Endogenous Group Formation in Contests: Unobservable Sharing Rules

We study contests in which players compete by expending irreversible effort to win a prize, the prize is awarded to one of the players, the winner shares the prize with other players in his group, if any, and each group's sharing rule is unobservable to the other groups and the singletons, if any, when the players expend their effort. The number of groups, their sizes, and the number of singletons are exogenous in the first model, whereas they are endogenous in the second model. We show that group formation occurs if the number of players is four or smaller, but does not occur otherwise. We examine the effect of endogenous group formation on total effort level and the profitability of endogenous group formation. In each of the two models, comparing the outcomes of the case of unobservable sharing rules with those of the case of observable sharing rules, we show that the two cases yield quite different outcomes.     

Contests with multiple alternative prizes: Public-good/bad prizes and externalities

We study contests in which there are multiple alternative public-good/bad prizes, and the players compete, by expending irreversible effort, over which prize to have awarded to them. Each prize may be a public good for some players and a public bad for the others, and the players expend their effort simultaneously and independently. We first prove the existence of a pure-strategy Nash equilibrium of the game, then establish when the total effort level expended for each prize is unique across the Nash equilibria, and then summarize and highlight other interesting and important properties of the equilibria. Finally, we discuss the effects of heterogeneity of valuations on the players’ equilibrium effort levels and a possible extension of the model.     

Winner’s effort maximization in large contests

We investigate the temporal structure that maximizes the winner’s effort in large homogeneous contests. We find that the winner’s effort ranges from a lower bound of 0 to an upper bound of one third of the value of the prize, depending on the temporal structure; the upper (lower) bound is approached with an infinite number of players playing sequentially (simultaneously) in the first periods (period). Nevertheless, when the number of players is large but finite, we show that winner’s effort is maximized when all players play sequentially except in the very last period and that, within the family of such optimal temporal structures, more players play simultaneously in the very last period than sequentially in all other periods. Furthermore, out of all players, the percentage of those playing simultaneously in the very last period goes to 100% as the number of players grows larger and larger.     

Optimal risk taking in an uneven tournament game with risk averse players

We analyze the optimal choice of risk in a two-stage tournament game between two players that have different concave utility functions. At the first stage, both players simultaneously choose risk. At the second stage, both observe overall risk and simultaneously decide on effort or investment. The results show that those two effects which mainly determine risk taking – an effort effect and a likelihood effect – are strictly interrelated. This finding sharply contrasts with existing results on risk taking in tournament games with symmetric equilibrium efforts where such linkage can never arise. Conditions are derived under which this linkage leads to a reversed likelihood effect so that the favorite (underdog) can increase his winning probability by increasing (decreasing) risk which is impossible in a completely symmetric setting.     

Two-stage contests with effort-dependent values of winning

We study two-stage all-pay contests in which synergy exists between the stages. The value of winning for each contestant is fixed in the first stage while it is effort-dependent in the second one. We assume that a player’s effort in the first stage either increases (positive synergy) or decreases (negative synergy) his value of winning in the second stage. The subgame perfect equilibrium of this contest is analyzed with either positive or negative synergy. We show, in particular, that whether the contestants are symmetric or asymmetric their expected payoffs may be higher under negative synergy than under positive synergy. Consequently, they prefer smaller values of winning (negative synergy) over higher ones (positive synergy).     

Sabotage in dynamic tournaments

This paper studies sabotage in a dynamic tournament. Three players compete in two rounds. In the final round, a player who is leading in the race, but not yet beyond the reach of his competitors, is sabotaged more heavily. As a consequence, if all players start off at the same position, they do not work productively or sabotage at all in the first round. Thus sabotage is not only directly destructive, but also depresses incentives to work productively. If players are heterogeneous ex ante, sabotage activities in the first round may be concentrated against an underdog, contrary to findings from static tournaments.     

Contests with investment

Perfectly discriminating contests (all pay auctions) are widely used as a model of situations where individuals devote resources to win some prize. In reality such contests are often preceded by investments of the contestants into their ability to fight in the contest. This paper studies a two stage game where in the first stage, players can invest to lower their bid cost in a perfectly discriminating contest, which is played in the second stage. Different assumptions on the timing of investment are studied. With simultaneous investments, equilibria in which players play a pure strategy in the investment stage are asymmetric, exhibit incomplete rent dissipation, and expected effort is reduced relative to the game without investment. There also are symmetric mixed strategy equilibria with complete rent dissipation. With sequential investment, the first mover always invests enough to deter the second mover from investing, and enjoys a first mover advantage. I also look at unobservable investments and endogenous timing of investments. Copyright © 2007 John Wiley & Sons, Ltd.     

Contests over joint production on networks

We consider a network of heterogeneous agents where each edge represents a two‐player contest between the respective nodes. In these bilateral contests, agents compete over an endogenous prize jointly produced using their own contest efforts. We provide a necessary and sufficient condition for the existence of Nash equilibrium and characterize the equilibrium total effort for every agent. Our model has insightful results regarding the network type, that is, depending on whether the network is bipartite or nonbipartite. Finally, considering the sum of all expected utilities as an efficiency notion, we investigate the optimal network structure.     

Investment in public infrastructure with spillovers and tax competition between contiguous regions

Two contiguous regions compete to attract a population of heterogeneous firms. They choose infrastructure levels in a first stage and compete in taxes in a second stage. We study the properties of Subgame Perfect Nash equilibria in this stage game depending on the extent to which the benefits of infrastructure spill over from one region to the other. First, we show that the presence of inter-regional spillovers allows jurisdictions to control for the intensity of tax competition and therefore affects the optimal levels of infrastructure selected at equilibrium. Second, by comparing the non-cooperative and cooperative outcomes, we show that the extent to which regions overinvest in infrastructures negatively depends on the intensity of the spillovers.     

Allocation of Prizes in Contests with Participation Constraints

We study all‐pay contests with an exogenous minimal effort constraint where a player can participate in a contest only if his effort (output) is equal to or higher than the minimal effort constraint. Contestants are privately informed about a parameter (ability) that affects their cost of effort. The designer decides about the size and number of prizes. We analyze the optimal prize allocation for the contest designer who wishes to maximize either the total effort or the highest effort. It is shown that if the minimal effort constraint is relatively high, the winner‐take‐all contest in which the contestant with the highest effort wins the entire prize sum does not maximize the expected total effort or the expected highest effort. Rather, a random contest in which the entire prize sum is equally allocated to all the participants yields a higher expected total effort as well as a higher expected highest effort.     

On optimal favoritism in all-pay contests

I analyze the optimal favoritism in a complete-information all-pay contest with two players, whose costs of effort are weakly convex. The contest designer could favor or harm some contestants using one of two instruments: head starts and handicaps. I find that any given player’s effort distribution is ranked in the sense of first-order stochastic dominance according to how (ex post) symmetric the players are in terms of competitiveness. Consequently, as long as the designer values effort from both contestants, “leveling the playing field” is optimal regardless of which instrument is used.     

Contests for catch shares

Review of Economic Design - We study contests in which fishers (or players), in a fishery managed under a catch-share program, compete over catch shares by expending irreversible effort, and the...     

Relative performance as a strategic commitment mechanism

Can managers' personality traits be of use to profit maximizing firm owners? We investigate the case where managers have a variety of attitudes toward relative performance that are indexed by their type. We consider two stage games where profit maximizing owners select managers in the first stage, and these managers, knowing each other's types, compete in a duopoly game in the second stage. The equilibria of various types of competition are derived and comparisons are made to the standard case where managers are profit maximizers. We show that managers' types can be used as a strategic commitment device that can increase firm profits in certain environments. Copyright © 2002 John Wiley & Sons, Ltd.     

Learning and decision costs in one-person games

This paper reports the results of a two-part data analysis of learning in a repeated costly decision experiment. In the first part we test payoff dominance under the hypothesis of expected payoff maximization. We utilize a dynamic probability distribution over decisions for each player, characterizing what each player has learned about the payoff function by the moments from these distributions. In the second part, we test the hypothesis of expected payoff maximization for players in each treatment group. Expected payoff maximization is supported but payoff dominance is not. Copyright © 1999 John Wiley & Sons, Ltd.     

Evolutionary stability of prospect theory preferences

We demonstrate that in simple 2×2 games (cumulative) prospect theory preferences can be (semi-)evolutionarily stable, in particular, a population of players with prospect theory preferences is stable against more rational players, i.e. players with a smaller degree of probability weighting. We also show that in a typical game with infinitely many strategies, the “war of attrition”, probability weighting is (semi-)evolutionarily stable. Finally, we generalize to other notions of stability. Our results may help to explain why probability weighting is generally observed in humans, although it is not optimal in usual decision problems.     

A statistical study of racism in English football

The English Premier League is often quoted as being the ‘best football league in the world’, a mix of British born and international stars showing off their skills almost every week. With around 25 % of Premier League players being from ethnic minority backgrounds, it is arguably one of the world’s most inclusive sports leagues—but is it really as fair as it seems? This analysis looks at the 2011–2012 Premier League season, to investigate whether there existed evidence of forms of racial discrimination against Black, non-White and non-European players during this period. We analyze player data collected from two hundred and ninety players and seventeen teams using logistic regression, to determine how being a Black, non-White or non-European player depends on variables like total number of appearances, number of fouls, number of yellow cards, number of red cards, number of cards awarded by English referees, number of cards awarded by non-English referees, total number of goals scored by the team, and total number of goals scored by the player. Our results show that there does seem to exist some discrepancy between the treatment of non-White and non-European and Black players compared to the sample of White players.     

Optimal information exchange in contests

We study optimal exchange of private information in a two-player all-pay auction contest with independent private binary values. A benevolent information center who is informed about the players’ values facilitates the exchange of information by disclosing a signal publicly. The informativeness of the signal determines the monotonicity of the unique symmetric equilibrium and the players’ expected payoff. We characterize the upper bound of players’ expected payoff and the corresponding optimal signals utilizing such a relation between the informativeness and the payoff. When the players are ex ante sufficiently heterogeneous, the optimal signals work through an information-rent channel by inducing allocative efficient contests. When the players are ex ante sufficiently homogeneous, the optimal signals work through an unlevel-playing-field channel by inducing asymmetric contests. In order to guarantee efficient allocation, a regulator can punish any exchange of information when the players are sufficiently homogeneous and impose no restrictions when they are sufficiently heterogeneous.     

The generalized Stackelberg equilibrium of the all-pay auction with complete information

We characterize the equilibrium of the all-pay auction with general convex cost of effort and sequential effort choices. We consider a set of n players who are arbitrarily partitioned into a group of players who choose their efforts ‘early’ and a group of players who choose ‘late’. Only the player with the lowest cost of effort has a positive payoff in any equilibrium. This payoff depends on his own timing vis-a-vis the timing of others. We also show that the choice of timing can be endogenized, in which case the strongest player typically chooses ‘late’, whereas all other players are indifferent with respect to their choice of timing. In the most prominent equilibrium the player with the lowest cost of effort wins the auction at zero aggregate cost. We thank Dan Kovenock and Luis C. Corchón for discussion and helpful comments. The usual caveat applies. Wolfgang Leininger likes to express his gratitude to Wissenschaftszentrum Berlin (WZB) for its generous hospitality and financial support.     

Network games with incomplete information

We consider a network game with strategic complementarities where the individual reward or the strength of interactions is only partially known by the agents. Players receive different correlated signals and they make inferences about other players’ information. We demonstrate that there exists a unique Bayesian-Nash equilibrium. We characterize the equilibrium by disentangling the information effects from the network effects and show that the equilibrium effort of each agent is a weighted combinations of different Katz–Bonacich centralities.     

Picking a loser: Strategic surprise in a design and development game

We take a setting in which upstream players produce design ideas and downstream players select among these ideas to develop finished products. Design diversity is valuable at the upstream stage and coordination is valuable at the downstream stage. However, this outcome is not always realized. We show that an intermediary between upstream and downstream can improve on equilibrium outcomes by acting as a coordination and commitment device whose optimal policy must sometimes reward inferior ideas. We apply the model to technology standards, trend‐driven industries, political primaries, and the management of process innovation. We discuss incentives to vertically integrate.