Optimal seedings in elimination tournaments

We study an elimination tournament with heterogenous contestants whose ability is common-knowledge. Each pair-wise match is modeled as an all-pay auction. Equilibrium efforts are in mixed strategies, yielding complex dynamics: endogenous win probabilities in each match depend on other matches’ outcome through the identity of the expected opponent in the next round. The designer seeds competitors according to their ranks. For tournaments with four players we find optimal seedings for three different criteria: (1) maximization of total tournament effort; (2) maximization of the probability of a final among the two top ranked teams; (3) maximization of the win probability for the top player. We also find the seedings ensuring that higher ranked players have a higher winning probability. We compare our predictions with data from NCAA basketball tournaments.     

Are sequential round‐robin tournaments discriminatory?

I examine sequential round‐robin tournaments with three and four symmetric players. Each player is matched once with each other player and each match is organized as a Tullock contest. A single prize is allocated to the player with the most matches won. I show that, depending on the position of their matches in the sequence of the tournament, the players' ex ante winning probabilities and expected payoffs differ. Compared to tournaments with matches organized as all‐pay auctions, however, discrimination is weaker and partially reversed. Moreover, aggregate effort in tournaments with suitable Tullock contests is larger than in tournaments with all‐pay auctions. The fairness of round‐robin tournaments may be improved by an endogenous sequence of matches or the requirement that players fix their effort ex ante.     

“Success breeds success” or “Pride goes before a fall”?: Teams and individuals in multi-contest tournaments

We study the impact of progress feedback on players' performance in multi-contest team tournaments, in which team members' efforts are not directly substitutable. In particular, we employ a real-effort laboratory experiment to understand, in a best-of-three tournament, how players' strategic mindsets change when they compete on a team compared to when they compete individually. Our data corroborate the theoretical predictions for teams: Neither a lead nor a lag in the first component contest affects a team's performance in the subsequent contests. In individual tournaments, however, contrary to the theoretical prediction, we observe that leaders perform worse—but laggards perform better—after learning the outcome of the first contest. Our findings offer the first empirical evidence from a controlled laboratory of the impact of progress feedback between team and individual tournaments, and contribute new insights on team incentives.     

Participation incentives in rank order tournaments with endogenous entry

Rank order tournaments, in which the payment to an agent is based upon relative observed performance, are a commonly used compensation scheme. In practice, agents often compete in some (but not all) events in a set of tournaments. The present study considers two mutually exclusive tournaments, in which agents themselves decide which event to enter. An agent bases this decision upon the combination of three distinct effects: a prize effect, a winning probability effect, and an effort cost effect. The precise impact of each of these effects is analyzed. Of particular interest is the possibility that a field of higher quality may be attracted to the event with smaller prizes.      

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.     

Tournaments with Prize‐setting Agents*

In some tournaments, it is the contestants themselves who determine reward allocation. Union members bargain over wage distribution, and some firms allow self‐managed teams to freely determine internal resource allocation, incentive structure, and division of labor. We analyze, and test experimentally, a tournament where heterogeneous agents determine the spread between winner prize and loser prize. We investigate the relationship between prize spread, uncertainty, heterogeneity, and effort. We find that a large prize spread is associated with a low degree of uncertainty and a high degree of heterogeneity, and that heterogeneity triggers effort. By and large, our real‐effort experiment supports the theoretical predictions.     

A note on modelling tournaments

Existing studies on tournaments utilize different specifications of the production function. Comparison of these results is difficult. This paper shows that the strategic behavior of agents in a tournament can be analyzed more generally using a specification of the production function along the lines of the standard agency model. I also show that in mixed contests, the more able contestant would in equilibrium have a higher probability of winning the contest despite attempts to use effort to compensate for ability by the less able contestant (Proposition 3).The author would like to thank Joseph Stiglitz for suggesting the idea for this paper, Jerry Green for reading an earlier version of the paper, and three anonymous referees for helpful comments and suggestions. The usual disclaimer applies.     

Tournaments of Weakly Heterogeneous Players

We consider three prominent tournament formats—contests, binary elimination tournaments, and round-robin tournaments—in the case when players are heterogeneous in their abilities but the heterogeneity is, in a well-defined sense, weak. Using linear approximation, we characterize equilibrium strategies and payoffs in the three tournament games and compare them to the benchmark symmetric case of identical players. We describe small deviations from the symmetric equilibria by elasticities of a player's equilibrium effort with respect to her own ability and the abilities of her rivals. Our results only require general symmetry and smoothness assumptions but not specific functional forms for the probabilities of winning in tournaments. We show that, in equilibrium: (i) a player's effort and payoff depend on her rivals' abilities in a model-independent way, either through the average ability of the field (for static games), or through the properly discounted average ability of the field (for dynamic elimination tournaments); (ii) players respond stronger to changes in their own relative abilities than to changes in their rivals' relative abilities; (iii) aggregate effort (dissipated rent) does not change compared to the benchmark case; (iv) it is not possible to manipulate aggregate effort through seeding in binary elimination tournaments, although optimal seeding schemes for other purposes can be identified; and (v) balanced seeding and a uniform distribution of relative abilities cancel out the dependence of a player's effort on her rivals' abilities in binary elimination tournaments.     

On the Competition of Asymmetric Agents

Abstract. Rank-order tournaments are usually implemented in organizations to provide incentives for eliciting employees' effort and/or to identify the agent with the higher ability, for example in promotion tournaments. We close a gap in the literature by experimentally analyzing a ceteris paribus variation of the prize spread – being the major design feature of tournaments – in a symmetric and an asymmetric setting. We find that effort significantly increases with the prize spread as predicted by standard theory. However, only for sufficiently large prize spreads weak players competing against strong players strain themselves all the more and sorting of agents is feasible.     

Tournaments, minimum wage, and money non-neutrality

To reconstruct the micro-foundation of Keynesian macro-economics, the efficiency wage theory has generally been considered a success in providing a plausible explanation for the existence of involuntary unemployment. However, little has been said about how monetary policy causes fluctuations in aggregate employment and output in the efficiency wage theory. This paper extends Lin and Yang’s [Econ. Inq. 39 (2001) 644] shirking-type efficiency wage model with tournaments to account for money non-neutrality. A distinct feature of our model is that, due to the adoption of tournaments, there will be a hierarchical wage structure rather than a flat wage in the economy. As will be argued, the labor market characterized in this paper is in a sense a reversion to Keynes’ General Theory, but also an improvement upon it.     

POSITIVE SELF‐IMAGE IN TOURNAMENTS*

This article analyzes the implications of worker overestimation of productivity for firms in which incentives take the form of tournaments. Each worker overestimates his productivity but is aware of the bias in his opponent's self‐assessment. The manager of the firm, on the other hand, correctly assesses workers' productivities and self‐beliefs when setting tournament prizes. The article shows that, under a variety of circumstances, firms can benefit from worker positive self‐image. The article also shows that worker positive self‐image can improve welfare in tournaments. In contrast, workers' utility declines due to their own misguided choices.     

How Does FDI Respond to the Size of Shadow Economy: An Empirical Analysis under a Gravity Model Setting

Using a panel study under a gravity model setting focusing on data from 1999 to 2007 for Organization for Economic Co-operation and Development countries, we show that the size of the shadow economy in the host economy relative to the investor economy can play a significant role in attracting Foreign Direct Investment (FDI) inflows. We further explore nonlinearities in how shadow economy differentials affect incoming FDI. Our results match the theoretical predictions which highlight that opportunities for tax evasion can have an impact on the activities of Multinational Corporations.     

代理人能力分布不均情形下首位晋升制与末位淘汰制的比较研究

在基于相对业绩比较的锦标赛激励机制中,首位晋升制与末位淘汰制是两种常见的竞赛激励机制。在代理人能力分布不均情形下,将代理人之间的拆台行为引入传统的锦标赛模型,可以研究代理人能力分布不同时首位晋升制与末位淘汰制下代理人的行为以及对委托人收益的影响。通过数理模型推导可获得以下结论:第一,首位晋升制下代理人的努力水平和拆台水平均比较高,委托人采用首位晋升制更有利,能力分布对委托人最优选择不产生影响。第二,低能力代理人占多数时,高能力代理人努力水平过高,受到的拆台也更多,高能力代理人占多数时,低能力代理人努力水平更高受到的拆台更多,能力分布对代理人的行为产生影响。     

Hybrid Contests

This paper examines hybrid contests where participants commit two types of resources to improve their probability of winning the prize. The first type is forfeited ex ante, before the prize is allocated, by winners and losers alike, while the second is committed ex ante by all contenders but expended ex post, after the prize is allocated, and only by the contestant that wins the prize. The model yields a number of interesting results. Among them is the finding that, as the number of contestants increases, the ex ante expenditures of individual contestants decrease while the ex post expenditure increases. Even more interesting, the total of the ex ante and ex post expenditures by the contenders in a hybrid contest may decrease with the number of competitors. The study also finds that there is no rent overdissipation, and compares the total expenditures in the contest and “all‐pay” allocation mechanisms.     

Honesty in tournaments

We apply the die rolling experiment of Fischbacher and Föllmi-Heusi (2013) to a two-player tournament incentive scheme. Our treatments vary the prize spread. The data highlights that honesty is more pronounced when the prize spread is small.     

Difference-Form Contests and the Robustness of All-Pay Auctions

In much of the existing literature on rent-seeking games, the outcome of the contest is either infinitely sensitive or relatively insensitive to contestants' efforts. The current paper presents a family of contest games that permit characterization of equilibrium for all levels of sensitivity of the outcome to contestants' efforts. Specifically, the outcome of the contest depends on the difference between efforts, which encompass the lottery and the all-pay auction as polar cases. The equilibrium converges to that of the all-pay auction as the probability of winning the prize grows infinitely sensitive to one's effort, and the main qualitative features of equilibrium persist over a large parameter region. Journal of Economic Literature Classification Numbers: D44, D72.     

The Effect of Child Care Subsidies: A Critique of the Rosen Model

In an influential article, Sherwin Rosen (1997) argues that Swedish subsidies of child care services lead to a substantial misallocation of resources that slows economic growth. We offer two major reasons why Rosen's approach is flawed. First and foremost, he ignores the positive externalities of increasing the quality of child care, despite their clear relevance to his general equilibrium model. Second, he overlooks distributional impacts, despite evidence that child care subsidies redistribute the costs of children between men and women, rich and poor, young and old. By exploring these and a number of smaller problems with Rosen's model we hope to encourage the development of more systematic efforts to explore the effects of state support for child rearing.     

Round‐Robin Tournaments with a Dominant Player

We analyze the subgame perfect equilibrium of the round‐robin tournament with one strong (dominant) and two weak players, and we compare this tournament and the one‐stage contest with respect to the players' expected payoffs, expected total effort, and their probabilities of winning. We find that if the contest designer's goal is to maximize the players' expected total effort, then – if the asymmetry between the players is relatively low – the one‐stage contest should be used. However, if the asymmetry is relatively high, then the round‐robin tournament should be used.     

Estimating British workers' demand for safety

This paper estimates workers' demand function for job safety using the British General Household Survey data. The estimation employs Rosen's two-stage procedure. The main difference between our study and those done in the past is that we estimate hedonic price equations with data sets from two labour markets. Our approach overcomes the usual identification problems associated with the application of Rosen's method. The estimation shows that there is a significant wage compensation for job risk in the UK. The willingness-to-pay for a 1/100 000 decrease of annual job fatal accident rate from our estimated workers' demand function is about £6 in 1973 prices. The estimation of a demand function for safety also enables the derivation of workers' willingness-to-pay for non-marginal change of job risk, and this can be used for cost-benefit analysis on projects involving such non-marginal changes.     

Changing rewards in contests: Has the three-point rule brought more offense to soccer?

In the mid-nineties FIFA decided to increase from two to three the number of points assigned to the winning team of a soccer match played under traditional round-robin national leagues. Since a game of soccer can be regarded as a contest, FIFA's measure provides an interesting case-study for analysing how a change in the system of rewards (from a zero to a non-zero sum rule) may affect the contestants' equilibrium behaviour. In this paper we try to assess, both theoretically and empirically, whether FIFA's new point rule has changed soccer towards a more offensive game, in which teams adopt more risky strategies. In particular, we evaluate the “na?ve hypothesis” according to which the measure would induce every team to play always more offensively, and we explore the extent to which the change in teams' behaviour may be affected by quality differentials between teams. Our most important hypothesis is that when the asymmetry between opposing teams is large enough, an increase in the reward for victory induces the weaker team to play more defensively, rather than the opposite. By looking at a subset of matches held in the Portuguese first division league, which approximate the conditions of our model, we find support for this hypothesis. First version received: July 1999/Final version received: May 2001