Rewards versus punishments in additive, weakest-link, and best-shot contests

https://doi.org/10.1016/j.jebo.2015.11.013Get rights and content

Highlights

  • I provide a theory to explain how punishment or reward enhances performance of team.

  • Additive, weakest-link and best-shot performance functions are considered.

  • I derive easy-to-check conditions to judge whether reward or punishment is better.

Abstract

In this study, we provide a theory to explain how bottom punishment or top reward enhances performance of teams when there is heterogeneity across players in cost–performance relationships. In contrast to the existing literature, we consider three types of performance functions: additive, weakest link (the performance is the min of performances of n contestants), and best shot (the performance is the max of performances of n contestants). For any of the three types, we derive easy-to-check sufficient conditions to judge whether reward or punishment is better. From the sufficient condition for the additive performance function, we know that punishment is better for less heterogeneous people and reward is better for more heterogeneous people. In addition, the sufficient conditions for the best-shot and weakest-link cases suggest that some unintuitive results hold; even under the best-shot (weakest link) performance function, the bottom punishment (top reward) becomes better when the gap in the abilities of contestants becomes very small (large).

Introduction

Assuming that the employees of two companies have the same capability, which company would have better productivity—the company that motivates employees through a system in which top performers are promoted (promotion-based incentive) or the company that motivates employees through a system in which bottom performers are demoted (demotion-based incentive)? Would the answer differ by type of work?

Effective use of promotion and demotion, in other words, reward and punishment, are important questions for incentive design in firms. In the seminal papers of Lazear and Rosen (1981), Green and Stokey (1983), and Nalebuff and Stiglitz (1983), it is argued that a rank-order tournament is useful to reduce the moral hazard problem in a firm. The literature on tournament theory and contest theory has explored promotion and reward for a long time but limited attention has been paid to demotion and punishment.

A recent study of Moldovanu et al. (2012) (hereafter, MSS) fills this gap using a framework of contest theory. Extending the previous work of Moldovanu and Sela (2001), MSS consider what the optimal scheme of reward and punishment is in terms of maximizing the sum of effort when allocating a fixed reward or punishment according to ranking of effort level. They find that (1) if a designer can use a reward only, the top reward is optimal, (2) if he can use only a punishment, the bottom punishment is optimal when the distribution function of the agent's ability parameter satisfies an increasing failure rate (IFR), and (3) if he can use both the reward and punishment, the top reward (bottom punishment) is optimal when the distribution function satisfies IFR and is a concave (convex) function. While their first result is in line with the existing literature,1 the second and third results are quite new because MSS are the first to investigate both the stick and the carrot theoretically in the framework of contest theory. Using a similar model to MSS, Thomas and Wang (2013) analyze the carrot and stick in an environment in which contestants endogenously join the race.

In addition, MSS reveal the difficulty of analyzing the optimal sanction from the set of any types of reward and punishment based on the relative ranking without imposing the restriction of the agent's ability parameter on the class of distribution functions (F). One way to avoid the difficulty is to focus on the two most well-known types of sanctions, that is, the top reward and bottom punishment, and to soften the restriction on F. Focusing on the top reward is not in doubt from (1) of MSS. By contrast, focusing on the bottom punishment may be controversial and needs some justification. A rationale is that punishing an employee is a very precarious event (Trevino, 1992) and the employer has to minimize the number of punished employees. Choosing which employee to punish needs good accounts from subordinates in order to identify the worst performer, and this decision must be the most acceptable and understandable to employees. In fact, many punishment studies in other areas focus on worst-performer punishment (Yamagishi, 1986, Dickinson, 2001, Andreoni and Gee, 2012, Kamijo et al., 2014). In addition, Akerlof and Holden (2012) find from their analysis of reward and punishment schedules based on tournament theory that the top reward and bottom punishment have special importance compared to reward and punishment of others. Furthermore, from the viewpoint of symmetry, the most comparable sanction to the top reward is the bottom punishment.

Based on and modifying the MSS model, we study and compare the top reward and bottom punishment incentives. In our model, a given number of individuals simultaneously choose an effort level, input amount, or performance (hereafter, “effort level”) and the individuals receive punishment or reward accordingly. Although the effort level is measured objectively, there is uncertainty about the type of people. The parameter related to the cost associated with the effort varies by individual and this is private information, following the common prior F.

Departing from most existing literature, including MSS, in which the designer's objective is to maximize the sum of efforts of team members, we consider the cases in which the objective is best shot (max effort of n contestants) and weakest link (min effort of n contestants), in addition to the additive case. The designer's objective is changed owing to the task trait. An additive case is the most common and suitable for tasks wherein the performance of team members substitute each other; the sales score of the sales department or some type of professional team sports are typical cases of additive tasks. The polar case of the substitute task is best shot, in which doing or success by one person is sufficient. When their work is complementary, the objective of the designer becomes a weakest-link form, which is very common in line departments in manufacturing facilities and modern organizations, in which the division of labor progresses deeply (Kremer, 1993).

This study makes the following contributions to the literature on contest theory. First, for any type of the three objective functions, we derive easy-to-check sufficient conditions on F to judge which incentive is better. In particular, the condition for an additive case is weaker than the case for convexity or concavity. Moreover, combined with the IFR condition, our condition becomes sufficient for when the top reward (bottom punishment) is the best among all punishment or reward schemes. Thus, we extend (3) of MSS to the case without convexity or concavity.

Second, the sufficient condition for the best-shot objective indicates that in line with an intuition, top reward becomes better to enhance the max of performances of n agents in a wide class of F. Furthermore, it is shown that as n increases, for most distribution functions, the top reward is better for enhancing the best-shot objective of the designer. On the other hand, the sufficient condition for the best shot states that there is a case in which an unintuitive result holds (i.e., the bottom punishment is better for enhancing the max of n performances). A similar story holds for the weakest-link objective.

Third, the sufficient conditions indicate that the optimal use of sanctions is explained by the degree of the ability gaps among team members. Separating the whole population into a low ability group and a high ability group, the punishment is better for the situation in which the ability gap between the two groups is small, and the reward is better when the ability gap is large. This tendency of “reward for heterogeneous and punishment for homogeneous people” holds for any of the three objectives.

Finally, we confirm what kinds of sanctions are effective when the ability of all group members increases evenly. Regardless of the initial distribution function, we show that punishment becomes better than reward if members’ abilities are improved significantly.

Our findings on the proper usage of the carrot and stick based on the degree of heterogeneity or improvement of group members’ ability have not been pointed out in previous studies. Under contest theory, it is known that increased diversity reduces the total sum of efforts (Schotter and Weigelt, 1992, Gradstein, 1995). This is because the presence of highly capable individuals demotivates less capable individuals and, as a result, those less capable individuals reduce their effort level, in turn, causing highly capable individuals to choose a half-hearted effort level. Because of this type of discouragement effect, the exclusion principle, which states that the total sum of efforts increases by excluding highly capable individuals, holds (Baye et al., 1993). However, these studies have not discussed the relationship between the carrot and stick.2 MSS is the first study that refers to the relationships between the distribution of capability and the optimal use of the carrot and stick, but their analysis is motivated more from the mathematical properties of the distribution function (convexity or concavity) and is unrelated to the extent of the capability gap in a focused group.

The rest of this paper is organized as follows. Section 2 describes the basic model. Section 3 derives the equilibrium strategy for each system and, by comparing the equilibrium strategies, examines the merits of the effect of effort promotion by each system. In Section 4, we analyze the optimal sanction under additive, best-shot, and weakest-link objectives of the designer. Section 5 concludes. All proofs of theorems appear in the Appendix A.

Section snippets

General setup

The general setup follows the model of all-pay auction by Moldovanu and Sela (2001) and MSS. Consider an effort selection game among n players. Let N = {1, 2, . . . , n} with n  2 be a set of participants of the game. Each i makes an effort xi  [0, ∞]. An effort can be interpreted as performance if it is in the context of workers’ decisions in an organization. An effort incurs a disutility or cost of xi/θi, where θi  (θL, θH) is the cost parameter for i and θH > θL  0. Thus, the cost function of an effort

Equilibrium analysis

An ability parameter of player i is private information to i. This is identically and independently distributed according to F. Let a distribution function F with its density f be continuous and increasing in its domain [θL, θH]. Thus, an effort selection game with punishment or reward is an incomplete information game.

A strategy of player i in this game is a function that associates his realized type θi with effort xi. Let βi be the strategy of player i. We adopt the symmetric Bayesian Nash

The optimal use of sanctions

The designer's problem is to choose the better sanction institution from reward or punishment based on the designer's objective and group characteristics. The designer's goal is to increase the performance of a team. We consider the three types of performance functions. The team performance under additive, best-shot, and weakest-link performance functions are ∑iNxi, max {x1, …, xn}, and min {x1, …, xn}, respectively.3 Thus, the

Conclusion

In this study, we provide a theory explaining how bottom punishment or top reward enhance performances when there is heterogeneity across players in cost-performance relationships. We consider additive, weakest-link, and best-shot performance functions of team tasks, and for each of these three functions, we derive easy-to-check sufficient conditions to judge whether reward or punishment is better. Although our focus is on the two outstanding incentives, our sufficient condition for the

Acknowledgements

The author is grateful to Asuka Komiya, Tatsuyoshi Saijo, Kan Takeuchi, Takeshi Harui, and participants of the 2014 Economic Science Association (ESA) International Meetings and the 2014 Asia-Pacific ESA Conference for their helpful comments. In addition, the author is grateful for financial support from the Japan Society for the Promotion of Science Grant-in Aid for Exploratory Research and Grant-in-Aid for Scientific Research (B).

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