Cheating and enforcement in asymmetric rank-order tournaments.

• Author: Stowe, C. Jill

1. Introduction

   Among the many issues surrounding incentives to cheat in rank-order tournaments and contestants' behavior when cheating is possible, one of the most important is the effect of asymmetry in the position of the contestants. If two contestants are competing for a prize but one has an advantage over the other (such as greater ability or talent, or simply being "ahead" because of the history of play up to that point in the contest), will the trailing player necessarily be more likely to cheat because he needs to close the gap and has less to lose from possible disqualification? Might there be circumstances under which the leader cheats to maintain her advantage, because if she does not, the trailing player can close the gap by cheating? This article presents a model to provide insights into how positional concerns influence cheating in rank-order tournaments. (1)

   Rank-order tournaments present an environment in which the incentive to cheat may be particularly strong for a few reasons. (2) First, tournaments are generally used in environments in which actions are difficult to monitor. Second, a small increase in a contestant's output can dramatically change his payoff if it increases his rank. Cheating in various forms has been observed in important competitive settings, such as corporate promotion tournaments and sports competitions, which can be broadly characterized as taking place in contests and can be modeled as rank-order tournaments of the sort first introduced by Lazear and Rosen (1981).

   In this article, we consider a game in which two heterogenous players simply choose whether or not to cheat. This can be thought of as the final stage of a tournament in which players have previously chosen effort and now find themselves having an opportunity to cheat while knowing their current position relative to each other. Or it can be considered as the final round of a multistage tournament in which one player is ahead of the other entering the last phase of competition. Our focus is on how equilibrium cheating behavior is affected by varying the intensity of enforcement, represented by the probability that contestants are audited to detect cheating, and on the conditions required to achieve complete deterrence of cheating and thus a "clean" contest. Consistent with intuition we find that the trailing player has a strong incentive to cheat, but in some circumstances, it is the leading player who is more likely to cheat.

   In addition, we explore how differences in the auditing regime affect cheating behavior. We find that employing "correlated" audits (both contestants are audited simultaneously with some probability and otherwise neither is) yields less frequent cheating in the mixed-strategy equilibrium of the game than if the two contestants are audited with an equal probability but as a result of independent random draws. Using a correlated audit regime achieves more effective deterrence of cheating than independent audits for any given expenditure of resources when resources are not sufficient to achieve full deterrence. Then, relaxing the restriction that players must be monitored with equal probability, we find that full deterrence can be achieved at a lower cost by auditing the contestants with different probabilities. Specifically, the "leading" contestant can be audited with a lower probability than the trailing contestant while maintaining no cheating in equilibrium. This suggests that conditioning enforcement on contestants' positions (ex ante) and also perhaps final rank (ex post) is advantageous and that uniform enforcement is not the least cost method of fully deterring cheating.

   Other authors have considered cheating in asymmetric tournaments or contests, and the most closely related works are Berentsen (2002) and Haugen (2004). Berentsen (2002) considers a game with two heterogeneous players who simultaneously decide whether or not to cheat (take performance-enhancing drugs) before competing. The author focuses on implementing a no-cheating outcome as a function of the penalty associated with getting caught cheating, whereas we analyze cheating behavior as a function of the probability of an audit. Experimental evidence suggests that contestants are more responsive to changes in the audit probability than changes in the size of the penalty, with observed behavior more closely tracking theoretical predictions when the audit probability is varied (Evans et al. 2008). This suggests that framing the problem with the audit probability as the main choice variable for the contest organizer may be more empirically relevant. Furthermore, this difference in the modeling choice leads to divergent results in cheating behavior. This is due to the fact that although the penalty and the audit probability both determine the expected penalty, the audit probability also affects the probability a player is disqualified, which is strategically important.

   Krakel (2007) analyzes a two-player asymmetric tournament in which contestants choose both effort and cheating, which are modeled as complements. Given this setup, the article focuses on a no-cheating equilibrium and identifies three effects that determine whether an individual decides to cheat: the likelihood effect (cheating improves the probability of winning), the cost effect (how cheating affects effort costs), and the base-salary effect (if a player cheats, he reduces his expected base salary since there is a chance he will get caught). Furthermore, Krakel examines whether ex ante (before the tournament begins) or ex post (after rankings are determined) testing leads to higher levels of legal input and finds that greater effort is exerted in ex ante testing. We also consider ex ante and ex post auditing, but we do it in the context of determining the least costly method of deterring cheating.

   Two articles model the cheating decision in a symmetric contest. Gilpatric (2009) presents a model in which contestants simultaneously choose cheating and effort, and as a departure from previous models, cheating is a continuous choice. Curry and Mongrain (2009) also model cheating in symmetric contests in which players make a dichotomous choice whether to cheat or not, but their model concentrates on the minimum audit probability required to deter all cheating. They focus on the important effect of "re-awarding" on cheating incentives, that is, whether the prize to the top-ranked contestant passes by default to the second-ranked contestant if the winner is found to have cheated.

   A small number of empirical articles have explored cheating behavior in addition to Evans et al. (2008). Nagin et al. (2002) discuss a natural experiment that enabled them to identify cheating behavior by workers receiving piece-rate compensation for telemarketing solicitations. They find that a significant portion of workers, but not all, are "rational cheaters" whose cheating behavior is responsive to the expected penalties. Duggan and Levitt (2002) find evidence of cheating in Sumo wrestling. This study explores behavior that is closer to the context of our model because it addresses cheating in a contest; however, the cheating they identified in Sumo is of the form of one contestant "throwing" a match for another's benefit in return for a future quid pro quo, which cannot arise in the context of a nonrepeated game. Of course, empirical investigation of cheating is hampered by the fact that cheating is concealed, and it is generally impossible to know how much cheating goes undetected (although Duggan and Levitt illustrate how clever techniques can at times be used to obtain inference about the frequency of cheating). This highlights the importance of gaining a better theoretical understanding of the factors that determine the extent of cheating.

   The article proceeds as follows. Section 2 presents a model of the decision to cheat or not in a two-player tournament when the players are heterogeneous. It explores both the pure and mixed-strategy equilibria, both of which depend on the probability of audit. In section 3, we show that when both contestants are subject to an equal audit probability, correlated audits more effectively deter cheating than independent audits (in which each contestant is audited or not as a result of an independent random draw). In section 4, we show that differential audit rules, whereby one player may be audited with higher probability than the other conditional on their relative positions, may be desirable if feasible. Section 5 concludes, and all proofs are collected in the appendix.

2. The Decision to Cheat

   In a Lazear-Rosen tournament setting, consider risk-neutral players j ("she") and k ("he"). The winner of the tournament, determined by the player with the highest level of output q, receives [w.sub.1], and the loser receives [w.sub.2], where [w.sub.1] > [w.sub.2] and [w.sub.1] > 0. Denote the spread between the winning and losing prizes S = [w.sub.1] - [w.sub.2].

   Consider a tournament in which players make a simple dichotomous choice to cheat or not; a player who cheats increases his average output by x. Moreover, one player may have an advantage over the other. Without loss of generality, suppose that player j is weakly ahead of player k by [alpha] [greater than or equal to] 0 ([alpha] = 0 corresponds to the case of a symmetric contest, and [alpha] > 0 corresponds to an asymmetric contest). (3) This difference in position is common knowledge to both contestants as well as the contest organizer. When asymmetry exists, it may be due to differences in talent, effort, or luck or may be a consequence of previous play (O'Keefe et al. 1984). We do not model the choice of effort or prior play, but simply analyze behavior when cheating is possible and prior choices or abilities may result in asymmetric positions.

   As is standard in tournament models, output is also affected by [epsilon], a random component symmetrically distributed around 0 that...