Simultaneous versus sequential all-pay auctions: an experimental study

While both simultaneous and sequential contests are mechanisms used in practice such as crowdsourcing, job interviews and sports contests, few studies have directly compared their performance. By modeling contests as incomplete information all-pay auctions with linear costs, we analytically and experimentally show that the expected maximum effort is higher in simultaneous contests, in which contestants choose their effort levels independently and simultaneously, than in sequential contests, in which late entrants make their effort choices after observing all prior participants’ choices. Our experimental results also show that efficiency is higher in simultaneous contests than in sequential ones. Sequential contests’ efficiency drops significantly as the number of contestants increases. We also discover that when participants’ ability follows a power distribution, high ability players facing multiple opponents in simultaneous contests tend to under-exert effort, compared to theoretical predictions. We explain this observation using a simple model of overconfidence.     

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.     

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).     

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.     

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.     

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.     

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.     

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).     

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.     

Disclosure policy in Tullock contests with asymmetric stochastic entry

We examine how disclosure policy can be optimally designed to incentivize contestants when their participation is exogenously stochastic. In a generalized Tullock contest setting with two players who are asymmetric in both their values and entry probabilities, we fully characterize the necessary and sufficient conditions under which no disclosure dominates full disclosure. We find that the comparison depends solely on a balance effect exercised by entry probabilities on the expected total effort. The optimal disclosure policy must better balance the competition. These conditions continue to hold when the precision r of Tullock contests is endogenously chosen by the designer.     

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.     

Entry in contests with incomplete information: Theory and experiments

This paper studies entry decisions in contests with private values. Potential contestants observe their value and the common opportunity cost of entry, and make entry decisions simultaneously. Theory predicts that whether or not contestants are informed of the number of entrants prior to choosing their expenditures has no effect on entry or aggregate expenditures. We test these assertions in our experiments. We find substantial over-entry in both information structures. However, entry is higher when contestants are informed. Since expenditures do not, on average, differ across information structures, aggregate expenditure is also higher when contestants are informed. Contestants earn on average less than the opportunity cost of entry.     

Unfair contests

Real-world contests are often “unfair” in the sense that outperforming all rivals may not be enough to be the winner, because some contestants are favored by the allocation rule, while others are handicapped. This paper analyzes an unfair, two-player discriminatory contest (all-pay auction) with private values. We characterize equilibrium strategies, provide closed form solutions, and illustrate additional strategic issues arising in such unfair contests.     

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.     

Using a Simple Contest to Illustrate Mechanism Design

This article describes a simple classroom activity that illustrates how economic theory can be used for mechanism design. The rules for a set of contests are presented; the results typically obtained from these contests illustrate how the prize structure can be manipulated in order to produce a particular outcome. Specifically, this activity is designed to show how changing the prize structure can impact both the contestants’ average efforts and the effort level of the hardest-working contestant. The activity can be run in a 50-minute class, has instructions that fit on a single piece of paper, and, although it can be run in large classes, requires only six students.     

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.     

Fair play in contests

The aim of this paper is to incorporate fair play norms into the analysis of contests where players have the ability to cheat in order to improve their chances of winning. We propose a utility function integrating fair play norms and apply it to a stylized model of rank-order tournament with cheating. We study how the set of equilibria is affected by the presence of fair play values. These values are shown to have an ambiguous impact on the incentives to cheat so that it may happen that there is more cheating with fair play values than without. We also study the impact of the number of contestants, with results suggesting that fair play norms may be effective in closed tournaments with few competitors (such as, for example, promotions in firms), but not in large-scale tournaments such as worldwide athletic or academic contests.     

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.     

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.     

Effort and performance in group contests

We study contest performance by focusing on the per capita payoffs of the contestants in an important class of asymmetric two-group contests. The group with the higher valuation has a greater chance of winning the contest whenever the rent-seeking technology is increasing and concave. We note that equilibrium efforts and payoffs per capita are uniquely defined and can be subjected to a comparative statics analysis. We show that payoff per capita for a group increases with its own valuation but decreases with the valuation of the opposing group. Per capita payoff for a group increases with its population.