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.     

Competitive prizes: when less scrutiny induces more effort

We consider a principal who is keen to induce his agents to work at their maximal effort levels. To this end, he samples n days at random out of the T days on which they work, and awards a prize of B dollars to the most productive agent. The principal’s policy (B, n) induces a strategic game Γ(B, n) between the agents. We show that to implement maximal effort levels weakly (or, strongly) as a strategic equilibrium (or, as dominant strategies) in Γ(B, n), at the least cost B to himself, the principal must choose a small sample size n. Thus less scrutiny by the principal induces more effort from the agents.The need for reduced scrutiny becomes more pronounced when agents have information of the history of past plays in the game. There is an inverse relation between information and optimal sample size. As agents acquire more information (about each other), the principal, so to speak, must “undo” this by reducing his information (about them) and choosing the sample size n even smaller.     

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.     

Promotion Tournaments and Individual Performance Pay

We analyze the optimal combination of promotion tournaments and linear individual performance pay in an employment relationship. An agent's effort is nonobservable and he has private information about his suitability for promotion. Thus, the two incentive schemes need to be combined to serve both incentive and selection purposes. If harder working agents respond less to intensified effort incentives, we find that the principal puts less emphasis on individual performance pay when selection becomes more important. Thus, we provide a possible explanation as to why, in practice, individual performance pay is less prevalent than promotion‐based incentives.     

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.     

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.     

Ex-post favoring ranks: a fairness notion for the random assignment problem

We introduce two notions of ex-post fairness, namely ex-post favoring ranks (EFR) and robust ex-post favoring ranks, which consider whether objects are received by those agents who have the highest rank for them. We examine their compatibility with standard properties of random assignments and state some impossibility theorems. We also propose and formalize a revised version of the Boston mechanism and prove that it provides an EFR random assignment.

     

Dynamic incentive effects of assignment mechanisms: Experimental evidence

Optimal assignment and matching mechanisms have been the focus of exhaustive analysis. We focus on their dynamic effects, which have received less attention, especially in the empirical literature: Anticipating that assignment is based on prior performance may affect prior performance. We test this hypothesis in a lab experiment. Participants first perform a task individually without monetary incentives; in a second stage, they are paired with another participant according to a pre‐announced assignment policy. The assignment is based on the first‐stage performance, and compensation is determined by average performance. Our results are largely consistent with a theory: Pairing the worst‐performing individuals with the best yields 20% lower first‐stage effort than random matching (RAM) and does not induce truthful revelation of types, which undoes any policy that aims to reallocate types based on performance. Perhaps surprisingly, however, pairing the best with the best yields only 5% higher first‐stage effort than RAM and the difference is not statistically significant.     

Pay,productivity and aging in Major League Baseball

Using panels of player pay and performance from Major League Baseball (MLB), we examine trends in player productivity and salaries as players age. Pooling players of all ability levels leads to a systematic bias in regression coefficients. After addressing this problem by dividing players into talent quintiles, we find that the best players peak about 2 years later than marginal players, and development and depreciation of performance appear to be more pronounced for players with the highest ability levels. Within-career variation, however, is less pronounced than between-player variation, and the performance level of players within a given quintile will typically remain lower than the talent level for rookies in the next higher quintile. We also find preliminary evidence that free agents are paid proportionately to their production at all ability levels, whereas young players’ salaries are suppressed by similar amounts.     

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.     

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.     

Is the Coach Paid too Much?: Coaching Salaries and the NCAA Cartel

Recently, a great deal of controversy has been generated from the salaries earned by head coaches in the NCAA. Although many figures in the world of sports earn high salaries, one important difference in the case of the NCAA is that the players do not get paid. We develop a model that shows that a cartel agreement to not pay the players raises the coach's salary if some players choose where to play based on the identity of the coach. The agreement not to pay the players improves competitive balance in the baseline model, but this result does not generalize.     

Principal‐agent and Peer Relationships in Tournaments

Effort provision in tournaments may depend on both social preferences towards the competitor and social preferences towards the principal who designed the tournament. In a laboratory experiment, we analyze agents' behavior in different tournament settings that vary the distribution of the prize between agents. The principal chooses between a low and a high guaranteed payment. Standard economic theory predicts the same effort provision in all situations. Instead, our results show that both the generosity of the principal and the strict separation between winner and loser increase the effort level in tournaments. Moreover, agents focus their preferences either on the principal or on the agent. Copyright © 2014 John Wiley & Sons, Ltd.     

Perseverance and suspense in tug-of-war

We study a tug-of-war game between two players using the lottery contest success function (CSF) and a quadratic cost (of effort) function. We construct a pure strategy symmetric Markov perfect equilibrium of this game, show that it is unique, and provide closed-form solutions for equilibrium strategies and values. In stark contrast to a model of tug-of-war with an all-pay auction CSF, players exert positive efforts until the very last battle in this equilibrium. We deliver a set of empirically appealing results on effort dynamics.     

Behavioral Decision Making in the (Q,R) Purchasing Model: An Experimental Study

This paper presents and analyzes the results of a decision‐making experiment in inventory management under uncertainty. The experiment included 81 participants who played the role of a small car importer facing random demand as in the (Q,R) model. The results show strong evidence of learning and convergence, and the average reorder point (R) closely approaches the optimal level for maximizing profits. However, the participants' decisions are still biased by realizations of extreme values of demand and loss of potential sales. We argue that participants are affected by recency, loss aversion, and, possibly, their own risk aversion. Copyright © 2013 John Wiley & Sons, Ltd.     

Survival and the art of profit maximization

We study the relationship between rationality and economic survival in a simple dynamic model, where agents from different populations interact repeatedly through random matching. An explicit criterion (“bankruptcy”) determines whether agents survive each interaction; all agents are presumed to be aware of this criterion. Survival in each interaction depends on two factors: the strategies agents adopt when they interact, and the wealth levels they bring to the game. The model is completely symmetric with the only difference between the agents of different populations being their objectives. We consider the case where there are two populations of agents in which all agents from one group have as their objective, maximizing the expected profits from each interaction, while all agents from the other attempt simply to maximize the probability of surviving (i.e., not going bankrupt in) the interaction. We show that under the equilibrium dynamics, the proportions of each group in the total population must be bounded away from zero, but the balance is in favor of the survival-probability maximizers in that their numbers as a fraction of total population must always exceed 1/2, and can be arbitrarily close to unity. On the other hand, the fraction of total wealth controlled by the expected profit maximizers must also be at least 1/2, and can asymptotically tend to unity.     

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.     

Equity‐based incentive contracts and behavior: Experimental evidence

Equity ‐ based incentive contracts provide managers with dual incentives, motivating both effort and fraud. We report the results from an experiment in which manager subjects make effort and fraud decisions that affect a firm's value. The main treatment variable is the incentive contract, which can be of either the simple equity or stock option type. We find that both effort and fraud are increasing in a manager's share of equity and decreasing in the strike price of an option. Interestingly, the stock option contract induces relatively more fraud than the simple equity contract, even though the two induce the same effort.     

Decision making under risk in Deal or No Deal

We analyse the choices of 399 contestants in the Australian version of the television game show Deal or No Deal. We calculate risk aversion bounds for each contestant, revealing considerable heterogeneity. We then estimate a structural stochastic choice model that captures the dynamic decision problem faced by contestants. To address individual heterogeneity, we nest the dynamic problem within the settings of both a random effects and a random coefficients probit model. Our structural model produces plausible estimates of risk aversion, confirms the role of individual heterogeneity and suggests that a model of stochastic choice is indeed appropriate. We find mixed evidence of greater risk aversion by females. We also examine generalizations to expected utility theory, finding that the rank‐dependent utility model adds non‐negligible explanatory power and indicates optimism in probability weighting. Finally, we test, but are unable to confirm, the existence of an endowment effect for lotteries. Copyright © 2009 John Wiley & Sons, Ltd.     

He who must not be named

Landsberger et al. have studied a sealed bid first price auction with two players in which the ranking of the valuations is known. They argue that such a situation can arise in a sequential auction where only the name of the winner is revealed. In this paper we consider sequential auctions where two identical goods are sold sequentially to N players who are interested in both objects. In sealed bid auctions, no information is a priori revealed by the mechanism, but the seller can in principle reveal whatever he wants. We restrict our attention to the case where only the name of the winner is revealed to be in the context of Landsberger et al. for the second auction. The aim of the paper is to compare such a sequential auction with a simultaneous auction where both goods are sold as a bundle or equivalently with a sequential auction where no information is revealed. We first show that there exists an equilibrium of the sequential game in pure and monotone strategies. Then, the comparison of the seller's expected revenue in the two cases allows us to conclude that contrary to Landsberger et al.'s predictions, the seller can not use the information to increase his revenue. This result is obtained using simulations for a large class of distribution functions. The seller must not reveal the name of the winner between the two auctions and instead sell both goods using a simultaneous auction.Received: 31 July 2001, Accepted: 5 February 2003, JEL Classification: B44I wish to thank Laurent Linnemer, Thomas Ricke, Michael Visser and Shmuel Zamir for helpful comments and suggestions. I am very grateful to the referees and the associated editors in charge of my paper.