Contest architecture

A contest architecture specifies how the contestants are split among several sub-contests whose winners compete against each other (while other players are eliminated). We compare the performance of such dynamic schemes to that of static winner-take-all contests from the point of view of a designer who maximizes either the expected total effort or the expected highest effort. For the case of a linear cost of effort, our main results are: (1) If the designer maximizes expected total effort, the optimal architecture is a single grand static contest. (2) If the designer maximizes the expected highest effort, and if there are sufficiently many competitors, it is optimal to split the competitors in two divisions, and to have a final among the two divisional winners. Finally, if the effort cost functions are convex, the designer may benefit by splitting the contestants into several sub-contests, or by awarding prizes to all finalists.     

Maximum efforts in contests with asymmetric valuations

Efforts may be reduced when players with different valuations participate in a contest. This paper considers the problem of designing a contest to elicit maximum aggregate effort from players with asymmetric valuations. Optimal designs for different classes of contest technologies are computed and characterized. A value weighted contest is optimal in the concave case. In the unconstrained case, the optimal contest is equivalent to a first price all-pay auction with a reserve price. The optimal design discounts the effort of the high valuation player in order to induce him to compete vigorously.     

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.     

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.     

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).     

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.     

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.     

Collusion and the organization of delegated expertise

We study the optimal design of incentive contracts for experts in different collusion environments, and explore implications for the organization of delegated expertise. We consider a principal relying on experts to gather and report two signals about a project's value. The principal can have a single expert gather both signals or two experts gather one signal each. We show that absent collusion, the multiexpert organization dominates the single expert organization. However, this ranking is reversed when the experts can collude among themselves (horizontal collusion) and with the principal (vertical collusion).     

ROLE OF RELATIVE AND ABSOLUTE PERFORMANCE EVALUATIONS IN INTERGROUP COMPETITION

We investigate the moral hazard problem in which a principal delegates multiple tasks to multiple workers. The principal imperfectly monitors their action choices by observing the public signals that are correlated with each other through a macro shock. He divides the workers into two groups and makes them compete with each other. We show that when the number of tasks is sufficiently large, relative performance evaluation between the groups accompanied by absolute performance evaluation results in eliminating unwanted equilibria. In this case, any approximate Nash equilibrium nearly induces the first‐best allocation.     

Contests where there is variation in the marginal productivity of effort

Summary. We provide a characterization of participants' behavior in a contest or tournament where the marginal productivity of effort varies across contestants and individual productivity is private information. We then consider the optimal design of such a contest. We first analyze contestant behavior for the usual type of contest, where the highest output wins. Abilities need not be independently distributed. We demonstrate that there is a unique symmetric equilibrium output function, that output is increasing in ability, and that marginal effort is increasing in ability, while effort decreases when the cost of effort increases. Next we consider the case where the highest output need not win, with independently distributed abilities. We analyze the contest designer's decisions in choosing contest rules optimal from her perspective. We show that the output produced, probability of winning, and contest designer's expected revenue are generally increasing in contestants' ability. We examine the relationship between the marginal cost of producing output and marginal utility per dollar of the net award for winning. Received: July 30, 1998; revised version: August 7, 2000     

Unemployment insurance with hidden savings

This paper studies the design of unemployment insurance when neither the searching effort nor the savings of an unemployed agent can be monitored. If the principal could monitor the savings, the optimal policy would leave the agent savings-constrained. With a constant absolute risk-aversion (CARA) utility function, we obtain a closed form solution of the optimal contract. Under the optimal contract, the agent is neither saving nor borrowing constrained. Counter-intuitively, his consumption declines faster than implied by Hopenhayn and Nicolini (1997) [1]. The efficient allocation can be implemented by an increasing benefit during unemployment and a constant tax during employment.     

Contest design and heterogeneity

I show that the effect of heterogeneity on contest investments depends on the structure of the competition, which implies that heterogeneity matters for optimal contest design. This insight helps to explain empirical evidence on the comparison of different contest structures.     

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.     

Asymmetric awareness and moral hazard

This paper introduces asymmetric awareness into the classical principal–agent model and discusses the optimal contract between a fully aware principal and an unaware agent. The principal enlarges the agentʼs awareness strategically when proposing a contract and faces a tradeoff between participation and incentives. Leaving the agent unaware allows the principal to exploit the agentʼs incomplete understanding of the world, relaxing the participation constraint, while making the agent aware enables the principal to use the revealed contingencies as signals about the agentʼs action choice, relaxing the incentive constraint. The optimal contract reveals contingencies that have low probability but are highly informative about the agentʼs effort.     

Job design with sequential tasks and outcome externalities revisited

We revisit job design with sequential tasks and outcome externalities from a different perspective, extending Schmitz (2013a). When two sequential tasks need to be performed by wealth-constrained agents, the principal can hire only one agent or two different agents. When there exists an outcome externality in terms of the fixed cost and such an externality is not too large, if the two tasks are conflicting (resp. synergistic), then it is optimal for the principal to hire two different agents (resp. only one agent). When there exists an outcome externality regarding the marginal cost, the opposite result holds.     

A contest with the taxman - the impact of tax rates on tax evasion and wastefully invested resources

We develop a moral hazard model with auditing where both the principal and the agent can influence the probability that the true state of nature is verified. This setting is widely applicable for situations where fraudulent reporting with costly state verification takes place. However, we use the framework to investigate tax evasion. We model tax evasion as a concealment-detection contest between the taxpayer and the authority. We show that higher tax rates cause more evasion and increase the resources wasted in the contest. Additionally, we find conditions under which a government should enforce incentive compatible auditing in order to reduce wasted resources.     

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.     

总体努力水平约束下的Tullock竞赛与最优约束机制设计:总体约束,个体约束还是无努力水平约束?

竞赛中参赛者往往面临各种参与约束,本文研究总体努力水平约束下的Tullock竞赛以及竞赛组织者如何最大化总努力水平。我们刻画了总体约束下的Tullock竞赛均衡,并将均衡努力水平与无约束及对称个体约束下的结果相比较。为了诱致出最高的总努力水平,最优总体约束水平和对称个体约束水平都应设定在比对奖品估值较高参与者的估值稍微低一点的水平,而最优非对称个体约束则要把估值较高参与者面临的约束设在比其估值稍微低一点的水平。     

Monitoring multiple agents: The role of hierarchies

This paper presents an analysis of principal/multiple-agent problems when there are monitoring possibilities. A principal/owner of a firm has to decide whether or not to employ one or more supervisors who can observe noisy signals on the workers/agents' actions. For both workers and supervisors there is a moral hazard problem. The optimal allocation of monitoring activities and determinants of the decision to hire a supervisor are discussed, and it is explored how multi-level monitoring-hierarchies are structured in such a firm.