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

Journal of Economic Behavior & Organization 122 (2016) 17–30

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Journal of Economic Behavior & Organization journal homepage: www.elsevier.com/locate/jebo

Rewards versus punishments in additive, weakest-link, and best-shot contests Yoshio Kamijo School of Economics and Management, Kochi University of Technology, 2-22 Eikokuji-cho, Kochi-shi, Kochi 780-8515, Japan

a r t i c l e

i n f o

Article history: Received 13 July 2015 Received in revised form 14 October 2015 Accepted 18 November 2015 Available online 8 December 2015 JEL classification: C72 L2 Keywords: Contest theory Heterogeneity Punishment Reward

a b s t r a c t 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). © 2015 Elsevier B.V. All rights reserved.

1. 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 (promotionbased 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

E-mail address: [email protected] http://dx.doi.org/10.1016/j.jebo.2015.11.013 0167-2681/© 2015 Elsevier B.V. All rights reserved.

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Y. Kamijo / Journal of Economic Behavior & Organization 122 (2016) 17–30

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

1 How to allocate a fixed amount of pie according to effort ranking is considered in Barut and Kovenock (1998) and Glazer and Hassin (1988). The common finding among these studies is that the “winner takes all” approach in which the top-ranked individual is given the entire reward becomes the optimal allocation under a moderate condition. Brookins and Ryvkin (2014) examine the theoretical prediction of contest theory in a laboratory setting.

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(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. 2. Basic model 2.1. 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 / i , where  i ∈ ( L ,  H ) is the cost parameter for i and  H >  L  0. Thus, the cost function of an effort is the product of the individual-specific component (preference, wealth, ability, skill, and so on) and the selected provision component. A player with a high  can make high effort by small cost. In an effort selection game without any sanction institution, players choose their efforts simultaneously. Let x = (x1 , x2 , x . . ., xn ) ∈ [0, ∞]n be a profile of efforts. The payoff of player i choosing xi is − i . i

2.2. Punishment and reward Let Nmax (x) and Nmin (x) be the number of highest performers and lowest performers in x, respectively. Let Iimax (x) be i’s

maximum indicator function with value 1 if xi is highest at x and 0 otherwise. Similarly, let Iimin (x) be i’s minimum indicator function with value 1 if xi is lowest at x and 0 otherwise. Let P > 0 be a fixed amount of punishment. In a bottom punishment institution, the payoff of player i in profile x is

vpi (x) = −

xi P − Iimin (x) i N min (x)

Thus, we assume that the punishment is directed toward the worst performer. If there is a tie, the amount of sanction is divided among the tied members, or one individual among the tied members is selected randomly to receive the sanction. Let R > 0 be the amount of reward. Similarly, the payoff of player i in a top reward institution is

vri (x) = −

xi R + Iimax (x) max N (x) i

The best performer gains a reward R, and if there is a tie, the reward is divided among the tied members or one member is chosen randomly and gains the entire reward amount. 3. 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 [ L ,  H ]. 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  i with effort xi . Let ˇi be the strategy of player i. We adopt the symmetric Bayesian Nash equilibrium (ˇ, ˇ, . . ., ˇ), in which every player uses the same strategy ˇ as a solution criterion in order to evaluate the performance of the sanction institutions. For an equilibrium strategy ˇ, we assume that ˇ is a continuous, differentiable, and increasing function with ˇ( L ) = 0. This is a standard assumption familiar to auction theorists. We now explore the conditions that should be satisfied by ˇ. First, we consider the bottom punishment institution. Suppose n − 1 players follow the strategy ˇ, except one player with type . Then, the expected payoff of the player when he chooses effort x is as follows:  n−1 x p (x, ) = − − P 1 − F(ˇ−1 (x)) . 

2 Although there are experimental studies on the heterogeneity of players (endowments and cost) in public goods games, including Buckley and Croson (2006), Chan et al. (1996), and Nitta (2014) for endowment heterogeneity and Fisher et al. (1995), Tan (2008), and Fellner et al. (2011) for provision cost heterogeneity, there is no study that compares reward and punishment.

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Y. Kamijo / Journal of Economic Behavior & Organization 122 (2016) 17–30

0.30

Equilibrium Strategy for Punishment

0.8

n=3

Equilibrium Strategy for Rewrad

0.25 0.6

n=5

0.20

0.4

0.15 0.10

n=10

n=3 n=5 n=10

0.2

0.05

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

Fig. 1. Equilibrium strategies for sanction institutions. The left panel corresponds to the equilibrium strategy for the punishment institution and the right panel corresponds to the equilibrium strategy for the reward institution. ( L ,  H ) = (0, 1), F(z) = z, and P = R = 1.

Note that F(ˇ−1 (x)) is the probability that a player following ˇ chooses an effort less than x and 1 − F(ˇ−1 (x)) is the probability that a player chooses an effort more than x. Thus, (F(ˇ−1 (x))) best performer among n players and (1 − F(ˇ−1 (x))) among n players. Using the first-order condition at ˇ(), we have



n−2

ˇ () = P(n − 1) 1 − F()

n−1

n−1

is the probability that a player choosing x is the

is the probability that a player choosing x is the worst performer

f ().

On integrating this equation by  with the condition ˇ( L ) = 0, we have the following:





p

ˇ () = P(n − 1)

(1 − F(z))

n−2

zf (z)dz.

(1)

L

Thus, we have a candidate for the equilibrium strategy of the punishment institution from the above equation. Next, we explore the necessary conditions of ˇ in the top reward institution. By a similar procedure, we obtain the candidate for the equilibrium strategy of the reward institution as follows:





r

ˇ () = R(n − 1)

(F(z))

n−2

zf (z)dz.

(2)

L

It is easy to check that ˇp and ˇr are Bayesian Nash equilibrium strategies for punishment and reward institutions, respectively. Fig. 1 shows the equilibrium strategies for punishment and reward institutions for different group sizes. From this, we easily find that the effort-improving effects of these two incentives are totally different in “the person who is more responsive to the incentives.” In the reward institution, a highly able player is strongly motivated and makes a very high effort, but the effort-improving effect on a person with low ability is limited. This is consistent with the finding from all-pay auctions, known as the discouragement effect, in which a more highly able person discourages a less able person (Schotter and Weigelt, 1992; Gradstein, 1995). The converse holds for the punishment institution; a person with low ability is strongly motivated while the effect on a highly able person is limited. Another important difference is that, while in punishment the difference in effort between low- and high-ability players is moderate, there is a considerable difference in the effort between them in the reward institution. Comparing ˇp and ˇr under the condition P = R, we find a useful insight regarding punishment and reward. Let  med be the parameter satisfying F( med ) = 1/2, that is, the median of F. Theorem 1. (i) (ii) (iii) (iv) (v)

Assume R = P. The following conditions hold:

for any    med , ˇp () > ˇr ()   for any  <  med , (ˇp ) () > (ˇr ) ()   med p r for any  =  , (ˇ ) () = (ˇ ) ()   for any  >  med , (ˇp ) () < (ˇr ) () ˆ ˇp () < ˇr () there exists ˆ < H such that for any  > ,

This indicates that median ability plays a key role in understanding the characteristics of the performance-improving effect of the two institutions. The first statement of this theorem is that less able players make more effort in punishment institutions than in reward institutions. Thus, a person with lower ability than the median is more strongly motivated by

Y. Kamijo / Journal of Economic Behavior & Organization 122 (2016) 17–30

21

the threat of punishment. In fact, the marginal increase in the efforts of less able players by punishment is greater than that by reward, indicating that for less able players, avoiding the stick is more effective than being shown the carrot ((ii) of the theorem). This marginal incentive is reversed for more highly able players, who are more motivated by the carrot ((iv) of the theorem), but their effort level for the reward may be smaller than that for the punishment because of the cumulative pushing-up or chain-reaction effects of punishment. In other words, the pushing-up effects on less able players contribute weakly to the progress of more highly able players. However, the equilibrium contribution of very high-ability players is higher with reward than punishment ((v) of the theorem).

4. 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 best-shot, and weakest-link performance functions of performance functions. The team performance under additive, 3 are i ∈ N xi , max {x1 , . . ., xn }, and min {x1 , . . ., xn }, respectively. Thus, the designer’s objective changes with the type of performance functions. We assume P = R and these are normalized to 1.4 The goal of this section is to demonstrate how the objective of the designer and the configuration of group members’ abilities affect the optimal choice of the punishment and reward. Thus, the designer has to use the two institutions based on the task and group characteristics. Let Gkn denote the distribution function of the k-th highest order statistic for n independent random variable following F,

and gkn be the density of Gkn . Note that Gnn (z) = 1 − (1 − F(z))n , gnn (z) = nf (z)(1 − F(z))n−1 , G1n (z) = F(z)n , g1n (z) = nf (z)F(z)n−1 .

By using this notation, we can write ˇp () =



L

n−1 zgn−1 (z)dz and ˇr () =



L

zg1n−1 (z)dz, respectively.

4.1. Additive performance function In this subsection, we consider the case in which the designer’s objective is to maximize the sum of the efforts of n contestants. The expected effort of one player under punishment is

 p



H

p

H



E = E[ˇ ()] =

n−1 zgn−1 (z)dz

ˇ ()f ()d = L





p

f ()d.

L

L

On interchanging the order of integration of Ep , we obtain the following:





H



H



H

n−1 f ()d zgn−1 (z)dz =

Ep = L

z

n−1 (z)dz, [1 − F(z)] zgn−1 L

which is equal to



H

(n − 1)

(1 − F(z))n−1 zf (z)dz.

L

By a similar calculation, the expected effort of one player under reward is





H

E r = E[ˇr ()] =

H

ˇr ()f ()d = (n − 1) L

(1 − F(z))(F(z))n−2 zf (z)dz.

L

Comparing Ep and Er , we obtain implications of how to use the stick and carrot. For x ∈ (0, 1/2), F−1 (1 − x)/F−1 (x) is the ratio of the ability of the 100x-th player from the top relative to the ability of the 100x-th player from the bottom. Thus, the value of F−1 (1 − x)/F−1 (x) measures how the abilities of top group players and bottom group players diverge. A larger value of F−1 (1 − x)/F−1 (x) means that the ability gap is large. The following is one of our main results on the optimal use of punishment and reward.

3

The terms “best shot” and “weakest link” are from Hirshleifer (1983). This assumption is justifiable only when the designer can use monetary sanctions. While evidently there is a situation in which an employer uses a salary cut or increase based on the employee’s record, we note that the symmetry of punishment and reward sizes may be a restrictive condition because, in some contexts, the use of monetary punishment is limited, and thus, the punishment involves a psychological factor. 4

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Y. Kamijo / Journal of Economic Behavior & Organization 122 (2016) 17–30

Theorem 2.

In the additive performance function,

(i) if F−1 (1 − x)/F−1 (x)  (1 − x)/x for all x ∈ (0, 1/2) and strict inequality holds for some x ∈ (0, 1/2), the bottom punishment is better than the top reward, (ii) if F−1 (1 − x)/F−1 (x)  (1 − x)/x for all x ∈ (0, 1/2) and strict inequality holds for some x ∈ (0, 1/2), the top reward is better than the bottom punishment, and (iii) if F−1 (1 − x)/F−1 (x) = (1 − x)/x for all x ∈ (0, 1/2), there is no difference between the two. The theorem suggests that when the ability gap between the top x player and bottom x player is smaller (greater) than some value, depending on x for any x ∈ (0, 1/2), the bottom punishment (top reward) is useful to enhance the sum of effort of n players. In other words, the punishment is better for a situation in which the ability gap between the low ability and high ability groups is small, and in a sense, the whole population is less heterogeneous in terms of their abilities. By contrast, the reward is better for the situation in which the ability gap between the two groups is large, and in a sense, the whole population is more heterogeneous in terms of their abilities. The intuition for this result is obtained from the pushing-up effects from the bottom under the punishment. When the low ability group is as bad as the high ability group, the pushing-up effects from the bottom on less able players induced by the punishment effectively contribute to the progress of more highly able players in the latter group. Thus, the punishment is likely to be better than the reward for a population in which the ability gap is small. By contrast, when the ability gap in the population is large, the pushing-up effect is limited for high ability players and the reward is likely to be better. For a better understanding of the theorem, it is useful to transform the condition in the theorem to other forms. For a distribution function F, let (x) be a F−1 (x)/x for x ∈ (0, 1], which is the slope connecting the origin to (x, F−1 (x)). For instance, let us consider distribution function F1 in Fig. 2. Then, the slopes of L1 and L2 are (a) and (1 − a), respectively. In this figure, (1 − a) > (a) holds. Using , for any x ∈ (0, 1/2), F −1 (1 − x)/F −1 (x)  (1 − x)/x ⇔ (x)  (1 − x), and F −1 (1 − x)/F −1 (x)  (1 − x)/x ⇔ (x)  (1 − x). Thus, according to Theorem 2, the punishment (reward) is better if (x) is greater (smaller) than (1 − x) for any x ∈ (0, 1/2). For instance, the graph of F1−1 of Fig. 2 is increasing and waving, implying that this is neither convex nor concave. Nonetheless, this satisfies (x) < (1 − x) for any x ∈ (0, 1/2), and thus, reward is better than punishment. The following shows the immediate consequences of the theorem. First, the sufficient conditions in the theorem relate to the property of increasing or decreasing of (x) in the domain of (0, 1). In fact, the theorem implies that if (x) is decreasing (increasing) in (0, 1), the punishment (reward) is better. Second, they can be connected with the concavity or convexity of a distribution function F. If F is a convex (concave) function with F(0) = 0, (x) apparently is decreasing (increasing), and thus, the punishment (reward) is better (see Table 1). Convexity or concavity of F is the condition that MSS focus on in order to compare punishment and reward. For instance, let us consider the class of distribution function F() =  ˛ on [0, 1] with ˛ > 0. When ˛ > 1, F−1 (x)/x = x(1/˛)−1 is decreasing and the punishment is better than the reward. When ˛ < 1, the converse holds, and thus, the reward is better than the punishment. 1.0

theta

L3

F_ 2 ^

1

0.8

F_ 1 ^

1 L2

0.6

0.4

L1 0.2

a 0.0

0.2

1-a

0.5 0.4

0.6

Fig. 2. Relationship of  and F−1 .

x 0.8

1.0

Y. Kamijo / Journal of Economic Behavior & Organization 122 (2016) 17–30

23

Table 1 Punishment versus reward when F is convex or concave function with F(0) = 0.

Convex Concave

Additive

Weakest-link

Best-shot

Punishment Reward

Punishment Reward or Punishment

Reward or Punishment Reward

The conditions specified in this theorem are easy to use because we need not restrict our attention to any class of distribution functions. The following corollary of the theorem provides a useful sufficient condition to check the optimal sanctions. Corollary 1. function,

For a distribution function F, consider the line connecting (0, 0) and (1/2, F−1 (1/2)). In the additive performance

(i) if F−1 (x) is over the line in the domain (0, 1/2) and beneath the line in the domain (1/2, 1), the bottom punishment is better than the top reward, and (ii) If F−1 (x) is beneath the line in the domain (0, 1/2) and over the line in the domain (1/2, 1), the top reward is better than the bottom punishment It is easy to check that F2 in Fig. 2 satisfies the first condition of the corollary and thus, the punishment is better than reward. To demonstrate how the ability gap of group members affects the optimal institution, we restrict our attention to the class of symmetric distribution on [0, 1]. Note that if F is symmetric, F−1 (1 − x) = 1 − F−1 (x) holds. Proposition 1.

Assume that F is a symmetric distribution on [0, 1]. In the additive performance function,

(i) if F−1 (x)  x for all x ∈ (0, 1/2] and strict inequality holds for some x ∈ (0, 1/2), the bottom punishment is better than the top reward, (ii) if F−1 (x)  x for all x ∈ (0, 1/2] and strict inequality holds for some x ∈ (0, 1/2), the top reward is better than the bottom punishment, and iii if F−1 (x) = x for all x ∈ (0, 1/2] and strict inequality holds for some x ∈ (0, 1/2], there is no difference between the two. Proof.

The statements are obvious from Theorem 2. 

Note that when F−1 (x) = x for all x, F is a uniform distribution on [0, 1]. F is less heterogeneous than a uniform distribution when F is symmetric and F−1 (x)  x for all x ∈ (0, 1/2]. In fact, the variance of a symmetric distribution F is



1/2

2× 0

1 2 ( − ) f ()d = 2 × 2

and this is less than 2 ×

 1/2 0

2



1/2

(F −1 (x) − 0

1 2 ) dx 2

(x − 12 ) dx if F−1 (x)  x and F −1 ( 12 ) =

1 . Thus, when F satisfies these conditions, the variance of F 2 heterogeneous than a uniform distribution when F−1 (x)  x

is less than that of a uniform distribution. By contrast, F is more for all x ∈ (0, 1/2]. We know there is indifference about enhancing the sum of effort for reward and punishment if F is a uniform distribution. If F is less heterogeneous than the uniform distribution, punishment is better than reward. On the contrary, if F is more heterogeneous than the uniform distribution, reward is better than punishment. Thus, the diversity of group members matters for the optimal choice of punishment and reward. In the last part of this subsection, we compare our results with those of MSS on the optimal punishment and reward structure. A reward and punishment structure is based on the relative effort of players and fixed amounts of reward and structure (V1 , V2 , . . . , Vn ), the k-th punishment are allocated to players in any way. To be precise, in a reward and punishment n highest performer obtains sanction Vk , where Vk may be positive or negative and |V |  1 must hold. Note that reward i i=i and punishment structure (1, 0, . . . , 0) is the top reward and (0, 0, . . ., 0, − 1) is the bottom punishment this study focuses on. The optimal reward and punishment structure is one that maximizes the sum of effort. We say that F satisfies IFR if f()/(1 − F()) is increasing. The following proposition concerns the optimal punishment and reward structure. Proposition 2.

Assume that F is a distribution function on [0, 1] and satisfies the IFR. In the additive performance function,

(i) if F−1 (1 − x)/F−1 (x)  (1 − x)/x for all x ∈ (0, 1/2) and strict inequality holds for some x ∈ (0, 1/2), the bottom punishment is the optimal reward and punishment structure, and (ii) if F−1 (1 − x)/F−1 (x)  (1 − x)/x for all x ∈ (0, 1/2) and strict inequality holds for some x ∈ (0, 1/2), the top reward is the optimal reward and punishment structure.

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Y. Kamijo / Journal of Economic Behavior & Organization 122 (2016) 17–30

Proof. From Proposition 3 of MSS, if F satisfies IFR, the optimal reward and punishment structure is either the bottom punishment or the top reward. Thus, the statements are obvious from Theorem 2.  This proposition generalizes the result of MSS (Proposition 4 of their study) by stating that if F satisfies convexity and the IFR, the optimal reward and punishment structure is the bottom punishment, and if F satisfies concavity and the IFR, the optimal structure is the top reward. In fact, consider a distribution function F(x) = (sin(2ax − a) − sin(−a))/(sin(a) − sin(−a)) on [0, 1] where 0 < a  /2. This satisfies the IFR and the conditions of (i) of the proposition, implying that the bottom punishment is the optimal under F. However, F is neither convex nor concave. On the other hand, consider a distribution function F(x) = (arcsin(2ax − a) − arcsin(−a))/(arcsin(a) − arcsin(−a)) on [0, 1] where 0 < a  1. Then, this satisfies the IFR when a is close to zero and the conditions of (ii) of the proposition, implying that the top reward is the optimal under F. However, F is neither convex nor concave. 4.2. Weakest-link performance function In this subsection, we consider the case in which the designer’s objective is to maximize min of the efforts of n contestants. We show the following results about better incentives when a performance function is weakest link. Theorem 3.

In the weakest-link performance function,

(i) if F−1 (1 − x)/F−1 (x)  ((1 − x)/x)n for all x ∈ (0, 1/2) and the strict inequality holds for some x ∈ (0, 1/2), the bottom punishment is better than the top reward, and (ii) if F−1 (1 − x)/F−1 (x)  ((1 − x)/x)n for all x ∈ (0, 1/2) and the strict inequality holds for some x ∈ (0, 1/2), the top reward is better than the bottom punishment Similarly to the additive performance function, the theorem suggests that the punishment is better for the situation in which the ability gap between the low ability group and the high ability group is small, and the reward is better for the situation in which the ability gap between the two group is large. However, the threshold that distinguishes the optimal sanctions differs between the additive and weakest-link objectives. Since (1 − x)/x < ((1 − x)/x)n for any n > 2 and for any x ∈ (0, 1/2), the punishment becomes better than the reward in a wider domain under a weakest-link objective, compared with an additive objective. The reason is simple. As explained after Theorem 1, the punishment enhances the effort of less ability players and, thus, meets the objective of enhancing the min of n players’ efforts. We obtain a slightly different implication on the effective sanction depending on the concavity or convexity of F from the case of the additive objective (see Table 1). First, the convexity with F(0) = 0 implies that the punishment is better under a weakest-link objective. This is because the convexity means that F−1 (1 − x)/(F−1 (x))  (1 − x)/x, implying that (i) of Theorem 3 holds. Second, the concavity of F does not imply that the top reward is better for the weakest-link objective. For instance, let us consider the class of distribution function F() =  ˛ on [0, 1] with ˛ > 0. Then, F−1 (1 − x)/F−1 (x) = ((1 − x)/x)1/˛ , and this is less than ((1 − x)/x)n if 1/˛ < n ⇔ ˛ > 1/n. Thus, the punishment is better than the reward if ˛ > 1/n. Since F() =  ˛ is a uniform distribution when ˛ = 1, the punishment is better under a uniform distribution. In addition, there are concave distribution functions (1/n < ˛ < 1) such that the punishment is better to enhance the performance, contrary to the fact that the concave distribution function is a sufficient condition for the reward being better than the punishment under the additive performance function. We define a function hw (.) on [0, 1/2] by hw (x) =

xn xn + (1 − x)n

∀x ∈ [0, 1/2].

The next proposition demonstrates the results in Theorem 3 under a symmetric distribution. Proposition 3.

Assume that F is a symmetric distribution function on [0, 1]. In the weakest-link performance function,

(i) if F −1 (x)  hw (x) for all x ∈ (0, 1/2) and the strict inequality holds for some x ∈ (0, 1/2), the bottom punishment is better than the top reward, and (ii) if F −1 (x)  hw (x) for all x ∈ (0, 1/2) and the strict inequality holds for some x ∈ (0, 1/2), the top reward is better than the bottom punishment. Proof. The statements are obvious from Theorem 3 and the fact that F−1 (1 − x) = 1 − F−1 (x) for any x ∈ (0, 1) since F is a symmetric distribution.  The left panel of Fig. 3 illustrates the statements of this proposition. Since hw is a convex function with hw (0) = 0 and hw (1/2) = 1/2, the area for the punishment being better is wider than that for reward being better. In fact, for F1−1 and F2−1 ,

which are located over hw , the punishment is better to enhance lowest performance of n members. By contrast, for F3−1 , which are located under hw and where the abilities of contestants are weighted in the lowest and highest (i.e., the variance is large), the reward is better than the punishment. However, the area for reward being better shrinks as n increases because

Y. Kamijo / Journal of Economic Behavior & Organization 122 (2016) 17–30 0.5

0.5

theta

25

theta

Punishment is better Punishment is better

0.4

0.4

F_3^{-1} F_2^{-1}

0.3

Reward is better

h^b

0.3

F_1^{-1} 0.2

F_1^{-1}

0.2

h^w 0.1

F_3^{-1}

F_2^{-1} 0.1

Reward is better x 0.1

0.2

0.3

0.4

x 0.5

0.1

0.2

0.3

0.4

0.5

Fig. 3. Distribution functions and optimal sanctions under weakest-link (left panel) and best-shot (right panel) performance functions.

hw (x) =

xn xn +(1−x)n

point-wise converges to the function h(.) such that h(x) = 0 for x ∈ [0, 1/2) and h(1/2) = 1/2 as n increases.

Thus, every symmetric and increasing distribution F satisfies (i) of this theorem if n is sufficiently large. 4.3. Best-shot performance function Next, we show the following results about better incentives when the performance function in an organization is best shot. We have the following theorem. Theorem 4.

In the best-shot performance function,

(i) if F−1 (1 − x)/F−1 (x)  (1 − xn )/(1 − (1 − x)n ) for all x ∈ (0, 1/2) and the strict inequality holds for some x ∈ (0, 1/2), the bottom punishment is better than the top reward, and (ii) if F−1 (1 − x)/F−1 (x)  (1 − xn )/(1 − (1 − x)n ) for all x ∈ (0, 1/2) and the strict inequality holds for some x ∈ (0, 1/2), the top reward is better than the bottom punishment In line with the results for the other two performance functions, the theorem suggests that the punishment is better for the situation in which the ability gap between the low ability group and the high ability group is small, and the reward is better for the situation in which the ability gap between the two groups is large. However, the threshold that distinguishes the optimal sanctions is more biased toward the direction in which the reward is likely to be better. Since (1 − x)/x > (1 − xn )/(1 − (1 − x)n ) for any n > 2 and for any x ∈ (0, 1/2),5 the reward becomes better than the punishment in a wider domain under a best-shot objective compared with an additive objective. While the concavity of F implies that the top reward is better to enhance the max of n players’ efforts, the convexity of F does not imply that the bottom punishment is better to enhance the best-shot objective. Let us consider the class of distribution function F() = 1 − (1 − )˛ on [0, 1] with ˛ > 0. Then, F−1 (1 − x)/F−1 (x) = (1 − x1/˛ )/(1 − (1 − x)1/˛ ), and for any x ∈ (0, 1/2), this is more than (1 − xn )/(1 − (1 − x)n ) if ˛ > 1/n because (1 − xa )/(1 − (1 − x)a ) is a decreasing function on a for any x ∈ (0, 1/2). Thus, the reward is better than the punishment if ˛ > 1/n. When ˛ = 1, F() = 1 − (1 − )˛ becomes a uniform distribution and thus, reward is better than punishment under a uniform distribution. Note that F() = 1 − (1 − )˛ is concave if ˛  1 and convex if ˛  1. Thus, even under a convex distribution function (1 > ˛ > 1/n), the reward is better, which is contrary to the result for an additive performance function wherein the convexity is a sufficient condition for the punishment being better. This is due to the fact that the reward enhances the higher ability player more than punishment and, thus, meets the best-shot objective. We define a function hb (.) on [0, 1/2] by hb (x) =

1 − (1 − x)n 2 − xn − (1 − x)n

∀x ∈ [0, 1/2].

The following Proposition 4 demonstrates the results in Theorem 4 in a symmetric distribution function. Proposition 4.

5

Assume that F is a symmetric distribution function on [0, 1]. In the best-shot performance function,

We can check that 

1−(1−x)n x 

−

1−xn 1−x

=

1−2x+xn+1 −(1−x)n+1 x(1−x)

(0) =  (1/2) = 0,  (0) > 0, and  (x) < 0 for x ∈ (0, 1/2).

is positive in (0, 1/2). In fact, (x) = 1 −2x + xn+1 − (1 − x)n+1 is positive in (0, 1/2) since

26

Y. Kamijo / Journal of Economic Behavior & Organization 122 (2016) 17–30

(i) if F−1 (x)  hb (x) for all x ∈ (0, 1/2) and the strict inequality holds for some x ∈ (0, 1/2), the bottom punishment is better than the top reward, and (ii) if F−1 (x)  hb (x) for all x ∈ (0, 1/2) and the strict inequality holds for some x ∈ (0, 1/2), the top reward is better than the bottom punishment. Proof. These statements are obvious from Theorem 4 and the fact that F−1 (1 − x) = 1 − F−1 (x) for any x ∈ (0, 1) since F is a symmetric distribution.  The right panel of Fig. 3 illustrates the statements of this proposition. Since hb is a concave function with hb (0) = 0 and the area for the reward being better is wider than that for punishment being better. In fact, for F1−1 and F2−1 ,

hb (1/2) = 1/2,

which are located under hb , the reward is better to enhance the highest performance of n members. By contrast, for F3−1 , which are located over hb and where the abilities of contestants are weighted in the median ability (i.e., the variance is small), the punishment is better than the reward. However, the area for punishment being better shrinks as n increases because n ˜ such that h(x) ˜ = 1/2 for x ∈ [0, 1/2] as n increases. Thus, every hb (x) = 1−(1−x) n point-wise converges to the function h(.) 2−xn −(1−x)

symmetric and increasing distribution F satisfies (ii) of this theorem if n is sufficiently large. 4.4. The effect of an increment of the abilities of contestants In this subsection, we investigate how the optimal institution changes if the abilities or skills of group members increase evenly. For A > 0 and a distribution function F on ( L ,  H ), F+A is defined by F+A () = F( − A) for  ∈ ( L + A,  H + A). F+A is said to be a right parallel shift of F. Thus, F+A is a distribution function after uniformly increasing the skill parameters of group members from an initial distribution F. ˆ > 0 exists such that for any A > A, ˆ under a distribution function F+A , the Theorem 5. For any distribution function F, some A bottom punishment is better than the top reward for each of the additive, best-shot, and weakest-link performance functions. This theorem shows that as the lowest ability increases, punishment tends to be chosen as the optimal sanction. This is consistent with the diversity view, in which less diversity implies punishment and more diversity implies reward. Fixing the shape of F, a large right parallel shift implies that the group becomes less heterogeneous with higher average ability. For example, imagine a select group of baseball players in some country. Their average skills are somewhat higher than the average of all baseball players in the country but their abilities are less heterogeneous. In such a case, punishment could improve their performance better than reward, even though their average performance is quite high. 5. 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 additive case with combing the IFR provides a sufficient condition that the top reward or bottom punishment is optimal among the set of any type of reward and punishment structures. Taking a uniform distribution as a reference point, we find that by and large, the bottom punishment is better when a distribution function is less heterogeneous than the uniform function, and the top reward is better when a distribution function is more heterogeneous than the uniform function. This tendency leads to unintuitive results as follows. Even though the objective is to maximize the max (min) of the performances of n contestants, the bottom punishment (top reward) can be better. Finally, we show that irrespective of the objectives of the designer, the punishment becomes better when the abilities of team members are improved significantly. Based on the abovementioned three findings, the appropriate usage of reward and punishment incentives is summarized as follows. First, punishment is better when the objective of implementing an incentive is that everyone is required to put out a certain level of effort or result (weakest link), while reward is better when it is sufficient if some individuals produce excellent results (best shot). Thus, a penalty-type incentive is suitable to enforce regulations or maintain order while a reward-type incentive is suitable for such activities as research and art. Furthermore, the capability gap within the group that is subject to the incentive is important. Generally speaking, the reward type works better for a group of individuals who have not been screened specially because the gap is probably large. By contrast, the punishment type could be effective for a group consisting of screened individuals. In terms of relationship to task, the punishment type is better for the kind of task that can be performed by anyone because the capability gap would probably be small, while the reward type is better for the kind of task that can be performed only by some people. It is interesting that this implication conforms with the conclusion based on the perspective of justice (i.e., it is wrong to motivate people by punishment when it is known there are individuals who cannot fulfill the requirement; see De Geest and Dari-Mattiacci, 2013). Furthermore, the implication is in accordance with the conclusion by Wittman (1984) regarding the cost of implementing sanctions, that the incentive should be designed to minimize the number of individuals on whom the sanction is imposed.

Y. Kamijo / Journal of Economic Behavior & Organization 122 (2016) 17–30

27

We present a variety of guidelines for the proper use of the carrot and stick; however, to some extent, these results should be treated with caution. First, our sufficient conditions are derived from the assumption that P = R. While the general / R, we cannot derive clear sufficient conditions to check whether tendency of our findings still holds, even in the case of P = reward or punishment is better. In such a case, we have to compute Ep and Er by numerical calculation and compare them. Second, as has been demonstrated in such fields as behavioral economics, people tend to become so afraid of punishment that they put too much effort in attempting to avoid punishment. This not only suggests that our results understate the effectiveness of the punishment system but also implies that it is extremely difficult to predict the reactions of people toward punishment (Trevino, 1992). Therefore, opportunities to use punishment may naturally be limited. Third, our model ignores the intrinsic motivation that people have for behavior. It is necessary to consider that extrinsic motivation, such as the carrot and stick, often ruins intrinsic motivation.6 Building a theory of sanctions incorporating these behavioral factors is a future topic. 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). Appendix A. Proof of Theorem 1 The statements (i) to (iv) are obvious from Eqs. (1) and (2). We prove (v) of the theorem. When P = R,

 ˇr (H ) − ˇp (H )



H

= R(n − 1)



H

=R



n−2

(F(z))



− (1 − F(z))n−2 zf (z)dz

L

(F(z))

n−1



+ (1 − F(z))n−1 zdz

L

=R



(F(z))

n−1

 H

+ (1 − F(z))n−1 z



H

= R(H − L ) − R





LH

H

1dz − R

=R

LH =R



H



(F(z))

L

+ (1 − F(z))n−1 dz

n−1

+ (1 − F(z))n−1 dz

(F(z))

n−1



+ (1 − F(z))n−1 dz



n−1

(F(z))



L

 −R



L



1 − (F(z))n−1 − (1 − F(z))n−1 dz > 0.

L

This means that for  close to  H , ˇr () > ˇp (). Proof of Theorem 2 (Additive performance function) From Section 4.1, we know



H

p

E = (n − 1)

(1 − F(z))n−1 zf (z)dz

L

and



H

r

E = (n − 1)

(1 − F(z))(F(z))n−2 zf (z)dz.

L

Let x = F(z), and we obtain



p

E = (n − 1)

(1 − x)(1 − x) 0

6



1 n−2 −1

F

(x)dx

and

1

(1 − x)xn−2 F −1 (x)dx.

r

E = (n − 1) 0

Discussions about an appropriate combination of intrinsic and extrinsic motivation are far beyond the objectives of this study.

28

Y. Kamijo / Journal of Economic Behavior & Organization 122 (2016) 17–30

Thus, we obtain Ep − Er



= (n − 1)

1





(1 − x)n−2 − xn−2 (1 − x)F −1 (x)dx

0 1/2









(1 − x)n−2 − xn−2 (1 − x)F −1 (x)dx

= (n − 1)

 01/2

+(n − 1)

(1 − x)n−2 − xn−2 xF −1 (1 − x)dx

0

1/2



(1 − x)n−2 − xn−2

= (n − 1)





(1 − x)F −1 (x) − xF −1 (1 − x) dx

. . .(A1)

0

Therefore, when (1 − x)F−1 (x) − xF−1 (1 − x)  0, Ep − Er  0. Thus, the proof of (i) ends. From this, (ii) and (iii) are proved immediately. Proof of Theorem 3 (Weakest-link performance function) The expected performance of the organization for punishment and reward under a weakest-link performance function are

 p Ew

p



H

p

= E[min{ˇ (1 ), . . ., ˇ (n )}] =

ˇ

p

()gnn ()d

H

= L

and

 r



H

r

= E[min{ˇ (1 ), . . ., ˇ (n )}] =

ˇ

r

()gnn ()d



 n−1 gn−1 (z)zdz

L

r Ew



H

0







g1n−1 (z)zdz

=

L

gnn ()d

gnn ()d,

0

L

respectively. By interchanging the order of integration and replacing z with F−1 (x) in a similar manner to the proof of Theorem 2, we obtain



1

(1 − x)n (1 − x)n−2 F −1 (x)dx,

p

Ew = (n − 1) 0



1

(1 − x)n xn−2 F −1 (x)dx.

r Ew = (n − 1) 0

Thus,

 p Ew

r − Ew

= (n − 1)

1





(1 − x)n−2 − xn−2 (1 − x)n F −1 (x)dx

0 1/2









(1 − x)n−2 − xn−2 (1 − x)n F −1 (x)dx

= (n − 1)

 01/2

+(n − 1)

0

(1 − x)n−2 − xn−2 xn F −1 (1 − x)dx

1/2



(1 − x)n−2 − xn−2

= (n − 1)





(1 − x)n F −1 (x) − xn F −1 (1 − x) dx

. . .(A2)

0

p

r Thus, when (1 − x)n F−1 (x) − xn F−1 (1 − x)  0 for all x ∈ (0, 1/2) and the strict inequality holds for some x ∈ (0, 1/2), Ew − Ew becomes positive. Thus, the proof of (i) ends. From this, (ii) is proved immediately.

Proof of Theorem 4 (Best-shot performance function) The expected performance of the organization for punishment and reward under a best-shot performance function are

 p Eb

p



H

p

= E[max{ˇ (1 ), . . ., ˇ (n )}] =

ˇ L

p

()g1n d

H





 n−1 gn−1 (z)zdz

= L

0

g1n ()d

Y. Kamijo / Journal of Economic Behavior & Organization 122 (2016) 17–30

and

 Ebr

r



H

r

= E[max{ˇ (1 ), . . ., ˇ (n )}] =

ˇ

r

()g1n d

L

H







g1n−1 (z)zdz

= L

29

g1n ()d,

0

respectively. By interchanging the order of integration and replacing z with F−1 (x) in a similar manner to the proof of Theorem 2, we obtain

 p Eb

= (n − 1)

1

01

(1 − xn )(1 − x)n−2 F −1 (x)dx, (1 − xn )xn−2 F −1 (x)dx.

Ebr = (n − 1) 0

Thus, we obtain p Eb

p − Eb

 = (n − 1)

1





(1 − x)n−2 − xn−2 (1 − xn )F −1 (x)dx

0 1/2









(1 − x)n−2 − xn−2 (1 − xn )F −1 (x)dx

= (n − 1)

 01/2

+(n − 1)

0

(1 − x)n−2 − xn−2 (1 − (1 − x)n )F −1 (1 − x)dx

1/2



(1 − x)n−2 − xn−2

= (n − 1)





(1 − xn )F −1 (x) − (1 − (1 − x)n )F −1 (1 − x) dx

. . .(A3)

0

Therefore, when (1 − xn )F−1 (x) − (1 − (1 − x)n )F−1 (1 − x)  0 for all x ∈ (0, 1/2) and the strict inequality holds for some x ∈ (0, p 1/2), Eb − Ebr becomes positive. Thus, the proof of (i) ends. From this, (ii) is proved immediately. Proof of Theorem 5 −1 Note that F+A (x) = F −1 (x) + A for any x. From (A1), Ep − Er under F+A is reduced to



1/2



(1 − x)n−2 − xn−2

(n − 1) 0





(1 − x)F −1 (x) − xF −1 (1 − x) dx



1/2





(1 − x)n−2 − xn−2 [1 − 2x] dx.

+ A(n − 1) 0

The first term is a finite number and the second term is always positive. Thus, for extremely large A, the equation above becomes positive. p p r under F Similarly, by using (A2) and (A3), we can show easily that Eb − Ebr and Ew − Ew +A becomes positive when A is sufficiently large. References Akerlof, R.J., Holden, R.T., 2012. The nature of tournaments. Econ. Theory 51, 289–313. Andreoni, J., Gee, L.K., 2012. Gun for hire: delegated enforcement and peer punishment in public goods provision. J. Public Econ. 96 (11), 1036–1046. Barut, Y., Kovenock, D., 1998. The symmetric multiple prize all-pay auction with complete information. Eur. J. Polit. Econ. 14, 627–644. Baye, M.R., Kovenock, D., de Vries, C.G., 1993. Rigging the lobbying process: an application of the all-pay auction. Am. Econ. Rev. 83 (1), 289–294. Brookins, P., Ryvkin, D., 2014. An experimental study of bidding in contests of incomplete information. Exp. Econ. 17 (2), 245–261. Buckley, E., Croson, R., 2006. Income and wealth heterogeneity in the voluntary provision of linear public. J. Public Econ. 90, 935–955. Chan, K.S., Mestelman, S., Moir, R., Muller, R.A., 1996. The voluntary provision of public goods under varying income distributions. Can. J. Econ. 29 (1), 54–69. De Geest, G., Dari-Mattiacci, G., 2013. The rise of carrots and the decline of sticks. Univ. Chicago Law Rev. 80, 341–392. Dickinson, D.L., 2001. The carrot vs. the stick in work team motivation. Exp. Econ. 4 (1), 107–124. Fellner, G., Iida, Y., Kröger, S., Seki, E., 2011. Heterogeneous productivity in voluntary public good provision: An experimental analysis, IZA Discussion Papers 5556. Fisher, J., Issac, R.M., Schatzberg, J.W., Walker, J.M., 1995. Heterogenous demand for public goods: behavior in the voluntary contributions mechanism. Public Choice 85, 249–266. Glazer, A., Hassin, R., 1988. Optimal contests. Econ. Inq. 26 (1), 133–143. Gradstein, M., 1995. Intensity of competition, entry and entry deterrence in rent seeking contests. Econ. Polit. 7 (1), 79–91. Green, J.R., Stokey, N.L., 1983. A comparison of tournaments and contracts. J. Polit. Econ. 91 (3), 349–364. Hirshleifer, J., 1983. From weakest-link to best-shot: the voluntary provision of public goods. Public Choice 41 (3), 371–386. Kamijo, Y., Nihonsugi, T., Takeuchi, A., Funaki, Y., 2014. Sustaining cooperation in social dilemmas: comparison of centralized punishment institutions. Games Econ. Behav. 84, 180–195. Kremer, M., 1993. The O-ring theory of economic development. Q. J. Econ. 108, 551–575. Lazear, E.P., Rosen, S., 1981. Rank-order tournaments as optimum labor contracts. J. Polit. Econ. 89 (5), 841–864. Moldovanu, B., Sela, A., 2001. The optimal allocation of prizes in contests. Am. Econ. Rev. 91 (3), 542–558. Moldovanu, B., Sela, A., Shi, X., 2012. Carrots and sticks: prizes and punishments in contests. Econ. Inq. 50 (2), 453–462. Nalebuff, B., Stiglitz, J., 1983. Prizes and incentives: towards a general theory of compensation and competition. Bell J. Econ. 14 (1), 21–43.

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