking contest. First, effectiveness of player's effort on the winning probabilities may differ among players. Secondly, players may evaluate the rent or prize of the rent‐seeking contest differently. Thirdly, players may face different financial constraints. This article proves under standard assumptions in the literature that there exists a unique pure‐strategy Nash equilibrium in a general asymmetric rent‐seeking contest with these three types of heterogeneity among players.     

Momentum and betting market perceptions of momentum in college football

Past research has demonstrated that bettors believe positive momentum carries over from contest to contest. This article examines whether there is any empirical support for this belief by testing for the presence of across-contest momentum effects in college football. We characterize momentum in multiple fashions and after controlling for between-team heterogeneity find no evidence that systematic relationships exist between the degree of momentum a team enters a contest with and the outcome of that contest. From a wagering market perspective, this indicates that there is no statistically significant advantage to betting on teams perceived to possess positive momentum. Our results also suggest that the combination of the opening betting line set by odds makers and the subsequent market movement of that line does not systematically overreact to teams on streaks.     

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.     

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.     

Lottery versus all-pay auction contests: A revenue dominance theorem

We allow a contest organizer to bias a contest in a discriminatory way; i.e., she can favor specific contestants by designing the contest rule in order to maximize total equilibrium effort (resp. revenue). The two predominant contest regimes are considered, all-pay auctions and lottery contests. For all-pay auctions the optimal bias is derived in closed form: It implies extreme competitive pressure among active contestants and low endogenous participation rates. Moreover, the exclusion principle advanced by Baye et al. (1993) becomes obsolete in this case. In contrast, the optimally biased lottery induces a higher number of actively participating contestants due to softer competition. Our main result regarding total revenue comparison under the optimal biases reveals that the all-pay auction revenue-dominates the lottery contest for all levels of heterogeneity among contestants. The incentive effect due to a strongly discriminating contest rule (all-pay auction) dominates the participation effect due to a weakly discriminating contest rule (lottery).     

Affirmative action in contest games

This paper analyzes the incentive effects of affirmative action in competitive environments. Competition is between heterogeneous players in a contest game where heterogeneity might be due to past discrimination. Two policy options are analyzed that tackle the underlying asymmetry: either it is ignored and the contestants are treated equally, or affirmative action is implemented to neutralize the disadvantages of discriminated players. Comparing the induced effort exertion under the two policies reveals that in a two-player contest game the normative neutralization objective of affirmative action coincides with higher effort exertion. However, in the multi-player contest affirmative action might have adverse incentive effects as the participation of additional weak players detrimentally affects effort incentives of other players. These results also obtain under imperfect information of the contest organizer.     

Effort maximization in asymmetric contest games with heterogeneous contestants

Contest rules are set up by administrators who frequently have discretionary power in specifying the details of these rules, i.e., they can bias the contest rules toward specific contestants in order to further their prime objective. We derive the optimal bias of the contest rule for a contest administrator, who is interested in maximizing the total efforts expended in the contest. The solution is obtained in closed form for a widely used class of n-person contest games. Setting the optimal bias has important implications: (i) there is never exclusion of strong players, instead there is (endogenously induced) inclusion of weak contestants; (ii) the contest administrator will optimally level the playing field by encouraging weak contestants, but he will not equalize the contestants’ chances unless they are identical; and (iii) at least three contestants will be active in equilibrium of the optimal contest, irrespective of heterogeneity.     

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.     

Membership in citizen groups

We analyze the coordination problem of agents deciding to join a group that uses membership revenues to provide a discrete public good and excludable benefits. The public good and the benefits are jointly produced, so that benefits are valued only if the group succeeds in providing the public good. With asymmetric information about the cost of provision, the static membership game admits a unique equilibrium and we characterize the optimal membership fee. We show that heterogeneity in valuations for the excludable benefits is always detrimental to the group. However, in a dynamic contest in which heterogeneity arises endogenously (returning members receive additional seniority benefits at the expense of junior members), we show that, in the ex-ante optimal contract, offering seniority benefits is beneficial for the group, despite the heterogeneity in valuations created.     

Higher cognitive ability is associated with lower entries in a p-beauty contest

“Beauty contests” are well-studied, dominance-solvable games that generate two interesting results. First, most behavior does not conform to the unique Nash equilibrium. Second, there is considerable unexplained heterogeneity in behavior. In this work, we explore the relationship between beauty contest behavior and cognitive ability. We find that subjects with high cognitive ability exhibit behavior that is closer to the Nash equilibrium.     

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.     

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.     

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.     

国外创新竞赛机制设计研究述评

对国外创新竞赛的研究文献进行了梳理,重点针对创新竞赛的设计要素和奖励机制两方面的研究进行了评述。提出如下未来的研究方向:良好评价机制的构建;竞赛者的参与意愿与竞赛效率的影响因素;网上创新竞赛欺诈行为的防范机制。     

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.     

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.     

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.     

Selective incentives and intragroup heterogeneity in collective contests

A group taking part in a contest has to confront the collective action problem among its members, and devices of selective incentives are possible means of resolution. We argue that heterogeneous prize‐valuations in a competing group normally prevent effective use of such selective incentives. To substantiate this claim, we adopt cost‐sharing as a means of incentivizing the individual group members. We confirm that homogeneous prize valuations within a group result in a cost‐sharing rule inducing the first‐best individual contributions. As long as the cost‐sharing rule is dependent only on the members’ contributions, however, such a first‐best rule does not exist for a group with intragroup heterogeneity. Our main result clarifies how unequal prize valuations affect the cost‐sharing rule and, in particular, the degree of cost‐sharing. If the relative rate of change of the marginal effort costs is decreasing, it is reduced by intragroup heterogeneity. If the rate is increasing, the cost is fully shared, but it cannot induce the first‐best contributions for the group.     

Doping and cheating in contest-like situations

Individuals who compete in a contest-like situation (for example, in sports, in promotion tournaments, or in an appointment contest) may have an incentive to illegally utilize resources in order to improve their relative positions. We analyze such doping or cheating within a tournament game between two heterogeneous players. Besides the direct and indirect costs of doping, three major effects are identified which determine a player's decision to deviate from a no-doping situation — a cost effect, a likelihood effect and a base-salary effect. Moreover, the influence of heterogeneity and exogenous performance risk on a no-doping outcome, and the influence of ex-ante and ex-post doping tests on the players' investments are discussed.     

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.