Heterogeneous Contestants Research Articles

Designing contests between heterogeneous contestants: An experimental study of tie-breaks and bid-caps in all-pay auctions

A well-known theoretical result in the contest literature is that greater heterogeneity decreases investments of contestants because of the “discouragement effect.” Levelling the playing field by favouring weaker contestants through strict bid-caps and favourable tie-breaking rules can reduce discouragement and increase the designer's revenue. We test these predictions in a laboratory experiment. Our data confirm that placing bid-caps and using favourable tie-breaking rules significantly diminishes discouragement of weaker contestants. However, its impact on revenues is muted by the fact that the encouragement of weaker contestants is offset by stronger contestants competing less aggressively, even when not predicted by theory. We discuss deviations from the Nash predictions in light of different behavioural approaches.

Effort maximization in contests under balance constraints

We study all-pay contests with complete information and two heterogeneous contestants who compete for a single prize. We combine two common designer's goals which are combined especially in sports contests; effort maximization and competitive balance. We focus on two cases when the designer's utility is either continuous or discontinuous in the contestants' efforts. We show that a maximum effort constraint is always efficient for increasing the designer's utility, namely, effort maximization under competitive balance.

Sabotage in dynamic tournaments with heterogeneous contestants: Evidence from European football

This paper analyses how sabotage, measured as the number of fouls and yellow and red cards, is affected by the relative performance of and asymmetries between teams, determined based on score margin and the odds gap between one team and its opponent. By applying detailed controls for within-match dynamics and differences in team quality, we first observe that badly losing favourites are more willing than losing underdogs to increase their unsporting behaviour in heterogeneous contests. There are more yellow cards and fouls as the game progresses. We further find that most sabotage (yellow cards) can be observed when the (absolute) goal difference is equal to 1. Teams decrease their level of sabotage (fouls) by increasing the number of goals of lead or trail. In addition, weaker teams have an incentive to engage in more unfair and destructive strategies, such as committing more fouls, including those penalized with yellow cards. However, the incidence of red cards is not influenced by the score margin or strength gap.

Expectations-based loss aversion in contests

This paper studies a multi-player lottery contest in which heterogeneous contestants exhibit reference-dependent loss aversion à la Kőszegi and Rabin (2006, 2007). We verify the existence and uniqueness of pure-strategy choice-acclimating personal Nash equilibrium (CPNE) under moderate loss aversion and fully characterize the equilibrium. The equilibrium sharply contrasts with that in the two-player risk-neutral symmetric case. Loss aversion can lead contestants' individual efforts to change nonmonotonically, while the total effort strictly decreases. Further, it always leads to a more elitist distributional outcome, in the sense that a smaller set of contestants remain active in the competition and stronger contestants' equilibrium winning probabilities increase. We demonstrate that loss aversion generates a fundamentally different decision problem than risk aversion and develop a rationale that explains the contrasting predictions from the two frameworks. Finally, our results are robust under the alternative equilibrium concept of preferred personal Nash equilibrium (PPNE).

Entry regulations and optimal prize allocation in parallel contests

In parallel contests, the contest organizer controls the entry of heterogeneous contestants by regulating access to the contests and determining the prize allocation across contests. The organizer can prevent a contestant from entering more than one contest. I show that the organizer allows entry to multiple contests and uniquely sets identical prizes across contests to maximize aggregate effort in all contests. Independent of the entry regulation, I find no sorting effects. Thus, a contest with a relatively high prize does not necessarily attract contestants with higher abilities. Furthermore, I discover interesting spillover effects of prizes between contests in the case of restricted entry regulations. For instance, the individual (aggregate) effort increases (decreases) in a contest if the prize in another contest increases. The endogeneity of contestants’ participation drives many of these results.

On the Optimality of Small Research Tournaments

I study two open questions concerning optimal tournament size in Fullerton and McAfee's (1999) model of research tournaments with heterogeneous contestants. For the case where firms' costs are commonly known prior to the tournament and the procurer can charge non-discriminatory entry fees, I prove a sharp, worst-case bound for the ratio between the cost of procuring a given total effort from the optimal number of contestants and the corresponding cost for a tournament of size two. The analysis confirms the attractiveness of the smallest possible tournament, with some notable exceptions. I then describe the problem of inducing a given expected total effort at the lowest expected cost, by choosing the optimal combination of prize and tournament size, when firms are privately informed about their i.i.d. marginal costs prior to the tournament and a set of lowest-cost contestants is selected by an entry auction. I show that this problem can be solved in closed form when marginal costs are uniformly distributed on [0,c], and that the result for this case strongly favors the smallest possible tournament.

Bid caps in large contests

We study the effect of rigid and flexible bid caps on contestants' aggregate costs and aggregate bids in all-pay contests with a large number of heterogeneous contestants and prizes.We show that rigid caps always decrease aggregate costs, whereas flexible caps have essentially no effect on aggregate costs. Rigid caps decrease aggregate bids when costs are linear or concave, but increase aggregate bids under some conditions when costs are convex. Flexible caps always decrease aggregate bids.

Optimal favoritism in all-pay auctions and lottery contests

We analyze the revenue-enhancing potential of favoring specific contestants in complete-information all-pay auctions and lottery contests with several heterogeneous contestants. Two instruments of favoritism are considered: head starts that are added to the bids of specific contestants and multiplicative biases that give idiosyncratic weights to the bids. In the all-pay auction, head starts are more effective than biases while optimally combining both instruments even yields first-best revenue. In the lottery contest, head starts are less effective than biases and combining both instruments cannot further increase revenue. As all-pay auctions revenue-dominate lottery contests under optimal biases, we thus obtain an unambiguous revenue-ranking of all six combinations of contest formats and instruments.

Sorted or pooled? Optimal tournament design for heterogeneous contestants

The optimal tournament for the heterogeneous contestants is described and developed in approach of game theory, which includes the sorted tournament between the strong contestants with high ability, the sorted tournament between the weak contestants with low ability, and the pooled tournament between the strong and the weak contestants. By comparing the equilibrium results of the incentive structure, the efforts level, and the expected profits in each kind of tournament, it is found that the strong contestants can get more utilities in the pooled tournament, and hence prefer to take part in the pooled tournament to compete with the weak competitors, while the weak contestants only can get the reservation utility in both the sorted and pooled tournament, and hence do not care about the abilities of competitors. The principal will choose and implement the sorted tournament because he can acquire more profits than that in the pooled tournament.

Heterogeneous Contestants and the Intensity of Tournaments

We empirically investigate whether tournaments between heterogeneous contestants are less intense. To test our hypotheses, we use professional sports data from the TOYOTA Handball-Bundesliga, the major handball league in Germany. Based on sports betting odds, we estimate the differences in winning probabilities of the competing teams and find evidence for a negative impact of the matchup’s heterogeneity on the intensity of the game. The decrease is significant not only at the beginning but also toward the end of the game. Further analysis shows that the overall intensity decrease is almost entirely driven by the reaction of the ex ante favorite team.

Designing Contests between Heterogeneous Contestants: An Experimental Study of Tie-Breaks and Bid-Caps in All-Pay Auctions

A well-known theoretical result in the contest literature is that greater heterogeneity decreases performance of contestants because of the “discouragement effect.” Leveling the playing field by favoring weaker contestants through bid-caps and favorable tie-breaking rules can reduce the discouragement effect and increase the designer’s revenue. We test these predictions in an experiment. Our data show that indeed, strengthening weaker contestants through tie-breaks and bid-caps significantly diminishes the discouragement effect. Bid-caps can also improve revenue. Most deviations from Nash equilibrium can be explained by the level-k model of reasoning.

Optimal Favoritism in All-Pay Auctions and Lottery Contests

We analyze the revenue-enhancing potential of favoring specific contestants in complete information all-pay auctions and lottery contests with several heterogeneous contestants. Two instruments of favoritism are considered: head starts that are added to the bids of specific contestants and multiplicative biases that give idiosyncratic weights to the bids. In the all-pay auction, head starts are more effective than biases while optimally combining both instruments even yields first-best revenue. In the lottery contest, head starts are less effective than biases and combining both instruments cannot further increase revenue. As all-pay auctions revenue-dominate lottery contests under optimal biases, we thus obtain an unambiguous revenue-ranking of all six combinations of contest formats and instruments.

Designing Contests between Heterogeneous Contestants: An Experimental Study of Tie-Breaks and Bid-Caps in All-Pay Auctions

A well-known theoretical result in the contest literature is that greater heterogeneity decreases performance of contestants because of the Leveling the playing field by favoring weaker contestants through bid-caps and favorable tie-breaking rules can reduce the discouragement effect and increase the designer's revenue. We test these predictions in an experiment. Our data show that indeed, strengthening weaker contestants through tie-breaks and bid-caps significantly diminishes the discouragement effect. Bid-caps can also improve revenue. Most deviations from Nash equilibrium can be explained by the level-k model of reasoning.

The Impact of Intermediate Information on Sabotage in Tournaments with Heterogeneous Contestants

Applying detailed within‐tournament information about intermediate scores and contestants' ability in rank‐order professional soccer tournaments, this study empirically analyzes the impact of interim results on the sabotage activities of heterogeneous contestants. Intermediate information that suggests that a contest is decided early decreases total sabotage. Splitting contestants into favorites and underdogs reveals that both contestants sabotage the most if intermediate information about the score compensates for or decreases ex ante heterogeneity between the teams. However, engaging in sabotage does not pay off for the contestants. Copyright © 2015 John Wiley & Sons, Ltd.

Participation, Contribution and Distribution in Parallel Contests

In parallel contests, the contest organizer is able to steer the entry of heterogeneous contestants by determining the prize distribution between the contests and regulating the access to the contests. Regulations can prevent contestants from entering multiple contests. We show in this article that the organizer faces a trade-off between maximizing the individual efforts and maximizing the aggregate efforts. Moreover, the organizer cannot attract high types by offering a relatively large prize in a contest. We also discover spillover effects between contests in the presence of restrictive entry regulations. The individual (aggregate) effort increases (decreases) in a contest if the prize in a neighboring contest is increased.

Multi-stage sequential all-pay auctions

We study multi-stage sequential all-pay contests (auctions) where heterogeneous contestants are privately informed about a parameter (ability) that affects their cost of effort. We characterize the perfect Bayesian equilibrium of these contests and analyze the effect of the number of contestants and their types on the contestants׳ expected highest effort.

Sabotage in Tournaments with Heterogeneous Contestants: Empirical Evidence from the Soccer Pitch

AbstractIn this paper, we address the problem of sabotage in tournaments with heterogeneous contestants. In a first step, we develop a formal model, which yields the prediction that favorites exert higher productive effort, while underdogs are more tempted to engage in destructive actions (sabotage). This is because favorites have a higher return on productive effort and both types of effort are substitutes. In a second step, we use data from German professional soccer to test this prediction. In line with the model, we find that favorite teams win more tackles in a fair way, while underdog teams commit more fouls.

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.

Attracting the Best and the Brightest: Asymptotics and Optimal Prize Structure of All-Pay Contests with Heterogeneous Participants

Tournaments are frequently adopted to procure goods and services for which spot markets do not exist, contracts are incomplete and outcomes cannot be verified in a court. Examples of such procurement contests include R&D tournaments held by the government agencies and large corporations. It is widely acknowledged that open participation is undesirable in such contests and tools such as entry fees and contestant selection auctions are suggested to restrict participation. With rapid evolution of the Internet and the Web, a new generation of procurement tournaments has emerged. The Web now allows almost any firm or individual to organize its own minor tournament, sometimes with just a few mouse clicks. A number of companies use this opportunity to run contests in which contestants develop and submit ideas or products, and prizes are awarded to one or more winners chosen by the company conducting the contest. All of these tournaments are characterized by the absence of entry fees, open participation, sunk efforts and a significant number of contestants. This paper presents an economic model of an all-pay contest with heterogeneous contestants. There is a contest sponsor that values K contest submissions. The contest offers L prizes and there are N risk-averse, wealth-constrained contestants who differ in their skills (their ability to produce quality solutions), and who turn in their submissions simultaneously. Our first result shows that every such game of incomplete information satisfying the Single Crossing Condition has a unique symmetric Bayes-Nash equilibrium in pure strategies, and that equilibrium effort levels are monotonic in ability. Thus, the equilibrium of the contest is efficiently ranking in that it ranks the winners in the decreasing order of their abilities. Next, we derive the asymptotic behavior of the bid function and the submission qualities. We show that, asymptotically, only the support of the type distribution and the cost function at the support determine the contest outcome, therefore, in the limit, the game has a particularly simple structure. Next, we show how that the speed of convergence to the asymptotics is of order 1/N and the asymptotic outcome can be used to construct an upper bound on the outcome in any finite sample game. Finally, we show how our result can be applied to derive the asymptotically optimal allocation of prizes for the profit-maximizing tournament sponsor and show that the optimal solution of the finite-sample allocation problem converges to the asymptotic one as the number of participants grows. In particular, it is shown that when the contestants are risk-neutral, the sponsor's optimal contest design places most of the budget on the top prize and sets the remaining prizes at their lowest possible levels (the minimum necessary to facilitate the transfer of the intellectual property). When the contestants are sufficiently risk-averse, the firm may optimally offer more prizes than there are the desired submissions, thus awarding prizes even to submissions it does not eventually want. As the aggregate surplus of contestants converges to zero when the number of participants grows, the profit maximizing prize structure is asymptotically socially optimal.

Potential, Prizes and Performance: Testing Tournament Theory with Professional Tennis Data

This paper tests two hypotheses from the theory of elimination tournaments: (i) that uneven tournaments, where the contestants are ex ante heterogeneous, entail lower effort exertion; this is a prediction from agency theory that has not been tested empirically before; and (ii) whether incentives set through prizes matter for effort exertion; this assumption underlies any agency theory about elimination tournaments, and has been empirically tested in other contexts. The evidence obtained with data from professional tennis tournaments supports both the assumption that incentives matter, as well as the theoretical implications concerning uneven tournaments among heterogeneous contestants.