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Relative deprivation in rank-order tournaments

Matthias Krakel

¨

)

Department of Economics, BWL II, UniÕersity of Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany

Received 19 January 1998; received in revised form 19 April 1999; accepted 10 January 2000

Abstract

Ž .

This paper examines the influence of relative deprivation RD , based on net income, on the strategy choice of workers in tournaments. The results show that for given tournament prizes, workers who experience RD exert more effort than workers who maximize their expected absolute incomes. These findings hold for productive effort as well as for

Ž .

counterproductive effort sabotage . In addition, the paper discusses various implications that arise when the employer can choose between different compensation schemes in the tournament.q_{2000 Elsevier Science B.V. All rights reserved.}

JEL classification: C72; J31; J33; J41

Keywords: Personnel policies; Rank-order tournaments; Relative deprivation; Sabotage

1. Introduction

Rank-order tournaments in organizations have already been discussed in many publications.1 _{The standard assumption in these analyses is that workers care only}

Ž .

about their own incomes. As an alternative Stark 1987, 1990 introduces the

Ž .

concept of relative deprivation RD in the discussion of tournaments. A person is relatively deprived when his income is lower than the income of a chosen reference group. RD will be great if there is a large income difference between the individual and the reference group and_ror the reference group is of considerable

)_{Tel.:q49-228-73-3914; fax:q49-228-73-9210.}

Ž .

E-mail address: m.kraekel@uni-bonn.de M. Krakel .¨

1

Ž . Ž .

See, for example, Lazear and Rosen 1981 , Green and Stokey 1983 , Nalebuff and Stiglitz

Ž1983 , O’Keefe et al. 1984 , Rosen 1986 , and McLaughlin 1988 .. Ž . Ž . Ž .

0927-5371r00r$ - see front matterq2000 Elsevier Science B.V. All rights reserved.

Ž .

size. In this context, RD means that a worker exerts effort to minimize the relative income difference between him or her and richer workers. The combination of the RD concept with rank-order tournaments is somewhat obvious: Tournaments are used by the employer as an incentive and sorting mechanism. On the other hand,

Ž .

tournaments also induce workers to compare themselves especially their incomes to each other and there are two groups of workers — the losers, who may feel relatively deprived, and the winners, who may form the relevant reference group for relatively deprived losers.

Ž . Ž .

This paper contains several extensions to the work of Stark 1987, 1990 : 1 The implications of RD, based on net income, for the workers’ effort choices are

Ž .

explicitly modeled. 2 This allows a direct comparison between the equilibrium efforts of RD workers and of those workers who maximize their absolute net

Ž . Ž .

incomes non-RD workers . 3 Workers decide under uncertainty; they want to Ž .

minimize their expected RD. 4 RD is also of great consequence for Lazear’s Ž1989 sabotage model..

In Section 2 a simple tournament model is described. This model is similar to

Ž . Ž

the benchmark model of Lazear and Rosen 1981 e.g., the individual output .

function, the i.i.d.-assumption , in order to examine the implications of RD for the standard tournament results. But there is also a difference from the approach

Ž .

chosen by Lazear and Rosen 1981 . Here, I assume that tournament prizes are given exogenously.2 _{Therefore, the approach taken here only represents a} partial-equilibrium exercise. I shall concentrate on the questions of how the behavior of RD workers differs from that of non-RD workers and whether for exogenously fixed prizes the employer prefers a homogeneous workforce of RD workers to a homogeneous workforce of non-RD workers. Questions concerning possible impli-cations for equilibrium wages and equilibrium distributions of workers across firms in the labor market are only briefly discussed.

Section 3 contains four propositions, which show the different behavior of RD and non-RD workers in tournaments. In this section and for the rest of the paper I will solely discuss symmetric Nash equilibria where each worker chooses the same effort.3

2

For example, wages are institutionally fixed by a collective agreement between an industry-wide union and the employers’ association. This assumption does hold to some extent for the German labor market. As an alternative, we can assume competitive markets that imply a zero-profit condition for the employer to derive optimal tournament prizes endogenously. As another alternative, we can assume that the employer has all the bargaining power and the workers have a given reservation utility. This approach is often used in principal agent models and is sketched here in the discussion of Proposition 1 and in Appendix B.

3

Intuitively, it makes sense to concentrate on symmetric Nash equilibria because each worker has to solve exactly the same decision problem under symmetric uncertainty. Therefore, most of the

Ž .

tournament literature focuses on symmetric solutions; see, for example, Lazear and Rosen 1981, 845 ,

Ž . Ž .

Section 4 shows that RD has consequences, not only for the tournament benchmark model, but also for variants of the basic model. Here, the idea of

Ž .

sabotage in tournaments see Lazear, 1989 is considered as an example.

2. Description of the model and notation

In this paper, I consider a rank-order tournament among n workers. Worker i Ži_s1, . . . , n produces an individual output Q according to Q. _se_q´_{. Here, e}

i i i i i

Ž describes the effort worker i chooses and ´ _{a random or luck component e.g.,}

i

.

activity-specific measurement error, individual ability . There is symmetric uncer-tainty for the employer and the workers, i.e. ´ _{is unknown to the employer and}

i

each of the workers. Any individual only knows that the ´ _{are identically and}

i

w x Ž .

independently distributed over 0,´ _{with cumulative distribution function F} ´

Ž . X

Ž . Ž .

and density function f ´ _sF ´ _{i.i.d.-assumption . The employer can only} observe Q but not e and ´_{, so a standard hidden action problem is considered.}

i i i

Workers are compensated according to the ranks of their outputs. I assume that the employer can choose one of two compensation schemes or tournament prize

Ž .

structures: i The worker with the highest output receives a large bonus or prize

Ž . Ž .

B , whereas each of the other n_{1 y}1 workers only gets B₂ -B₁ scheme s_s1 ;

Ž .ii each of the best n_y1 workers gets B and the worst worker receives B_{1 2} Ž-B₁. Žscheme s_s2 . The schemes 1 and 2 describe two polar cases of worker. compensation plans. In the following, the two schemes will be labeled on the basis of the number of large prizes. Therefore, scheme s_s1 is referred to as the

scheme with a single prize, and scheme s_s2 as the scheme with n_y1 prizes.4

So there is more total compensation paid out under the scheme with n_y1 prizes. Ž .

It is assumed that effort entails costs for the workers. c ei denotes these costs for X

Ž . Y

Ž . each worker with c ei )0 and c ei )0.

Two different cases are considered in Section 3: On the one hand, the general Ž .

case is discussed in which the probability distribution F ´ _{is not specified} 5

w x

further. Later, I assume that the ´ _{are uniformly distributed over 0,}´ _{. In the}

i

)s _Ž )s_.

following, enon-RD eRD denotes the Nash equilibrium effort of a non-RD worker ŽRD worker , when the employer chooses compensation scheme s s. Ž s1,2 . All. workers are assumed to be risk neutral. I will exclusively discuss symmetric Nash

4

As an alternative, the first scheme can be characterized as bonus scheme, because only the most successful worker is given an additional bonus DB[B1yB , and the second scheme as penalty2

scheme, because the worst worker receives a tournament prize which is lower than all the other prizes

by the amountDB.

5

It is well known that calculating with order statistics implies some difficulties. Thus, the simplifying assumption of uniformly distributed random components is not unusual. Additionally, the assumption of uniformly distributed random components can be motivated by the principle of

Ž .

equilibria and homogeneous tournaments, in which all workers are characterized

Ž )s _{. Ž .}

by non-RD or RD, respectively. Let EU ei non-RD denote worker i’s is1, . . . , n

Ž .

expected utility i.e., his expected income net of effort costs in equilibrium for a

Ž .

homogeneous non-RD tournament and compensation scheme s s_s1,2 , and Ž )s_.

ERD e_{i RD} worker i’s expected RD in equilibrium for a homogeneous RD tournament and compensation scheme s. Expected RD is calculated as net income difference between the winner of the tournament and worker i, times the relative

Ž Ž . .

quantity of tournament winners i.e., 1_rn for s_s1, and n_y1_rn for s_s2 , times the probability of losing the tournament.

3. Results

Comparing the workers’ equilibrium efforts for non-RD tournaments with those for RD tournaments, the following result can be derived for symmetric Nash equilibria and exogenous tournament prizes:

Proposition 1. e)s₎

e)s

for ss1,2.

R D n o n - R D

Proof. See Appendix A.B

In words, RD workers work harder than non-RD workers under both incentive

Ž .

schemes. From the discussion of standard non-RD tournaments we know various reasons for why workers choose high effort levels: there is a large prize spread D_{B, workers have low marginal costs of effort, luck plays only a minor role in the}

Ž

tournament, or workers self-sort in heterogeneous contests see Lazear and Rosen, .

1981 . Proposition 1 shows that there is another possible reason for high effort levels, namely the existence of RD workers. Comparing this result with the findings of the experimental literature, we see that on average predicted behavior of non-RD workers is generally supported by the experiments, but in single — even in symmetric — tournaments subjects tend to supply more effort than

Ž .

predicted see Bull et al., 1987; Schotter and Weigelt, 1992 . RD may be one possible argument for explaining this tendency.

The proposition has also an implication for recruiting. In reality, tournament prizes are not exogenous but chosen optimally by the employer. So if there exist both non-RD and RD workers in the labor market, it will be profitable for the employer to search for RD workers because for a given compensation scheme RD workers exert higher efforts than non-RD workers, or the other way round, for a given effort level the employer’s labor costs are less for RD workers than for non-RD workers. Thus, if the employer can choose between two types of homogeneous workforces she will always prefer a homogeneous workforce of RD

Ž

workers this general conclusion need not hold in the case of a principal agent .

this recruiting policy the employer has to identify correctly a worker’s type. To solve this problem the employer can use a self-selection device. Assume that each

Ž .

worker has private information about his type RD or non-RD . Perhaps, the employer is able to choose a proper pair of tournaments with different prize structures so that the RD workers prefer to participate in one tournament whereas the non-RD workers prefer participating in the other tournament.6

Additionally, we can speculate what might happen in a market equilibrium. When there are many employers who compete for a limited group of RD workers, RD workers’ wages will be bid up to a higher level. Now it can become attractive for employers to hire non-RD workers or a mixture of RD and non-RD workers. In market equilibrium there will be different pairs of tournament prizes and effort levels for non-RD workers and RD workers, respectively, so that no employer and no worker will have an incentive to revise his decision.

Finally, consider the case of mixed firms with heterogeneous workforces consisting of both non-RD and RD workers. Proposition 1 indicates that in this case the employer may have an incentive to create two different tournaments, one for the non-RD workers and the other for the RD workers. Otherwise, work incentives may be distorted because of heterogeneity.

Proposition 2. With exogenous tournament prizes and uniformly distributed ´ _,

i

( )1_{) (} )2_{) (} )s₎ )1

ERD ei R D sERD ei R D withEERD ei R D rEn-0 for ss1,2, butEeR DrEn-0 whereasE_e)2

rE_n)0.

R D

Proof. See Appendix A.B

Proposition 2 shows that as the number of competing workers increases, the equilibrium effort of RD workers rises under the scheme with n_y1 prizes, but falls under the scheme with a single prize. On the other hand, expected RD is identical for each RD worker under either incentive scheme. This result can be explained by the fact that expected RD is composed differently in the two schemes. Under the scheme with a single prize there is a high probability of

Ž .

becoming relatively deprived RD-probability; i.e., probability of losing , but only

Ž .

a low amount of RD in the case of losing the tournament RD-amount . Under the scheme with ny1 prizes, however, there is a low RD-probability, but a high RD-amount. Under the scheme with a single prize the RD-probability amounts to Žny1.rn, which is increasing in n, whereas under the scheme with ny1 prizes

6

Note that RD workers do not care about the absolute values for B and B and that expected1 2

relative deprivation is increasing inDB. Therefore, RD workers prefer a tournament with a low prize

spread DB. On the other hand, non-RD workers’ expected utility depends on DB as well as on B .2

the RD-probability is only 1rn and is decreasing in n. Intuitively, we can imagine that when competition becomes very strong, under the scheme with a single prize workers have no realistic chance of winning the tournament.7 Therefore, they do not even try to win the tournament and exert only little effort to avoid high costs

Ž .

c e . Moreover, the consequences of losing the tournament are negligible because_i Ž .

of the low RD-amount, which further decreases as c e_i becomes small. Under the scheme with n_y1 prizes, however, the chance of winning is very high when n is high and there is a high RD-amount in the case of losing the tournament. In this situation for each worker it pays to exert considerable effort.

Finally, we can contrast this intuitive interpretation with the theoretical and empirical evidence for non-RD workers. The proofs of Propositions 1 and 2 show that, for uniformly distributed luck, the equilibrium efforts of non-RD workers are independent of n.8 _{Therefore, non-RD workers’ equilibrium efforts do not vary} with increased competition n or increased number n_y1 of large prizes B in1

Ž .

scheme s_s2, respectively. Orrison et al. 1997 tested the effect of an increased number of large prizes in several experiments. Interestingly, their results show that effort significantly decreases when the number of large prizes increases. The authors suppose that the tested persons were subject to a decision anomaly,

Ž

because they incorrectly chose their efforts on the basis of the total and not of the .

marginal probability of winning. This probability rises when the number of large prizes increases and hence it seems plausible not to waste effort. Compared to the theoretical result for RD workers and the scheme with n_y1 prizes the experimen-tal findings become even more puzzling. Perhaps, there were no RD persons tested in the experiments, or the RD effect was dominated by the decision anomaly.9

Proposition 3. If the tournament prizes are exogenously giÕen and the ´ are

i

uniformly distributed, e)1_-e)2 _{and e})1

se)2 _{. In the general case,}

R D R D n o n - R D n o n - R D

e)1

Ge)2 _{whereas there is no clear ranking between e})1 _{and e})2_.

n o n - R D n o n - R D R D R D

Proof. See Appendix A.B

Proposition 3 means that with uniformly distributed luck RD workers work harder under the scheme with n_y1 prizes than under the scheme with a single

7

Ž . Ž .

But note that the marginal probability of losing is fY 0sfX 0s1r´ in equilibrium, which is independent of n. Therefore, the marginal probability of winning is independent of n, too. So if competition becomes stronger the effect of additional effort on tournament outcomes will remain the

Ž .

same. See also Orrison et al. 1997, 9 .

8

This follows from the symmetry of the equilibrium and the characteristics of the uniform distribution.

9

Another possible explanation may be that the subjects in the experiments do not calculate with uniformly distributed luck. If we compare the non-RD workers’ equilibrium efforts under both incentive schemes, we will see that for the case where luck is not distributed uniformly workers’ efforts are greater under the scheme with a single prize than under the scheme with ny1 prizes, because then

Ž . Ž .

prize, whereas non-RD workers exert the same effort under both schemes. On the other hand, without a restriction on the luck distribution there is no clear ranking between the efforts of the RD workers, whereas the non-RD workers work at least as hard under the scheme with a single prize than under the scheme with ny1 prizes. An explanation for the results with uniformly distributed luck follows immediately from the discussion of Proposition 2: there are higher work incentives for RD workers under the scheme with n_y1 prizes than under the scheme with a single prize, because workers are faced with a higher RD-amount when there are

Ž

n_y1 large prizes i.e., in the case of losing the tournament there are more better . paid workers under the scheme with n_y1 prizes than under the other scheme . For non-RD workers, however, work incentives are identical under both schemes, because the effort choices are independent of the number of large prizes, which follows from the symmetric equilibrium and the special form of the uniform distribution. In the general case, the RD workers’ efforts cannot be ranked clearly. The proof of Proposition 3 shows that each worker’s effort is increasing in the quotient of the marginal and the total probability of losing under both incentive schemes. There is no clear ranking, because both the marginal and the total

Ž .

probability of losing are larger or equal under the scheme with a single prize

Ž Ž . Ž . Ž . Ž .

than under the scheme with ny1 prizes i.e., fY 0 GfX 0 and F 0Y )F 0 ;X

.

see Appendix A . The non-RD workers’ efforts only depend on the marginal

Ž .

probability of losing under both schemes and, of course, on D_{B . As effort} increases in the marginal probability in both cases, non-RD workers choose greater or equal efforts under the scheme with a single prize than under the scheme with n_y1 prizes. To sum up, Proposition 3 emphasizes the different behavior of RD and non-RD workers in tournaments: intuitively, for RD workers the scheme with n_y1 prizes is more competitive because of the higher RD-amount, whereas for non-RD workers the other scheme is more competitive, because there is only one large winner prize for the most successful worker.

Proposition 4. For giÕen tournament prizes B and B , the compensation scheme_{1 2}

s_s1 dominates the scheme s_s2 from the employer’sÕiewpoint, when non-RD

holds for the workers. In the case of RD there is no clear ranking between the two schemes for the employer.

Proof. See Appendix A.B

prize than under the scheme with ny1 prizes. Therefore, the employer prefers the incentive scheme with a single prize. This result about non-RD workers replicates

Ž .

the one of Corollary 2 in Glazer and Hassin 1988, 142 . Additionally, for the non-RD case the result sheds some light on the question of whether it is better for the employer to use a carrot or a stick for worker motivation. In a sense, the scheme with a single prize is a kind of carrot scheme, because the most successful worker is the only one that is rewarded with a large prize. Under the scheme with n_y1 prizes, however, the worst worker is punished in the sense that only he gets a low prize. This incentive scheme can be characterized as a kind of stick scheme. Hence, Proposition 4 shows that for the employer a carrot is better than a stick.10 But note that this comparative result is not universally valid, because in this model total compensation costs are not identical under both incentive schemes. With RD workers no clear results can be derived. For example, if luck is uniformly distributed, the scheme with a single prize will induce a lower equilibrium effort at lower labor costs than the scheme with n_y1 prizes.

In this paper, I focus on two extreme prize structures. Additionally, we can take a look on intermediate prize structures consisting of some small and some large prizes.11 To begin with, the case of non-RD workers is considered. With uni-formly distributed luck, the workers’ equilibrium efforts will remain the same for any intermediate prize structure, because only for this distribution the density is a constant and thereby the marginal probability of losing always amounts to 1_r´_,

Xy1

) _{Ž . Ž .}

which implies: e _sc D_Br´ _{see Appendix A . For any other luck} non-RD

distribution, equilibrium efforts change depending on the chosen intermediate prize structure. The proof of Proposition 3 shows that for any luck distribution that

Ž . Ž . Ž . Ž Ž ..

is not the uniform distribution f_Y 0 )f_X 0 , as f_Y 0 f_X 0 represents the mean

Ž . Ž .

of the function f P of the highest lowest of n_y1 order statistics. If we have an intermediate prize structure, the more large prizes B the employer chooses, the1 lower will be the rank of the order statistic that is relevant for the equilibrium

Ž .

effort. For example, if there are m 1-m-n_y1 large prizes B a non-RD1

Ž . w Ž ) _.x

worker’s objective function will change to EU e _sB _qD_{B 1yF e ye y}

i i 2 Z i

Ž . Ž ) _. ) ₄

c e with F e _ye _sPr Z-e _ye , Z_s´ _y´ _and ´

i Z i i i Žnym.;Žny1. Žnym.;Žny1.

Ž . Ž . Ž

as the nym th of ny1 order statistics i.e., ´_{Žny1.;Žny1.})´_{Žny2.;Žny1.}) .

. . .)´ ) . . .)´ _{. Thus, a worker’s equilibrium effort can be}

Žnym.;Žny1. Ž1.;Žny1.

) X_y1_{Ž Ž .. Ž .} X_{Ž .}

described by enon-RDsc D_{Bf 0}Z _{with f 0}Z [F 0 as the mean of theZ

Ž . Ž .

function f P of the nym th of ny1 order statistics. If the employer increases

Ž . )

m, the number of large prizes, f 0 will fall and thereby eZ non-RD, too. Altogether,

10 _{Ž .}

Cf. in another context Varian 1990, 155–157 , but in the context of tournaments also Nalebuff

Ž .

and Stiglitz 1983, 33–34 .

11

For theoretical and experimental results about such intermediate cases for non-RD workers see

Ž .

a prize structure with many large prizes has two consequences for the employer: Workers’ equilibrium efforts decrease and labor costs increase. From the em-ployer’s viewpoint the optimal prize structure is the scheme ss1 with only one large prize, which will be chosen by the employer even if she is able to choose an intermediate prize structure.

Now, we have to consider the RD case. For uniformly distributed luck and an intermediate prize structure with m large prizes, in equilibrium a worker exerts

Xy1 Xy1

) _{Ž Ž . Ž .. Žw Ž .x wŽ . x.}

effort e _sc D_{B f 0rF 0 sc} D_{B 1r}´ _{r nymrn . Therefore,}

RD Z Z

the greater m is, the greater e) _{is, too. But labor costs also increase with m.} RD

Thus, there is a trade-off for the employer. In the general case, equilibrium effort ) X_y1 _Ž for an intermediate prize structure can be described by e _sc D_B

RD

Ž . Ž ..

f 0_rF 0 . When the´ _{are not uniformly distributed no clear statement can be}

Z Z i

Ž . Ž .

made, because both f 0 and F 0 decrease with an increasing m.Z Z

4. Sabotage in rank-order tournaments

The results of Section 3 show that non-RD and RD workers behave rather differently in the benchmark tournament model. In this section, I will sketch that the introduction of RD leads also to consequences in variants of the basic tournament model. As an example, the problem of sabotage in tournaments is

Ž . Ž .

considered cf. Lazear, 1989 . Lazear 1989 assumes that the workers have to decide about two kinds of effort variables — productive effort, which raises the

Ž .

individual output of the worker, and counterproductive effort ‘‘sabotage’’ , which reduces the individual outputs of other workers. Both kinds of effort can be used by a worker to reach a better rank in the tournament. If the probability of being detected and punished as a saboteur is small and the costs for sabotage are lower than the costs of exerting productive effort, the employer will face a serious sabotage problem. There is a close connection between the discussion of sabotage in tournaments and the problem of mobbing_rbullying at the workplace. During the last years, many companies have cut back their hierarchies so that better paid positions at higher stages of the hierarchy become a scarce resource for employ-ees. This may induce employees to engage more in winning promotion tourna-ments by counterproductive actions like sabotage or bullying. Lazear’s sabotage model recommends a personnel policy that assigns lower bonus or prize differ-ences for tournaments with sabotage than for those without the possibility of counterproductive actions.

The model in Section 2 can be supplemented by a second kind of effort variable, too, that stands for the amount of sabotage each worker chooses. Analogously to Lazear’s sabotage model individual output can now be described

by Q_se _q´ _{yÝ e . Here, e denotes worker i’s productive effort, which}

i i1 i j/i j i2 i1

Ž .

raises his output. The variable e_{j i} stands for worker j’s j_s1, . . . , n; j/i

2

)s _Ž )s_.

sabotage effort against worker i. Let e_non-RD e_RD denote the productive effort

1 1

Ž .

Ž . )s _Ž )s _{. Ž}

ss1,2 and enon-RD2 eRD2 the sabotage effort of a non-RD worker RD

. Ž . Ž .

worker . At last, the cost function cP has to be modified. Let now c e1 i1 denote

Ž .

worker i’s costs for productive effort and c2 Ýj/iei j2 his costs for sabotage with

X_{Ž .} Y_{Ž .} X_{Ž .} Y_{Ž .}

c₁P )0, c₁ P )0, c₂P )0 and c₂ P )0. For this variant of the benchmark model the following result can be derived for symmetric Nash equilibria and exogenous tournament prizes:

Proposition 5. e)s ₎

e)s

and e)s ₎

e)s

for ss1,2.

R D1 n o n - R D1 R D2 n o n - R D2

Proof. See Appendix A.B

According to Proposition 5, RD workers choose greater productive efforts and greater sabotage efforts than non-RD workers for both incentive schemes. The proof of Proposition 5 shows that for both schemes in equilibrium the sabotage effort depends on the marginal cost of sabotage, rises with increasing bonus

Ž

difference D_{B, rises with decreasing influence of luck i.e. with decreasing}

.12 _Ž

variance of Y and X, respectively , and rises with decreasing competition i.e. .

with decreasing n . These findings have some implications for an employer’s personnel policies. First, she can compress worker’s rewards, which has already

Ž .

been suggested by Lazear 1989 . Second, she can directly influence the workers’

Ž .

expected marginal costs for counterproductive effort by the threat of punishment. Third, she can increase the number of contestants, n, which makes sabotage more

Ž . Ž

expensive under the scheme with a single prize or less effective under the .

scheme with n_y1 prizes for the workers. At last, she can lower her monitoring precision to raise the variance of Y and X, respectively. Of course, these personnel policies are not costless for the employer; they directly or indirectly entail additional costs.13

The proof of Proposition 5 also shows that sabotage behavior is rather different under the two incentive schemes. We can suppose that the total amount of sabotage from the n workers also differs for the two schemes. Proposition 4 has shown that in the case of non-RD workers the scheme with a single prize is preferred from the employer’s viewpoint considering labor costs and productive efforts. From the sabotage model with exogenous tournament prizes another aspect arises for the employer when choosing her wage policy s:

Proposition 6. For non-RD workers under ss1 the total sabotage from the n

workers is at least as great as under ss2. For RD workers the same holds, if

( ) ( )

f_Y 0 _rf_X 0 _Gn_y1.

12

The less Y and X vary the more Y ’s and X ’s probability mass concentrates near the origin, i.e.

Ž . Ž . Ž .

the greater are fY 0 and fX0 , respectively. See also Lazear 1995, 29 .

13

Proof. See Appendix A.B

From Proposition 6 we see that in the case of non-RD workers the total sabotage is greater or equal under the scheme with a single prize than under the scheme with n_y1 prizes. Therefore, the employer will choose the scheme with n_y1 prizes, if she wants to minimize sabotage. With an additional assumption the same result also holds for RD workers. But the more the employer increases competition to lower the workers’ sabotage efforts the more likely the opposite holds. If luck is uniformly distributed, the opposite even holds for nearly any

14

Ž . Ž . Ž .

nontrivial tournament i.e. for n_G3 , because f 0 _sf 0 _s1_r´_.

Y X

Two additional results can be derived by specifying the workers cost functions

Ž . Ž .

c1P and c2 P. For the rest of the paper, I assume that the workers have quadratic

Ž . 2 _{Ž . Ž .}2

cost functions: c e1 i1 _sk1Pe and ci1 2Ýj/iei j2 _sk2P Ýj/iei j2 with k , k1 2)0. Now, a critical reflection on the implication of Proposition 1 is possible. Proposi-tion 1 shows that for a given compensaProposi-tion scheme and given tournament prizes RD workers work harder than non-RD workers, which implies for the employer that she should prefer RD over non-RD workers in recruiting decisions. But from Proposition 5 we know that RD workers also choose greater sabotage efforts than non-RD workers. Therefore, we have to check the employer’s preferences once

Ž .

again. For this purpose, the following result can be derived at which E QiNs,t

Ž .

will denote worker i’s expected output, if he is of type t t_snon-RD, RD and

Ž .

the employer chooses compensation scheme s s_s1,2 :

( )

Proposition 7. If n_y1 k )k , for giÕen tournament prizes and quadratic cost

2 1

( ) ( )

functions: E Qi_Ns,RD )E Qi_Ns,non-RD for s_s1,2.

Proof. See Appendix A.B

Proposition 7 means that the expected output of RD workers will exceed non-RD workers’ expected output under both compensation schemes, if the number of competing workers is sufficiently large or sabotage effort is sufficiently costly compared to productive effort. Then, the employer always prefers RD workers when recruiting, no matter which compensation scheme she chooses. The

Ž .

result of Proposition 7 will be interesting, if the inequality ny1 k2)k1 is analyzed in its contents. This inequality ensures that in the long run the firm is able to survive, because productive effort dominates counterproductive effort of

Ž . 15

any RD and non-RD worker. So if the employer can influence this inequality — which at least holds for the parameter n — then rational personnel policies

14

See the proof of Proposition 2.

15_{When checking this assertion, note that a worker’s whole sabotage effort amounts to nŽ .} )s

y1 et₂

Ž . )s _{Ž .}

must meet the minimum standard that the inequality holds. But then the employer always prefers RD workers in her recruiting decisions!

At last, we can assume that a worker’s total — productive and counterproduc-tive — effort is restricted to a given capacity e. That means a worker has to decide how to allocate e units of effort to productive effort e and sabotage effort_i

1

e :_i

2

Proposition 8. If a worker’s total effort is restricted to e, for giÕen tournament

( ) ( ( ) )

prizes and quadratic cost functions: EE QiNs,t rEe-0 EE QiNs,t rEe)0 if

( )

k2-k1 k2)k1 for tsnon-RD, RD; ss1,2.

Proof. See Appendix A.B

In words, when a worker’s total productive and counterproductive effort is

Ž .

restricted to a certain amount, weakening this restriction will increase decrease

Ž .

the worker’s expected output, if counterproductive effort is more less costly than productive effort. This result holds for both types of workers and both incentive schemes given that the workers have quadratic cost functions. The result of Proposition 8 can be explained intuitively as follows: when sabotage is compara-tively costly for workers, weakening the effort restriction will lead to a higher increase in productive than in sabotage effort. This implies that the workers’ expected outputs will increase. The opposite argumentation will hold if productive effort is more costly than sabotage effort.

5. Conclusions

In conclusion, we can state that the behavior of workers who experience RD differs significantly from the behavior of those workers who maximize their absolute incomes in rank-order tournaments, which becomes obvious by explicitly modeling RD, based on net income, under risk. The tenor of this result is quite

Ž .

different from that in Stark 1987, 1990 . In his work, Stark argues that standard results of the tournament literature can also be derived under RD. This paper points to another direction; it stresses the individual differences between workers who minimize their RD and workers whose utility functions do not exhibit RD.

The following results can be derived when tournament prizes are exogenously given:

Ø Workers who minimize RD exert more productive and counterproductive

Ž .

effort sabotage than workers who maximize absolute income.

Ø Suppose that luck is uniformly distributed and that the employer can choose between two incentive schemes in the tournament. The first one rewards the winner of the tournament with a large prize whereas all the other workers get the

Ž .

Ž .

same large winner prize ‘‘scheme with ny1 prizes’’ . The results show that as the number of competing workers increases, the equilibrium effort of workers who minimize RD rises under the scheme with ny1 prizes, but falls under the scheme with a single prize.

Ø When luck is uniformly distributed, workers who minimize RD exert more effort under the scheme with n_y1 prizes than under the scheme with a single prize, whereas workers who maximize absolute income choose the same amount of effort under both schemes. Without restriction on the luck distribution there is no clear ranking between the efforts of workers who minimize RD, whereas the income maximizing workers work at least as hard under the scheme with a single prize than under the scheme with n_y1 prizes.

Ø When taking labor costs into account the employer prefers the scheme with a single prize for income maximizing workers, whereas from the employer’s view-point there is no clear ranking between the two incentive schemes for workers who minimize RD.

Ø The total sabotage activities in the tournament are at least as great under the scheme with a single prize than under the scheme with n_y1 prizes when workers maximize absolute income. The same will also hold for workers who minimize RD if an additional assumption is met.

Ø Suppose that workers have quadratic cost functions. If the number of competing workers is sufficiently large or sabotage effort is sufficiently costly compared to productive effort, the expected output of workers who minimize RD will be greater than the expected output of the income maximizing workers for either incentive scheme.

Ø Suppose that workers have quadratic cost functions and a worker’s total productive and counterproductive effort is restricted to a certain amount. When the

Ž .

effort restriction is weakened a worker’s expected output will rise fall , if

Ž .

counterproductive effort is more less costly than productive effort. This result holds for either type of worker and either incentive scheme.

Acknowledgements

I would like to thank Silke Becker, Rainer Gob, Christian Grund, Peter Kuhn,

¨

Bernd Schauenberg, Andrew Schotter, Gunter Steiner and two anonymous referees for helpful comments.

Appendix A. Proofs of the propositions

Ž .

symmetric Nash equilibrium. e) _{denotes the effort of each competitor. Worker i} will only feel relatively deprived if his output is less than the highest output among his ny1 competitors, i.e. if eq´ -e)_{q´ with ´}

i i Žny1.;Žny1. Žny1.;Žny1. as the highest of n_y1 order statistics. From worker i’s viewpoint the probability

) _{4 Ž} ) _.

for this event is Pr Y-e _ye _sF e _ye with Y[´ _y´ _and

i Y i i Žny1.;Žny1.

Ž . 16

F_Y P as distribution function of Y. Worker i’s expected RD can now be

Ž . Ž Ž )_{. Ž ..Ž . Ž} ) _.

characterized by ERD e_{i i s} D_{Byc e qc ei 1rn FY e yei with} D_B[ B_1yB . The first term in brackets describes the net income difference between₂

Ž Ž )_{.. Ž Ž ..}

the winner of the tournament B1_yc e and worker i B2_yc ei , i.e. the amount of deprivation that i feels. Multiplying this term with 1_rn gives the

Ž ) _.

amount of relatiÕe deprivation. F_Y e ye_i describes the probability of becoming

relatively deprived. Worker i chooses his Nash equilibrium effort to minimize expected RD. The assumption of a symmetric equilibrium means that e_se)

i

holds for i_s1, . . . , n. This symmetry condition together with the necessary condition for a Nash equilibrium leads to the following characterization of worker

X

Ž )1_{. Ž . Ž .}

i’s equilibrium effort for the scheme with a single prize: c e sD_{Bf 0rF 0}

RD Y Y

Ž . X_{Ž .} 17

with fY P [FY P as Y ’s density function.

Next, consider the scheme with a single prize for non-RD workers. Here, worker i

Ž . _w Ž ) _{.x Ž .}

maximizes his expected utility EU e _sB _qD_{B 1yF e ye yc e . The}

i i 2 Y i i

equilibrium effort e)1 _{can be derived in the same way as in the RD case. It is} non-RD

X_{Ž )}1 _{. Ž .}

characterized by c e _sD_{Bf 0 . Comparing this expression with the one}

non-RD Y

)1 )1 )1 _{Ž . Ž .} X_{Ž .}

for e_RD we see that e_RD)e_non-RD, because F 0_{Y g} 0,1 and c P )0.

Ž .

Analogous results hold for the scheme with n_y1 prizes s_s2 . Here, the

Ž . Ž Ž )_{. Ž ..Ž}

objective of a RD worker is to minimize ERD e_{i s} D_{Byc e qc ei ny}

. Ž ) _{. Ž .}

1_rn FX e _yei whereas a non-RD worker wants to maximize EU ei i _sB2_q

w Ž ) _{.x Ž . Ž .}

D_{B 1yF}X _{e ye}i _{yc e}i _{with F}X P as cumulative distribution function of the random variable X[´ _y´ _and ´ _{as the lowest of ny1 order}

i Ž1.;Žny1. Ž1.;Žny1. X

Ž )2_{. Ž . Ž .}

statistics. The equilibrium efforts are described by c e _sD_{B f 0rF 0 and}

RD X X

X

Ž )2 _{. Ž . Ž .} X

Ž .

c enon-RD _sD_{B f}X _{0 with f}X P [FX P as X ’s density function.

Proof of Proposition 2. To show that Proposition 2 holds we have to derive the

Ž . Ž . Ž . Ž . Ž .

explicit probabilities fY 0 , F 0 , fY X 0 and F 0 . F 0 stands for the probabil-X Y

ity of losing the tournament in equilibrium under the scheme with a single prize.

Ž . Ž

This event happens in n_y1 of n cases because of symmetry. Thus, F 0_{Y s} n_y

. Ž .

1_rn. F 0 represents the probability of losing the tournament in equilibrium_X

16

For the distribution of order statistics and functions of random variables see, for example, Mood et

Ž .

al. 1974 .

17

The second order condition can only be checked when the distribution function is fully specified in numerical terms, but this is not the case in this model. Therefore, here as in what follows I can only assume the existence of an equilibrium. This unsatisfactory fact is already known from the discussion

Ž .

of rank-order tournaments. See, for example, Lazear and Rosen 1981, 845, fn. 2 ; Nalebuff and

Ž . Ž .

under the scheme with ny1 prizes. Because of symmetry this happens only in 1 Ž .

of n cases. Thus, F 0X s1rn. These two expressions do not only hold for uniformly distributed luck, but also for the general case.

Ž . Ž .

Next, we can derive the density functions fY y and fX x : The density function of the highest of n_y1 order statistics ´ _{can be written as}

Žny1.;Žny1.

n_y1 !

Ž

.

_n

y2

f

Ž

´

.

_{s F}

Ž

´

.

Žny1. Žny1.;Žny1. Žny1.;Žny1. n_y2 !0!

Ž

.

= 1_yF

Ž

´

.

0_f

Ž

´

.

Žny1.;Žny1. Žny1.;Žny1.

s

Ž

n_y1 F

.

ny2

Ž

_´

. Ž

_{f ´}

.

Žny1.;Žny1. Žny1.;Žny1.

w x Ž . Ž .

with ´ _g0,´ _{. F} ´ _{and f} ´ _{are the distribution function and the}

Žny1.;Žny1.

Ž

density function of the n random components ´_{, respectively each component}

i

.

has the same probability distribution; see the i.i.d.-assumption above . When the´

i

w x

are uniformly distributed over 0,´ _{, the density can be simplified to}

´ny2

Žny1.;Žny1.

f

Ž

´

.

_s

Ž

_ny1

.

_.

Žny1. Žny1.;Žny1. _´ny1

Next, we can construct the density function for the composed random variable Y_s´ _y´ _{. Because} ´ _and ´ _{are stochastically independent,}

i Žny1.;Žny1. i Žny1.;Žny1. we can write the wanted density function as

f

Ž .

y _s

H

f

Ž .

´ _f

Ž

´_{yy d}

.

´

Y Žny1.

Y_s´_iy´_Žn

y1.;Žny1.

ny2

´_yy

Ž

.

s

H

Ž

n_y1

.

d´_.

n

´

Y_s´_iy´_Žn

y1.;Žny1.

At last, we need the concrete limits of the integral to determine the exact density function. We have to consider that

0F´_F´

Ž

_A1

.

and

0_F´ _F´

m

_0F´_yyF´

m

_yF´_F´_qy.

Ž

_A2

.

Žny1.;Žny1.

w x

The random variable Y is distributed over the interval y´_,´ _{, because Ys}´ _y

i

´ _{, and each of the random variables} ´ _and ´ _{can be 0 or}´ _in

Žny1.;Žny1. i Žny1.;Žny1.

w x

the worst or in the best case, respectively. Now, the interval y´_,´ _{can be}

Ž . Ž .

divided into two subintervals: I y´_{FyF0 and II 0}-yF´_{. The subinterval} Ž .I says that y_F0. Together with Eqs. A1 and A2 this implies that 0Ž . Ž . _F´_F´

Ž . Ž .

ŽA1 and A2 this implies that y. Ž . F´_F´ _{in the subinterval II . Therefore, it}Ž . holds that

´_qy _ny2

°

Ž

´_yy

.

n_y1 d´ _{if y}´_FyF0

Ž

.

H

_´n

0

~

fY

Ž .

y _s

´ _ny2

´_yy

Ž

.

n_y1 d´ _{if 0}-y_F´

Ž

.

H

n

¢

´

y

ny1

°

1

Ž

_yy

.

y if _y´_FyF0

n

´ ´

~

s _{´ ny}1

yy

Ž

.

if 0-y_F´_.

¢

_´n

Ž .

The density function f x for the random variable X[´ _y´ _with

X i Ž1.;Žny1.

´ _{as the lowest of ny1 order statistics can be derived in an analogous}

Ž1.;Žny1.

manner. For independently, identically, and uniformly distributed ´ _{it holds that}

i

´ _{has the density function}

Ž1.;Žny1.

f

Ž

´

.

Ž1. Ž1.;Žny1.

n_y1 !

Ž

.

₀ ny2

s F

Ž

´

.

_1yF

Ž

´

.

_f

Ž

´

.

Ž1.;Žny1. Ž1.;Žny1. Ž1.;Žny1. 0! n

Ž

_y2 !

.

ny2

´_y´

Ž

Ž1.;Žny1.

.

s

Ž

n_y1

.

_´ny1 .

Therefore

f

Ž .

x _s

H

f

Ž .

´ _f

Ž

´_{yx d}

.

´

X Ž1.

X_s´_iy´_Ž1.;Žn

y1.

ny2

´_y´_qx

Ž

.

s

H

Ž

n_y1

.

d´_.

n

´

Xs´_iy´_Ž1.;Žn

y1.

In analogy to the derivation considered above we find that

xq´ _ny2

°

Ž

´_y´_qx

.

ny1 d´ _{if y}´_FxF0

Ž

.

H

_´n

0

~

fX

Ž .

x s

´ _ny2

´_y´_qx

Ž

.

ny1 d´ _{if 0}-xF´

Ž

.

H

n

¢

´

ny1

°

Ž

´_qx

.

if y´_FxF0

n

´

~

s

ny1 1 x

y if 0-x_F´_.

¢

_{´ ´}n

Ž . Ž .

In equilibrium y_sx_s0, so that f 0 _sf 0 _s1_r´_{. These two expressions and}

Y X

Ž . Ž . )1

the ones for F 0 and F 0 lead to the following equilibrium efforts: e_{Y X RDs}

X_y1_{Ž Ž Ž ... )}1 X_y1_{Ž .} )2 X_y1_{Ž .} )2

c D_Bnr ´ _{ny1 , e sc} D_Br´ _{, e sc} D_Bnr´ _{, e s}

non-RD RD non-RD

X_y1_{Ž .} X_y1 _{Ž .}

c D_Br´ _withD_BsB1_{yB and c}2 P as the inverse function of the first

X_y1_{Ž .}

derivative of the cost function, at which c P is monotonely increasing because X

Ž . Y

Ž .

of c ei )0 and c ei )0. Expected RD in the two schemes can be written as

Ž )1_{. Ž} )2_{. Ž .} 2

follows: ERD e _sERD e _sD_{B ny1rn . The statements in}

Proposi-i RD i RD

tion 2 follow directly from these expressions.

Proof of Proposition 3. The results e)1 -e)2

and e)1

se)2

for uni-RD RD non-RD non-RD

formly distributed ´ _{follow from the proof of Proposition 2. Next, we have to}

i

consider the general case without the assumption of a special distribution for the random components ´_{. From the proof of Proposition 2 we know that}

i

´

ny2

f

Ž .

0 s

H

f

Ž . Ž

´ _{ny1 F}

.

Ž . Ž .

´ _f ´ _d´

Y

0

´

ny2

f

Ž .

0 s

H

f

Ž . Ž

´ _ny1

.

_1yF

Ž .

´ _f

Ž .

´ _d´_.

X

0

Ž . Ž .

We see that fY 0 represents the mean of the function f P of the highest of n_y1

Ž . Ž .

order statistics, whereas fX 0 describes the mean of the function f P of the

Ž . Ž .

lowest of ny1 order statistics. So, in general, it holds that fY 0 )fX 0 ; only if Ž .

f ´ _{is a constant — like in the case of uniformly distributed} ´ _{— we can factor}

i

Ž . Ž . Ž . Ž .

f ´ _{out of the integral, which then sums up to 1 so that f 0 sf 0 sf} ´ _s

Y X

Ž . Ž .

const. Thus, in the general case fY 0 GfX 0 and the results in the proof of Proposition 1 imply that e)1

Ge)2 _{, whereas the relation between e})1 _and

non-RD non-RD RD

e)2 _{is indefinite in general, because both the numerator and the denominator are} RD

)1 Xy1 _{Ž Ž . Ž .. )}2 Xy1 _{Ž Ž . Ž ..}

greater in e _sc D_{Bf 0rF 0 than in e sc} D_{Bf 0rF 0 .}

RD Y Y RD X X

Proof of Proposition 4. The statement of Proposition 4 follows immediately from

Ž .

the previous proofs and the fact that the labor costs are B1_q n_y1 B under the2

Ž .

scheme with a single prize, and n_y1 B1_qB2 under the scheme with n_y1 prizes.

Proof of Proposition 5. First, the equilibrium efforts under the scheme with a

Ž .

e22s. . .sei2s. . .se . The same symmetry condition holds for productiven2

effort: e11se21s. . .sei1s. . .se . Worker i’s expected RD can now ben1

Ž . Ž Ž . ŽŽ . . Ž . ŽŽ .

written as ERD e , e s D_{Byc e yc ny1 e qc e qc ny1}

i i1 i2 1 )1 2 )2 1 i1 2

..Ž . Ž ) .

ei2 1rn FY e)1yÝj/ ej2yei1qÝj/iej2 with e)1and e)2as a competitor’s productive and counterproductive effort, respectively. The first order conditions together with the symmetry conditions result in the following characterization of

18 )1 Xy1_Ž

w Ž . _{x w x}. )1

the equilibrium efforts: e _se _sc D_{Bf 0 nr ny1 and e se s}

i1 RD1 1 Y i2 RD2

Ž . Xy1_Ž

w Ž . _{x w x}2_{. Ž .}

1_r n_y1 c D_{Bf 0 nr ny1 is1, . . . , n . Non-RD workers’ expected}

2 Y

Ž . w Ž )

utility can be written as EU e , e _sB _qD_{B 1yF e yÝ e ye q}

i i1 i2 2 Y )1 j/ j2 i1

.x Ž . ŽŽ . .

Ýj/iej2 _yc e1 i1 _yc2 n_y1 ei2 . The corresponding equilibrium efforts are

)1 X_y1_{Ž Ž .. )}1 _{Ž .} X_y1_Ž

w Ž .x w x.

e_{i s}e_{non-RD s}c₁ D_{BfY 0 ei senon-RD s1r ny1 c2} D_{BfY 0 r ny1}

1 1 2 2

Ži_s1, . . . , n ..

Ž .

Next, the scheme with n_y1 prizes s_s2 has to be considered. Because of the special form of this compensation scheme workers’ sabotage behavior signifi-cantly differs from the one under the scheme with a single prize. Under the last scheme, a worker only receives the large bonus B when he wins against all the1 other workers. Thus, in symmetric tournaments a worker’s sabotage behavior will be identical against each of his competitors. Under the scheme with ny1 prizes, however, a worker already receives the large bonus when at least one other worker produces a lower individual output. Thus, it is rational for a worker to concentrate his sabotage behavior on only one competitor to win the direct comparison against this competitor. In this symmetric tournament a worker has no preferences which one he should sabotage. Therefore, he chooses his victim at random.19 _{By that}

Ž .

means, the probability of being sabotaged is 1_r n_y1 for each worker, and

Ž . Ž . ŽŽ

worker i’s i_s1, . . . , n expected RD can be written as ERD e , e _s D_B

i i1 i2

Ž . Ž . Ž . Ž . .Ž . . Ž )Ž Ž ..

yc1 e)1 yc2 e)2 qc1 ei1 qc2 ei2 ny1rn FX e)1yÝj/ ej2r ny1

Ž . Ž ..

yei1qÝj/i ej2 r ny1 . This expression leads to the equilibrium efforts

)2 X_y1 _{Ž Ž . .} )2 X_y1 _Ž

w Ž . _{x w x}.Ž

e se sc D_{Bf 0 n and e se sc} D_{Bf 0 nr ny1 is1,}

i1 RD1 1 X i2 RD2 2 X

.20

. . . , n . Under this incentive scheme, non-RD workers’ expected utility is

Ž . _w Ž )Ž Ž ..

described by EU e , e _sB _qD_{B 1yF e yÝ e r ny1}

i i1 i2 2 X )1 j/ j2

Ž Ž .._x Ž . Ž .

ye_i1_qÝ_j/i ej2_r n_y1 _yc e1 i1 _yc e2 i2 and their corresponding equilibrium

)2 Xy1_{Ž Ž .. )}2 Xy1_Ž

w Ž ._x

efforts by e _se _sc D_{Bf 0 and e se sc} D_{Bf 0 r}

i1 non-RD1 1 X i2 non-RD2 2 X

wn_y1x. Ži_s1, . . . , n . Proposition 5 holds, because under the scheme with a.

Ž .

single prize s_s1 the arguments in the inverses of the marginal cost functions

Ž .

are greater for RD than for non-RD workers by the factor n_r n_y1 and under the

Ž .

scheme with n_y1 prizes s_s2 by the factor n.

18

Ž . Ž . )

Note that the sumÝj/ ej2contains the variable ei2and that FY 0s ny1rn.

19

Of course, in this situation coalitions between workers against another worker seem to be rational but such outcomes cannot be stable, because the sabotaged worker immediately destroys the given coalition by offering a little money more than one of the actual coalition members to the rest of the

Ž .

coalition. There cannot be a coalition-proof Nash equilibrium in the sense of Bernheim et al. 1987 , because we have only one candidate for a Nash equilibrium here.

20

Ž .

Proof of Proposition 6. The total counterproductive effort from n sabotaging

non-RD workers will be greater under the scheme with a single prize than under

Ž .Ž Ž .. X_y1 _Ž

w Ž ._{x w x}.

the scheme with ny1 prizes, if n ny1 1r ny1 c₂ D_{BfY 0 r ny1} ) X_y1_Ž

w Ž ._{x w x}. Ž . Ž .

nc2 D_BfX _{0 r ny1}

m

_fY ₀ )fX 0 . From the proof of Proposition 3 we know that this inequality usually holds. Only for uniformly distributed´ _{we have}

i

Ž . Ž . Ž .

f 0 _sf 0 _sf ´ _{sconst. When we consider n sabotaging RD workers the}

Y X

total sabotage will be at least as great under the scheme with a single prize than

Ž .Ž Ž .. Xy1_Ž

w Ž . _x

under the scheme with n_y1 prizes, if n n_y1 1_r n_y1 c D_{Bf 0 nr}

2 Y

w x2_. X_y1_Ž

w Ž . x w x. Ž . Ž . Ž .

n_y1 _Gnc D_{Bf 0 nr ny1}

m

_{f 0 G ny1 f 0 .}

2 X Y X

Proof of Proposition 7. To check the employer’s preferences we have to compute

the workers’ expected outputs, because for given tournament prizes only the expected outputs are relevant for the employer’s recruiting decisions. For the two compensation schemes and two types of workers the following four expected outputs can be computed when the cost functions are quadratic:

D_Bf

Ž .

₀ D_Bf

Ž .

₀

Y Y

E _sE Q

Ž

_Ns_s1, non-RD

.

_{s y q}E

Ž .

´ _,

I i i

2 k1 2 k2

Ž

n_y1

.

D_Bf

Ž .

₀ D_Bf

Ž .

₀

X X

E sE Q

Ž

Nss2, non-RD

.

s y qE

Ž .

´ _,

II i i

2 k1 2 k2

Ž

ny1

.

D_Bf

Ž .

_{0 n} D_Bf

Ž .

_{0 n}

Y Y

E _sE Q

Ž

_Ns_s1, RD

.

_{s y q}E

Ž .

´ _,

III i 2 i

2 k1

Ž

n_y1

.

2 k2

Ž

n_y1

.

D_Bf

Ž .

_{0 n} D_Bf

Ž .

_{0 n}

X X

E _sE Q

Ž

_Ns_s2, RD

.

_{s y q}E

Ž .

´ _.

IV i i

2 k1 2 k2

Ž

n_y1

.

RD workers will have greater expected outputs than non-RD workers under each compensation scheme, if EIII)E and EI IV)E . These two inequalities holdII

Ž .

when ny1 k2)k .1

Proof of Proposition 8. Now we have to consider the additional assumption that

each worker has a fixed amount of effort units e, which can be used for productive and counterproductive effort. This implies that under the scheme with a single

Ž .

prize each worker’s efforts are restricted to ei1q ny1 ei2se and under the scheme with ny1 prizes to ei1qei2se. With these restrictions and quadratic cost functions the following Nash equilibrium efforts can be computed:

)1

e_{non - RD s} 2 k₂

Ž

n_y1 e

.

_qD_{B n}

Ž

_{y2 f}

.

_Y

Ž .

_{0 r 2 n}

Ž

_y1

. Ž

_k1qk2

.

_,

1

2 )1

e_{non - RD s} 2 k₁

Ž

n_y1 e

.

_yD_{B n}

Ž

_{y2 f}

.

_Y

Ž .

_{0 r 2 n}

Ž

_y1

. Ž

_k1qk2

.

_,

2

)2

e _s 2 k

Ž

n_y1 e

.

_qD_{B n}

Ž

_{y2 f}

.

Ž .

_{0 r 2 n}

Ž

_y1

. Ž

_{k qk}

.

_,

)2

e s 2 k

Ž

ny1 e

.

yD_{B n}

Ž

_{y2 f}

.

Ž .

_{0 r 2 n}

Ž

_y1

. Ž

_{k qk}

.

_,

non - RD2 1 X 1 2

2 2

)1

e s 2 k

Ž

ny1

.

e