Tournaments as Musical Chairs:

Why Punishing the Bottom Can be Better

Than Rewarding the Top1

Robert J. Akerlof

Harvard University

Littauer Center

1875 Cambridge Street

Cambridge, MA, 02138

akerlof@fas.harvard.edu

Richard T. Holden

MIT

E52-410

50 Memorial Drive

Cambridge, MA, 02142-1347

rholden@mit.edu

July 14, 2006

1We are grateful to Philippe Aghion, George Akerlof, Edward Glaeser, JerryGreen, Oliver Hart, Bengt Holmström, Emir Kamenica, Lawrence Katz, and EmilyOster for helpful comments and discussions.

Abstract

Within-�rm job promotions, wage increases, bonuses, and CEO compensation

are often interpreted as resulting from Lazear-Rosen-style tournaments. They

are prizes for top performers in rank-order tournaments set up to deal with

moral hazard. We �nd that, because agents are risk averse, it is generally

optimal to punish poor performers rather than to reward top performers. This

result holds with surprising generality. We describe a few quali�cations to

our result, but they do not seem to be su¢ cient to explain the pervasiveness

of prizes for winners. Our �ndings thus pose a puzzle for the theory of

compensation.

JEL Codes: D20, J33, L22.

Keywords: Hierarchies, moral hazard, organizations, prizes, tournaments.

1 Introduction

Lazear and Rosen (1981) argue that rank-order tournaments help solve a moral

hazard problem. This paper characterizes such rank-order tournaments.

Lazear-Rosen tournaments have often been interpreted as explaining within-

�rm job promotions, wage increases, bonuses, and CEO compensation (Pren-

dergast (1999)). This view is based on the premise that such tournaments

should give large rewards to top performers relative to average performers.

We �nd, in contrast that, with a surprising degree of generality, the best

way to structure a Lazear-Rosen tournament is like a round of musical chairs.

It is best to give punishments to a few of the worst performers rather than

give rewards to a few of the best. Furthermore, the di¤erence in pro�ts to the

principal between a tournament that rewards winners and one that punishes

losers can be considerable. This suggest that within-�rm job promotions,

wage increases, and other bene�ts that accrue to winners may arise in spite of

the Lazear-Rosen theory of tournaments rather than because of it.

Some of these prizes for top performers have been explained in other ways.

For example, within-�rm job promotions may occur because employees of the

�rm acquire �rm-speci�c human capital. High compensation for CEOs may

be the result of an agency problem (such as a board of directors appointed

by the CEO) rather than the result of an optimal tournament. Our �ndings

suggest that these alternative explanations may be more important.

Our result implies that Lazear-Rosen tournaments would be consistent with

laying o¤ workers for poor relative performance. A practice called �rank and

yank,�which is practiced by as many as twenty percent of US �rms (Grote

(2005)), including Microsoft, Ford, PepsiCo, Cisco, and General Electric, has

this as one of its components. Jack Welch was one of the �rst practitioners of

�rank and yank�and his implementation at General Electric was somewhat

typical. There was a system of periodic forced ranking, in which the top 20

percent (A�s), the middle 70 percent (B�s) and the bottom 10 percent (C�s)

1

of workers were identi�ed. Most C�s were �red. While the layo¤s were a

prominent element of this system, it is worth noting that it was not the only

element. There were also rewards for top performers. And while, as Welch

has noted, workers dreaded being put in the bottom category and worked hard

to avoid it, there were other bene�ts of the system. Welch argues that one of

the main bene�ts of �rank and yank�is that it forces ranking on a curve and

hence prevents managers from in�ating the performance of their subordinates

in order to avoid di¢ cult interactions (Welch (2001)).

Our result that rank-order tournaments should be structured like musi-

cal chairs is due to one of the fundamental assumptions of the moral hazard

problem: that agents are risk averse. Consider two tournaments set up by

a principal: one which gives a special prize to the best performer and the

same payo¤ to others (a �winner-prize tournament�) and one which gives a

special punishment to the worst performer and the same payo¤ to others (a

�loser-prize tournament�). Suppose these tournaments are set up to induce

the same e¤ort from the agents participating in the tournament. We show

that the winner-prize tournament is a riskier tournament for the agent than

the loser-prize tournament. As a result, the principal must pay the agents

more if she uses a winner-prize structure in order to compensate the agents

for the additional risk. Therefore, loser-prize tournaments are a more power-

ful way of providing incentives to agents than winner-prize tournaments in a

moral hazard context.

While our general �nding is that Lazear-Rosen tournaments should punish

losers rather than reward winners, we �nd two reasons why this might not be

the case. When the noise distribution is su¢ ciently skewed to the left or agents

are heterogeneous, for reasons we shall make clear in due course, the principal

may want to reward winners.1 But, it is doubtful that these quali�cations are

su¢ cient to explain the ubiquity of within-�rm job promotions, wage increases,

or bonuses.1By �skewed to the left,�we mean that the mode of the distribution is on the right.

2

We begin by considering a principal who is restricted to awarding only two

types of prizes in a tournament. That is, the principal can give one prize to

the top j performers and another to the bottom n� j performers. When j islow, the tournament gives a special prize to a few winners and when j is close

to n, it punishes a few losers. We show that, under quite general conditions,

the optimal choice of j for the principal is greater than n=2; and when we

further assume that the noise in individuals�outputs is uniformly distributed,

the optimal choice of j is n� 1: That is the exact opposite of winner-take-all.

Our result does not require unrealistically harsh punishments for the worst

performers. We discuss the consequences of adding a limited liability con-

straint that restricts the extent to which the agents can be punished. We

�nd that, when such constraints change the optimal structure at all, they are

likely to lead the principal to punish a larger number of poor performers less

harshly rather than reward a few top performers.

Risk aversion gives the principal a strong desire to punish losers even when

more than two types of prizes can be awarded. It does, however, lead the

principal to slightly smooth the prize schedule. In utils, we �nd that, like

the two-prize case, virtually the same reward is given to all but a few of the

worst performers. In money, though, we �nd that the best performer is given

a higher prize than the second-best performer. But, our result still translates

in this context. We �nd that, if there is a di¤erence in the prize given to the

best and the second-best performers, it is similar to the di¤erence in the prize

given to the second-best and third-best performers. This is not true at the

bottom of the distribution. The di¤erence in the prize given the worst and

second-worst performers is considerably greater than the di¤erence in the prize

given to the second-worst and third-worst performers. In this sense, there are

especially large punishments at the bottom but there are not especially large

rewards at the top.

The paper proceeds as follows. Section 2 provides a brief review of the

existing literature. Section 3 gives the basic setup of the model and states the

3

problem of the principal designing the tournament. Section 4 establishes the

main results of the paper, considering the case where agents are homogeneous

and the principal is limited to awarding two types of prizes. It is shown

that the loss in pro�ts from rewarding winners rather than punishing losers is

substantial and that the worst performers do not need to be �boiled in oil�

to obtain the result. It is also shown how the results from the two-prize case

translate to the many-prize case. Section 5 considers the assumptions that

might make it optimal to have a tournament that rewards winners. Section 6

contains some concluding remarks.

2 Brief Literature Review

Classic treatments of tournaments are given by Lazear and Rosen (1981),

Green and Stokey (1983) and Nalebu¤ and Stiglitz (1983). The two papers

most closely related to this one are Krishna and Morgan (1998) and Moldovanu

and Sela (2001). They also focus on optimal prize structures in rank-order

tournaments. While the assumptions made in these papers di¤er from those

of Lazear and Rosen (1981), the results are generally seen as applicable to

the Lazear-Rosen context. Both of these papers conclude that optimally

designed tournaments give a special prize to top performers. In fact, they

suggest that roughly the same prize should be given to all but a few of the best

performers. We see our di¤erence in results as due to important deviations

that Krishna and Morgan (1998) and Moldovanu and Sela (2001) have made

from the standard moral hazard setting.

Krishna and Morgan (1998) have an ex post participation constraint rather

than the standard ex ante participation constraint. This means that the prin-

cipal must make sure that an agent receives a certain number of utils ex post

rather than a certain number in expectation. With an ex ante participation

constraint, giving more to agents with high rank allows the principal to give

less to agents with low rank.2 This is not the case with an ex post participa-2Since an agent has some chance of achieving any rank, giving more for high rank in-

creases an agent�s expected utility. Since the PC is binding in equilibrium, this allows the

4

tion constraint.

Suppose we �x the amount given to the lowest ranked agent. It turns out

that, to induce a particular e¤ort level agents need to be paid less (beyond the

�xed amount) with a winner-take-all structure than a musical chairs structure.3

Since an ex post participation constraint �xes the level given to the lowest

ranked agent, this clearly implies that a winner-take-all structure dominates a

musical chairs structure in the Krishna-Morgan case. This is not necessarily

true with an ex ante participation constraint. If more is paid to agents above

what is given to the lowest ranked agent, less can be given to the lowest ranked

agent.

In an elegant paper, Moldovanu and Sela (2001) seek to explain prize struc-

tures in tournaments within the framework of private value all-pay auctions.

This is formally similar to models analyzed by Weber (1985), Glazer and Has-

sin (1988), Hillman and Riley (1989), Krishna and Morgan (1997), Clark and

Riis (1998) and Barut and Kovenock (1998). Moldovanu and Sela (2001) ana-

lyze a model where contestants have di¤erent costs of exerting e¤ort, which is

private information. The contest designed seeks to maximize the sum of the

e¤orts by determining the allocation of a �xed purse among the contestants.

They show that if the contestants have linear or concave cost of e¤ort functions

then the optimal prize structure involves allocating the entire prize to the �rst-

place getter. With convex costs, entry fees, or minimum e¤ort requirements,

more prizes can be optimal.

With this approach, there is a deterministic relationship between action

choice and output. Therefore, agents�attitudes to risk play no role. This con-

trasts sharply with the stochastic relationship which is present in the Lazear-

Rosen framework.

principal to give less for low rank.3This follows from our Lemma 1.

5

3 The Model

In this section, we will give the setup of the problem. The assumptions that

we make should be very familiar: they are the same assumptions as those

made by Lazear and Rosen (1981), Green and Stokey (1983) and Nalebu¤ and

Stiglitz (1983).

3.1 Statement of the Problem

We will consider a world in which there are n people available to compete in

a rank-order tournament. This tournament is set up by a principal whose

goal is to maximize her expected pro�ts. The principal pays a prize wi to the

individual who places ith in the tournament. The pro�ts which accrue to the

principal are equal to the sum of the outputs of the participating individuals

minus the amount she pays out: � =Pn

i=1(qi � wi): At least for now, we

will assume that individuals are homogeneous in ability. If individual j exerts

e¤ort ej; her output is given by qj = ej + "j + �, where "j and � are random

variables with mean zero and distributed according to distributions F and G

respectively. We assume that F and G are statistically independent. We

will refer to � as the �common shock�to output and "j as the �idiosyncratic

shock�to output. Since rank-order tournaments �lter out the noise created

by common shocks but individual contracts do not, rank-order tournaments

are considered most advantageous when common shocks are large.4

We will assume that individuals have utility that is additively separable in

wealth and e¤ort5. If individual j comes in ith in the tournament, her utility is

given by: u(wi)� c(ej) where u0 � 0; u00 � 0; c0 � 0; c00 � 0: Individuals have4See Holmström (1982) for a de�nitive treatment of relative performance evaluation

individual contracts. He shows that an appropriately structured individual contract witha relative performance component dominates a rank-order tournament for n �nite. Greenand Stokey (1983) prove convergence of optimal tournaments to the individual contractsecond-best and n!1:

5This implies that preferences for income lotteries are independent of action and thatpreferences for action lotteries are independent of income (Keeney (1973)). Among otherthings, this rules out the possibility that stochastic contracts are optimal.

6

an outside option which guarantees them �U; so unless the expected utility

from participation is at least equal to �U , individuals will not be willing to

participate.

The timing of events is as follows. Time 0: the principal commits to a prize

schedule fwigni=1: Time 1: individuals decide whether or not to participate.Time 2: if everyone has agreed to participate at time 1, individuals choose

how much e¤ort to exert. Time 3: output is realized and prizes are awarded

according to the prize schedule set at time 0.

3.2 Solving the Model

We will restrict attention to symmetric pure strategy equilibria (as do Green

and Stokey (1983) and Krishna and Morgan (1998)). While we cannot rule

out the possibility of other equilibria, a unique symmetric pure strategy equi-

librium will always exist, provided that the distribution of idiosyncratic shocks

is �su¢ ciently dispersed�.6 In a symmetric equilibrium, every individual will

exert e¤ort e�: Furthermore, every individual has an equal chance of winning

any prize. Thus, her expected utility is

1

n

Xi

u(wi)� c(e�)

In order for it to be worthwhile for an individual to participate in the tourna-

ment, it is necessary that

1

n

Xi

u(wi)� c(e�) � �U

An individual who exerts e¤ort e while everyone else exerts e¤ort e� receives

6See Nalebu¤ and Stiglitz (1983) and Krishna and Morgan (1998) for a discussion of thiscondition. The appendix to Nalebu¤ and Stiglitz (1983) contains a detailed discussion ofmixed strategy equilibria in tournaments.

7

expected utility

U(e; e�) =Xi

'i(e; e�)u(wi)� c(e)

where 'i(e; e�) = Pr(ith placeje; e�);

that is, the probability of achieving place i given e¤ort e while all other agents

choose e¤ort e�: The problem faced by an individual is to choose e to maximize

U(e; e�): The �rst-order condition for this problem is

c0(e) =Xi

@

@e'i(e; e

�)u(wi)

In order for the �rst-order approach to be a valid,7 it is necessary for the max-

imization problem of the agent to be convex. It turns out that this problem is

not convex for all distribution functions, F , but if F is �su¢ ciently dispersed,�

then the problem will indeed be convex. Many common distributions, includ-

ing the normal distribution with su¢ cient variance, meet this condition. We

will assume that F is su¢ ciently dispersed for the remainder of this paper.

Since we know that the solution to the maximization problem is e = e�, it

follows that

c0(e�) =Xi

�iu(wi)

where �i =@

@e'i(e; e

�)

����e=e�

Note that �i does not depend upon e� but simply upon the distribution

function F . Routine results from the study of order-statistics imply that the

formula for �i is

�i =

�n� 1i� 1

�ZR

F (x)n�i�1(1� F (x))i�2 ((n� i)� (n� 1)F (x)) f(x)2dx

7That is, for the agent�s �rst-order condition to be always equivalent to her incentivecompatability constraint.

8

SinceP

i 'i = 1; it follows thatP

i �i = 0: It is also easily shown that if F is

a symmetric distribution (F (�x) = 1� F (x)), that �i = ��n+1�i:

Now that we have elaborated the agent�s problem, we turn to the principal�s

problem. The principal�s object is to maximize

E(�) =Xj

ej �Xi

wi = n

e� � 1

n

Xi

wi

!:

The problem of the principal can therefore be stated as follows

maxwi

e� � 1

n

Xi

wi

!subject to

1

n

Xi

u(wi)� c(e�) � �U (IR)

c0(e�) =Xi

�iu(wi) (IC)

4 Results

We will now show that, when the principal is restricted to awarding two types

of prizes, she will generally choose to punish a few losers rather than reward

a few winners. One might believe that a restriction on the severity of the

punishment for the worst performers would lead to a prize structure that

rewards winners to be optimal. We will argue that this is generally not the

case. Finally, we will show how these results translate to the case where the

principal can award more than two prizes.

4.1 Two-Prize Tournaments

So far, we have considered the case where the principal can set n distinct

prizes. Suppose, instead, that the principal is limited to two prizes. That is,

suppose she can only give a prize w1 to the top j performers and a prize w2

9

to the bottom n� j performers. When the principal is restricted in this way,where would she like to set j? One possibility would be to set j = n=2, so

that the top half earns one prize and the bottom half earns another. Another

possibility would be to set j = 1; which gives a special prize to the best

performer. The opposite would be to set j = n� 1, so that there is a specialpunishment in store for the worst performer. We will now establish some

terminology.

De�nition 1 We will call a tournament a �j tournament�when the principalpays a prize w1 to the top j performers and a prize w2 to the bottom n � j

performers. Let u1 = u(w1) and u2 = u(w2): We will call a tournament a

�winner-prize tournament� if j < n2and a �strict winner-prize tournament�

if j = 1: We will call a tournament a �loser-prize tournament� is j � n2and

a �strict loser-prize tournament� if j = n� 1:

Is it better for the principal to implement a winner- or loser-prize tourna-

ment? We will show that under relatively general conditions, a loser-prize

tournament will be preferred. We will also give a condition under which the

optimal loser-prize tournament will be the strict one.

The argument will be made in the following way. We will consider a j

tournament and an n� j tournament that induce the same level of e¤ort andboth meet the individual rationality constraint. We will show that, when

agents are risk averse and j < n2, the payment made to agents by the principal

is greater when she uses the j tournament. The reason is that a j tournament

is riskier than an n�j tournament, and therefore the principal must give moreto the agents to compensate them for the additional risk.

First, we must know when a j tournament and an n�j tournament inducethe same e¤ort. The following lemma provides the answer.

Lemma 1 For F symmetric and �su¢ ciently dispersed�, a j tournament andan n � j tournament for which u1 � u2 is the same induce the same level of

10

e¤ort. This level of e¤ort is given by

c0(e) =

jXi=1

�i

!(u1 � u2)

Proof. See appendix.

Using this lemma, we will now establish the main result.

Proposition 1 Suppose the principal is restricted to use a j tournament (buthas a choice over w1 and w2). Let �j denote the expected pro�ts from the

optimal choice of w1 and w2: Suppose that F is a symmetric and �su¢ ciently

dispersed�distribution and that the following holds for u:

3 (u00)2 � u0u000 � 0 (*)

Then, �j � �n�j for j � n2:

Proof. See appendix.

Corollary 1 Suppose the principal is restricted to implementing a j tourna-ment, but can choose whatever j she likes. If F is symmetric and u satis�es

property (*), then the optimal j tournament is a loser-prize tournament (a

tournament with j � n2:)

Property (*) is a statement about the risk aversion of agents. While it

looks complex, it is met by a large class of utility functions. In particular,

property (*) holds for all CARA utility functions and CRRA utility functions

with � � 12: The condition amounts to a condition that the utility function

be su¢ ciently concave. It can also be related to standard economic measures

of risk aversion. Speci�cally, it requires that the Arrow-Pratt measure be

weakly greater than one third of the Coe¢ cient of Absolute Prudence (the

latter concept due to Kimball (1990)).

11

So far, we have managed to give conditions under which the optimal two-

prize tournament is a loser-prize tournament. We can go further and make

comparisons between loser-prize tournaments when we assume that the idio-

syncratic noise distribution is uniform.

Proposition 2 Suppose the principal is restricted to use a j tournament, thatF is a symmetric uniform distribution, and that the following holds for u:

3 (u00)2 � u0u000 � 0 (*)

Then, the optimal j tournament is the strict loser-prize tournament:

Proof. See appendix.

We generally �nd that the optimal two-prize tournament is the strict loser-

prize tournament even when the noise distribution is not uniform. However,

we cannot rule out pathological cases unless the noise distribution is uniform.

We will consider a number of numerical examples below in which we use a

normal noise distribution. In all of these examples, we have found the strict

loser-prize tournament to be the optimal two-prize tournament.

Typically, the optimal choice of j in a two-prize tournament depends upon

an interaction between the weights, �i, and the utility of wealth, u. For the

uniform distribution, �1 = ��n = 1 and �i = 0 for 1 < i < n. This spe-

cial property of the weights eliminates this interaction term, and allows us to

conclude that the optimal two-prize tournament is the strict loser-prize tour-

nament for any u meeting property (*). For other noise distributions, there

will be an interaction between the weights and the utility function. We can say

something, though, about when we expect this interaction to be large. There

are two necessary conditions for the interaction to be large: (i)��� �iPi

j=1 �j� i��� is

large for 1 < i < n2and (ii) 3u0(u00)2�(u0)2u000

(u0)6 is very small relative to �u00(u0)2 :

8 In

fact, even if (i) and (ii) are both met, it is still possible for the interaction to

be small.8For a derivation of these conditions, see the proof of Proposition 2 in the Appendix.

12

For most of the symmetric distributions that are commonly used, we �nd

that �i is relatively small for intermediate values of i and spikes in the neigh-

borhoods of i = 1 and i = n � 1. This means that��� �iPi

j=1 �j� i��� will be quite

small for 1 < i < n2: Below, we see this structure in a graph of the weights for

normally distributed noise with standard deviation of 1 and n = 100.

­0.026

­0.016

­0.006

0.004

0.014

0.024

1 11 21 31 41 51 61 71 81 91

Figure 1: Weights for the Normal Distribution

Furthermore, we rarely �nd that condition (ii) holds for standard choices

of utility functions. We therefore believe that a case where the optimal j

tournament sets j far from 1 is a perverse case.

4.2 Numerical Examples

We have shown that, under the conditions of Corollary 1, the optimal j tour-

nament is a loser-prize tournament. We have further shown that, under the

conditions of Proposition 2, the optimal j tournament is the strict loser-prize

tournament. We will now show, by way of example, that the losses to the

principal from choosing the wrong j are signi�cant.

We will show this by looking at the pro�ts of the principal when she uses a

winner-prize tournament rather than a loser-prize tournament in some numer-

ical examples. We will consider examples in which the optimal tournament

is the strict loser-prize tournament, and compare the pro�ts of the principal

13

when she uses the strict loser-prize, the strict winner-prize, or the j = n=2

tournament.

We will make the following assumptions. We assume that F is a normal

distribution with standard deviation of 1. We assume that c(e) = e2

2and

�U = �2: We will assume a CRRA utility function of the form: u(w) = w1���11��

with � = 3. We will allow the number of participants, n, to vary, and consider

the change in the principal�s payo¤. In order to have a sense of what a

given level of pro�ts means, we will also calculate what the pro�ts of the

principal would be in the �rst-best case, where e¤ort is observable. In all of

the cases considered the strict loser-prize tournament is the optimal two-prize

tournament. The results are as follows:

Table 1: Prize Structuresn �first�best �loser �winner �n=2

2 1.05 0.22 0.22 0.22

4 1.05 0.45 0.06 0.31

6 1.05 0.52 -0.04 0.34

8 1.05 0.55 -0.11 0.36

10 1.05 0.57 -0.15 0.37

20 1.05 0.63 -0.27 0.39

50 1.05 0.68 -0.35 0.40

100 1.05 0.70 -0.39 0.41

500 1.05 0.76 -0.44 0.41

1000 1.05 0.77 -0.44 0.41

We see that the strict loser-prize tournament does considerably better than

the other tournaments. The j = n=2 tournament also does considerably

better than the strict winner-prize tournament. The pro�ts from the loser-

prize tournament become larger as n increases, as does the di¤erence in pro�ts.

This is related to Proposition 3 of Green and Stokey (1983). They show that,

when the principal can use n distinct prizes (rather than two), the principal�s

pro�ts in the tournament converge to the individual contract second-best level

14

of pro�ts as n increases. In our case, the principal�s pro�ts improve for the

strict loser-prize tournament, but do not converge to the individual contract

second-best.

The results we have presented here suggest that the di¤erence in the prin-

cipal�s pro�ts from choosing a winner-prize tournament rather than a loser-

prize tournament, are quite substantial. We do not believe, therefore, that a

winner-prize tournament can be described as �close to optimal.�

4.3 Are Losers Boiled in Oil?

One might wonder, based upon what we have shown so far whether optimal

rank-order tournaments require giving horri�c punishments to the worst per-

formers. Does the introduction a limited liability constraint that reduces the

ability to punish losers lead to a di¤erent conclusion? In a range of numerical

examples, we �nd that limited liability constraints often lead to punishing a

larger number of losers less, rather than switching to a winner-prize structure.

Mirrlees (1975) describes schemes in which the punishment could become

very large, but with su¢ ciently rapidly decreasing probability such that the

expected punishment remains bounded away from in�nity. Loser-prize tour-

naments do not have this character, provided that there are a �nite number

of players. The punishment for the worst performer is bounded away from

negative in�nity. Since every player has a 1=n probability of winning each

prize, giving negative in�nity to the worst performer means that the expected

payo¤ to a player is also negative in�nity so long as the number of players

is �nite. Therefore, an in�nitely harsh punishment for the worst performer

violates the individual rationality constraint.

What are the consequences of imposing a limited liability constraint? One

possibility is that the limited liability constraint will not bind. In some con-

texts, the principal�s ability to punish may be more than su¢ cient. If the

limited liability constraint does bind, however, might the principal choose to

implement a winner-prize tournament?

15

The principal has two ways to slacken the limited liability constraint. The

principal can choose to implement a tournament that induces less e¤ort. For a

given j, the amount given to the worst performer rises as less e¤ort is induced.

The principal might also choose to change j. Lowering j slackens the limited

liability constraint since the amount received by the worst performer rises

monotonically as j falls. Lower j means that more agents are punished less

harshly.

We cannot conclude that the principal will use the �rst method of slacken-

ing the constraint to the exclusion of the second. The principal may lower j in

response to a limited liability constraint. But, will the principal ever choose

to lower j to the point that the limited liability constraint does not bind?

Suppose, for example, that the unconstrained optimal strict loser-prize tour-

nament does not meet the limited liability constraint but the unconstrained

optimal n� 2 tournament meets the limited liability constraint. Is it possiblethat the principal, facing the limited liability constraint, would implement a

strict winner-prize tournament?

We have used �j to denote the pro�ts from the optimal j tournament

without a limited liability constraint. Suppose that �j is u-shaped in j. It

might be the case, if �j is u-shaped, that �n�2 is less than �n�1. This could

give the principal a reason for choosing to implement a strict winner-prize

tournament rather than an n � 2 tournament when faced with the limitedliability constraint.

If �j is monotonic increasing, however, this does not give a reason to im-

plement a strict winner-prize tournament. If �j is monotonic increasing, we

see only one reason why the principal would choose the strict winner-prize

tournament rather than the n � 2 tournament. The principal can reduce theimplemented e¤ort level to slacken the limited liability constraint. If reducing

the implemented e¤ort level slackens the constraint considerably more in the

strict winner-prize tournament than the n � 2 tournament, this could poten-tially give a reason for choosing the strict winner-prize tournament. However,

16

we do not believe that this is the case. It is our belief, therefore, that when

�j is increasing, the principal will never choose to lower j beyond the point

where the limited liability constraint is just met.

In numerous numerical examples, we �nd that �j is monotonic increasing.

And, we have con�rmed our intuition that the principal generally chooses to

lower j just enough to meet the limited liability constraint. Figure 2 shows

an example of �j with normally distributed noise with standard deviation 1,

CRRA utility with coe¢ cient of 2, and �U = �2. We see that in this case �jis monotonic increasing.

­0.2

­0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 10 11 12 13 14

j

prof

its

Figure 2: Pro�ts for a j-tournament

4.4 More Than Two Prizes

How do the results that we have obtained in the two-prize case translate to

the case where the principal can award n prizes? The general logic�that

risk aversion gives the principal a reason to punish at the bottom rather than

reward at the top�should certainly transfer. The principal will now choose a

somewhat smoother prize schedule, however.

17

The question we need to ask is: what does this smoothing look like? We

have examined this issue by looking at the prize schedules in numerical exam-

ples. We have found in these examples that removing the two-prize restriction

has very little e¤ect on the prize schedule when considered in utils. The prize

schedule in utils is nearly �at except at the bottom of the performance distri-

bution. The worst performer receives a considerably lower prize than anyone

else.

Since we can only observe prizes in money, it is more relevant what the

prize schedule looks like in money than utils. While we do not �nd that

the money-prize schedule is �at at the top of the performance distribution,

our results do translate. We will now present a numerical example and show

precisely how our results translate.

The numerical example we present here gives results that we consider to

be typical. We allow the principal to set n distinct prizes. We will set

n = 20. This large tournament size gives us a prize schedule that is relatively

smooth, and therefore allows us to see the results more clearly. We assume

that F is a normal distribution with standard deviation of 1. We assume that

c(e) = e2

2and �U = �2: We consider a CRRA utility function with a coe¢ cient

of relative risk-aversion of 2. In this case, the optimal two-prize tournament

is the strict loser-prize tournament.

As mentioned, the prize schedule in utils looks very similar to the prize

schedule in the two-prize case (see �gure 3). The worst performer receives a

much smaller prize, and the schedule is virtually �at at the top of the distrib-

ution.

18

­14

­12

­10

­8

­6

­4

­2

0

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Tournament Placement

Figure 3: Prizes in Utils

When we look at the prizes in money, we get a slightly di¤erent schedule

(see �gure 4). The reason for the di¤erence is the concavity of the utility

function.

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Tournament Placement

Figure 4: Prizes in Money

There are several things to note about the prize schedule in money. First,

the prize schedule is not entirely �at. More is awarded to the top performer

19

than the median performer. However, the prize schedule is close to a straight

line except at the bottom of the distribution. Therefore, the extra amount

given to the best performer over the second-best performer is very similar

to the extra amount given to the second-best performer over the third-best

performer. At the bottom of the distribution, this is not the case. In this

sense, there is no special prize for top performers. But, there is a special

punishment for the bottom performers.

We get a better sense of the prize schedule by looking at a �rst-di¤erencing

of the schedule. Figure 5 shows wk � wk�1. If wk � wk�1 is constant, this

corresponds to a linear schedule. We see that the schedule is virtually constant

except at the top and bottom of the distribution. At the top, there is a very

slight increase in wk � wk�1 and at the bottom, there is a very large increase

in wk � wk�1. This means that there is a very small extra prize for the best

performer but a very large extra punishment for the worst performer.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Tournament Placement

Figure 5: Changes in Prizes in Money

While the results that we obtain in the many-prize case are not as sharp

as in the two-prize case, they still give a view that is contrary to a number

of intuitions that readers may hold about optimal prize structures. It is not

20

the case that optimal tournaments have a �winner-take-all� character. To

the extent that the top performer gets an additional payo¤ for ranking �rst

rather than second, it is not considerably larger than the payo¤ for placing

second rather than third or third rather than fourth. And, while the optimal

tournament does not have a �loser-lose-all�character, it is optimal to give a

special punishment at the bottom.

5 Robustness

There are two reasons why a principal might want to give a special prize to

top performers. If the idiosyncratic noise distribution is su¢ ciently skewed

to the left, the principal might want to give a special prize to top performers.

If agents are heterogeneous in ability and that heterogeneity takes a multi-

plicative form, the principal might also want to give a special prize to top

performers.

Tournaments that punish losers give incentives to an agent through a fear

of low rank if she gets a bad draw from the noise distribution. Tournaments

that reward winners give incentives to an agent through a desire for high rank

if she gets a good draw from the noise distribution. When agents are as likely

to receive a bad draw as a good draw and when the principal cares equally

about the e¤ort of agents with good and bad draws, we �nd that a loser-prize

tournament is best. But, if these conditions fail, it may be better to reward

winners than punish losers.

When the noise distribution is skewed to the left, there are more agents

with good draws than bad draws in expectation. Since there are likely to be

relatively few agents with bad draws, an agent with a bad draw is likely to rank

low in the tournament even with considerable e¤ort. As a result, punishment

for low rank gives little incentive to exert e¤ort. This gives a reason to reward

winners instead. The principal may also wish to reward winners when agents

are heterogeneous in productivity. The principal is better o¤ giving incentives

21

to high productivity agents than giving incentives to low productivity agents

since the e¤ort of high productivity agents is more valuable.

5.1 Symmetric Distribution

Like Krishna and Morgan (1998), we have used a symmetric distribution. In

the previous section, the symmetry of the distribution allowed us to conclude

that the j tournament and the n � j tournament which pay out the same

prizes w1 and w2 induce the same level of e¤ort. This was the key result that

allowed us to establish that the optimal two-prize tournament was a loser-

prize tournament. When the distribution, F , is not symmetric, however,

we cannot conclude that the same e¤ort is induced by the same prizes w1and w2: When F is skewed to the left and j < n=2, the e¤ort is greater in

the j tournament (a winner-prize tournament) than the n� j tournament (a

loser-prize tournament).

When, for the same prizes w1 and w2; the e¤ort induced in the j tournament

(j < n=2) is su¢ ciently greater than that in the n�j tournament, the optimalj tournament (a winner-prize tournament) will dominate the optimal n � j

tournament (a loser-prize tournament). It is thus no longer clear that the

optimal two-prize tournament is a loser-prize tournament.

Let us discuss the intuition for this result. Suppose the noise distribution

is skewed considerably to the left and the principal implements a strict loser-

prize tournament. Only a few agents are likely to get bad draws from the

noise distribution, unlike a symmetric distribution in which a relatively large

number of agents are likely to get bad draws. An agent calculates that, if

she gets a bad draw, she is very likely to lose the tournament since there are

unlikely to be others with an equally bad draw. Therefore, it is not worthwhile

for such an agent to put in e¤ort: she will likely lose the tournament whether

she exerts e¤ort or not. Similarly, an agent with a good draw is unlikely

to lose the tournament even when little e¤ort is exerted. Therefore, agents

have very little reason to exert e¤ort under a loser-prize tournament. Under

22

a strict winner-prize tournament, there is a greater incentive to exert e¤ort.

Many agents are clumped at the top of the noise distribution, so the payo¤

from exerting e¤ort to try to win the tournament is considerable.

So, a parameter which is important in determining the optimal tournament

structure is the skewness of the noise distribution. When it is skewed to the

right, this favors a loser-prize tournament. When it is skewed to the left, this

favors a winner-prize tournament.

5.2 Heterogeneity

In thinking about heterogeneity in tournaments, we will ignore issues that

arise as the result of asymmetric information (i.e. when the agents know more

about their types than the principal.) We will assume that agents only learn

their types after signing a contract with the principal. This leads to the same

participation constraint for all types and eliminates the possibility of screening

on the part of the principal.

While we will not analyze the asymmetric information case in which the

principal�s problem is partly a screening problem, we do not believe that this

is a reason to give special rewards to top performers. Furthermore, if rewards

are given to top performers for screening reasons, this is very di¤erent from

the moral hazard reason given by Lazear and Rosen.

We will assume that agents learn their types before deciding how much

e¤ort to exert. Otherwise, the heterogeneous case is virtually identical to the

homogeneous case. The timing will therefore be as follows. Time 0: the

principal o¤ers a contract. Time 1: the agent decides whether to accept the

contract. Time 2: the agent learns her ability. Time 3: the agent chooses an

e¤ort level. Time 4: output is realized and prizes are awarded according to

the contract.

If we assume that the ability distribution is asymmetric, this is similar to

assuming that the noise distribution is asymmetric. For example, suppose

23

there are many high ability types and one low ability type. In this case, the

low ability type considers it almost certain that she will place last and the

high ability types consider it almost certain that they will not place last. As

a result, a strict loser-prize tournament does not give much incentive to exert

e¤ort. This is analogous to the case where the noise distribution is skewed to

the left.

In order to examine a separate issue, we will assume in this section that

the ability distribution is symmetric. However, it is important to remember

that if the ability distribution, like the noise distribution, is skewed to the left,

this gives a reason to reward winners.

In this context, introducing heterogeneity into the model can strengthen

or weaken the results of the previous section, depending upon the form that

the heterogeneity takes. If the heterogeneity takes an additive form, where

output for high ability types is qj = ej+�+"j+� with � > 0; this strengthens

the results of the previous section. If the heterogeneity takes a multiplicative

form, so that the output of high ability types is qj = �ej + "j + � with � > 1;

this weakens the results of the previous section.

Let us begin by considering the additive heterogeneity case. We will

compare the strict winner-prize tournament (j = 1) to the strict loser-prize

tournament (j = n � 1). It will be assumed that there are an equal numberof high ability types and low ability types. We will show that in such a

context, heterogeneity actually seems to strengthen our case. We will limit

our attention here to the case where c(e) = e2

2; but we believe that the result

may extend to a much more general class of e¤ort cost functions. Let us

establish this result, �rst de�ning the type of heterogeneity in the tournament.

De�nition 2 A tournament is called a �-Additive tournament when: there

are 2n individuals, n of whom are low ability and n of whom are high ability.

Agents learn their ability after signing a contract but before choosing an e¤ort

level. The output of a low ability individual exerting e¤ort ei is qi = ei+"i+�

where "i is an idiosyncratic shock and � is a common shock. The output of

24

a high ability individual exerting e¤ort ej is qj = ej + � + "j + � where "jis an idiosyncratic shock and � is a common shock, and � � 0. All of the

idiosyncratic shocks are independent and are distributed according to F. For all

individuals participating in the tournament,utility is given by u(w)�c(e) wherew is the prize received, e is the e¤ort exerted, and u0 � 0; u00 � 0; c0 � 0; c00 � 0:All individuals have the same outside option, which gives them utility �U:

The following result and its corollary establish this.

Proposition 3 Consider a �-Additive tournament with c(e) = e2

2. For F

symmetric and �su¢ ciently dispersed�, the strict loser-prize tournament which

pays out prizes w1 and w2 induces a higher level of e¤ort than the strict winner-

prize tournament which pays out prizes w1 and w2.

Proof. See appendix.

Unlike the homogeneous case, which induces the same level of e¤ort when

the prizes are the same in the loser- and winner-prize tournaments, the het-

erogeneous case results in greater e¤ort in the loser-prize case. In this sense,

additive heterogeneity strengthens the result. It follows from the same logic

as that in the proof of Proposition 1, that the strict loser-prize tournament

dominates the strict winner-prize tournament in this setting.

Corollary 2 Consider a �-Additive tournament with c(e) = e2

2. Suppose that

F is symmetric and �su¢ ciently dispersed,� and that u meets property (*).

Then, the principal�s pro�ts are greater when she implements a strict loser

prize tournament than when she implements a strict winner prize tournament.

Let us now de�ne a tournament in which the form of heterogeneity is

multiplicative. We will show that if there is multiplicative heterogeneity, the

principal may want to give special prizes to top performers.

De�nition 3 A tournament is called a �-Multiplicative tournament when:

there are 2n individuals, n of whom are low ability and n of whom are high

25

ability. Agents learn their ability after signing a contract but before choos-

ing an e¤ort level. The output of a low ability individual exerting e¤ort ei is

qi = ei+"i+� where "i is an idiosyncratic shock and � is a common shock. The

output of a high ability individual exerting e¤ort ej is qj = �ej+"j+� where "jis an idiosyncratic shock and � is a common shock, and � � 0. All of the idio-syncratic shocks are independent and are distributed according to F. For all

individuals participating in the tournament,utility is given by u(w)�c(e) wherew is the prize received, e is the e¤ort exerted, and u0 � 0; u00 � 0; c0 � 0; c00 � 0:All individuals have the same outside option, which gives them utility �U:

As mentioned before, a winner-prize tournament induces more e¤ort among

high-ability types and a loser-prize tournament induces more e¤ort among low

ability types. In a �-Additive tournament, the principal is equally well o¤ by

an increase in e¤ort of�e among high or low ability types. This is not the case,

however, in a �-Multiplicative tournament. In this case, an increase in e¤ort

of �e among high types leads to higher expected output than a corresponding

increase among low types. This means that when � is very large, it is better

for the principal to implement a winner-prize tournament, which encourages

competition among high-types. In this case, too, we will limit our attention

to a small class of e¤ort cost functions, but again, the theorem is likely to hold

for a somewhat broader class. The following theorem states the result.

Proposition 4 Consider a �-Multiplicative tournament with c(e) = de�; d >

0 and � > 1. If F symmetric and �su¢ ciently dispersed�, then for su¢ ciently

large �, the principal will prefer a strict winner-prize tournament to a strict

loser-prize tournament.

Proof. See appendix.

6 Conclusion

This paper gives a framework and an intuition for thinking about how prizes

should be structured in rank-order tournaments created to deal with moral

26

hazard. Because agents are risk averse, when the principal cares equally about

incentivizing agents with high and low output, she should punish losers. When

the principal cares especially about incentivizing agents with high output,

which occurs when agents di¤er in productivity or when the noise distribution

is skewed to the left, she may wish to give rewards to winners.

These results have considerable practical signi�cance. They allow us to

test whether aspects of employee compensation arise because of or in spite

of the moral hazard theory of tournaments. Within-�rm job promotions,

wage increases, bonuses, and CEO compensation have often been interpreted

as prizes for top performers in Lazear-Rosen rank-order tournaments. Our

results indicate that this intuition is false.

In casting doubt on the moral hazard story, our �ndings create a puzzle for

understanding wage increases and bonuses based upon relative performance.

27

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Lazear, Edward P. and Sherwin Rosen, �Rank-Order Tournaments asOptimum Labor Contracts,�Journal of Political Economy, 1981, 89, 841�

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29

7 Appendix

Proof of Lemma 1. In a j tournament, the e¤ort exerted by the players is

given by

c0(ej) =Xi

�iu(wi)

=

jXi=1

�i

!u1 +

nX

i=j+1

�i

!u2

SincePn

i=1 �i = 0;Pn

i=j+1 �i = �Pj

i=1 �i: Thus,

c0(ej) =

jXi=1

�i

!(u1 � u2)

Now, let us consider the e¤ort exerted in an n� j tournament.

c0(en�j) =

n�jXi=1

�i

!u1 +

nX

i=n�j+1�i

!u2

= �

nXi=n�j+1

�i

!u1 +

nX

i=n�j+1�i

!u2

When F is a symmetric distribution, we know that �i = ��n+1�i: Thus, forF symmetric,

Pni=n�j+1 �i = �

Pji=1 �i and

c0(en�j) =

jXi=1

�i

!(u1 � u2)

This proves the lemma.

Proof of Proposition 1. The argument will proceed as follows. We will

consider a j tournament (j � n2) and an n � j tournament that both meet

the IR constraint and lead to the same exertion of e¤ort by players in the IC

constraint. We will show that for two such tournaments, the sum of prizes

30

paid by the principal in the j tournament exceeds the sum of prizes paid by

the principal in the n � j tournament. Given this, it follows immediately

that the optimal j tournament is dominated by the optimal n� j tournament.Consider the optimal j tournament. Given this result, we know that we

can obtain the same e¤ort with an n � j tournament while meeting the IR

constraint and paying out less in prizes. Following this argument, we will

now consider a j tournament and an n� j tournament that both meet the IRconstraint and lead to the same e¤ort exertion. Let w1 and w2 denote the

prizes paid in the j tournament and let ui = u(wi): Similarly, let ~w1 and ~w2denote the prizes paid in the n � j tournament and let ~ui = u( ~wi): Further,

let � = jn: The IR constraints for the j and n� j tournaments imply that

�u1 + (1� �)u2 = �U

(1� �)~u1 + �~u2 = �U

Lemma 1 tells us that e¤ort is the same in the j and n� j tournaments whenu1 � u2 = ~u1 � ~u2: These three equations tell us that

~u1 =�

1� �u1 +

1� 2�1� �

�U

~u2 = 2�U � u1

u2 =��1� �

u1 +1

1� ��U

Let W denote the sum of prizes in the j tournament and ~W denote the sum

of prizes in the n� j tournament. Also, let h = u�1: Then

W = �w1 + (1� �)w2 = �h(u1) + (1� �)h(��1� �

u1 +1

1� ��U)

~W = � ~w2 + (1� �) ~w1 = �h(2 �U � u1) + (1� �)h(�

1� �u1 +

1� 2�1� �

�U)

Let g(x) = �h(x) + (1 � �)h(�U��x1�� ) and � = u1 � �U � 0: We need to show

31

that, for � � 12; W � ~W; or

g( �U +�)� g( �U ��) � 0 (*)

We see that

g0(x) = �(h0(x)� h0(�U � �x

1� �))

h00(y) = �u00(h(y))[u0(h(y))]3

: Since u is concave, h00 � 0. Observe that g0(x) � 0 for

x � �U and g0(x) � 0 for x � �U since h00 � 0. Let '(�) � g( �U+�)�g( �U��):A su¢ cient condition for (*) is that: '0(�) � 0 8� � 0 since '(0) = 0: Wesee that

'0(�) = g0( �U +�)� g0( �U ��)= �(h0( �U +�)� h0( �U � �

1� ��)) + �(h0( �U ��)� h0( �U +

�

1� ��))

= �(h0( �U +�)� h0( �U +�

1� ��))� �(h0( �U � �

1� ��)� h0( �U ��))

Let !(�; x; y) = � [(h0(x+ �)� h0(x))� (h0(y + �)� h0(y))]. Then,

'0(�) = !(1� 2�1� �

; �U +�

1� ��; �U ��)

Observe that 1�2�1�� � 0 since � �

12and �U + �

1��� � �U ��: We would liketo show that@!

@�(�; x; y) � 0 when x � y:

@!

@�(�; x; y) = �(h00(x+ �)� h00(y + �))

A su¢ cient condition therefore for @!@�(�; x; y) � 0 is h000 � 0: Thus, h000 � 0 is

a su¢ cient condition for our result.

h000(y) =3u0 (u00)2 � (u0)2 u000

(u0)6

Thus, h000 � 0 if and only if 3 (u00)2 � u0u000 � 0: This proves the result.

32

Proof of Proposition 2. Let us consider a j tournament and a j0 tourna-

ment with j0 > j � n=2: Let � = jnand �0 = j0

n: As in the proof of Proposition

1, we will compare j and j0 tournaments that lead to the same level of e¤ort

exertion and consider the amounts paid out in prizes by the principal. Let w1and w2 denote the prizes paid in the j tournament and ~w1 and ~w2 denote the

prizes paid in the j0 tournament. Let ui = u(wi), ~ui = u( ~wi) and let W and~W denote the sum of prizes in the j and j0 tournaments respectively. Before

we proceed, we need to de�ne two functions:

�(x) = �dnxe

(x) = n

Z x

0

�(x)dx

We see that ( jn) =

Pji=1 �i: Thus, the incentive compatibility constraints for

the j and j0 tournaments can be written as

c0(e) = (�)(u(w1)� u(w2))

c0(e) = (�0)(u( ~w1)� u( ~w2))

Individual rationality implies that

�u1 + (1� �)u2 = �U

�0~u1 + (1� �0)~u2 = �U

Combining these four constraints, we can solve for W and ~W in terms of u1:

Let us de�ne a few functions:

�(�0) = (�0)� (�)

(�0)+ �0

(�)

(�0)

h = u�1

g(x; �0) = �h(�0) + (1� �0)h(�U � �0x

1� �0)

(�0) = g(1� �(�0)1� �

u1 +�(�0)� �

1� ��U; �)

33

Then, we �nd that W and ~W can be expressed as follows:

~W = (�0)

W = (�)

Let us consider 0(x): If we �nd that 0(x) < 0 for x 2 [�; �0]; then it

follows that ~W < W: This implies that the j0 tournament dominates the j

tournament.

0(x) =��U � u1

��0(x)

�1

1� x

��xh0(u1) + (

1

�0(x)� x)h0(

�U � xu11� x

)

�+

�h(u1)� h(

�U � xu11� x

)

�Let us de�ne

�(u) =��U � u

��0(x)

�1

1� x

��xh0(u) + (

1

�0(x)� x)h0(

�U � xu

1� x)

�+

�h(u)� h(

�U � xu

1� x)

�We see that �( �U) = 0: Since u1 > �U; 0(x) = �(u1) < 0 if �0(u) < 0 for

u > �U:

�0(u) =

� �U � u

1� x

�x

h00(u)�

1�0(x) � x

1� xh00(

�U � xu

1� x)

!

+

�1� x(1 + �0(x))

1� x

��h0(u)� h0(

�U � xu

1� x)

�Suppose it were the case that �0(x) = 1: Then,

�0(u) =

� �U � u

1� x

�x

�h00(u)� h00(

�U � xu

1� x)

�+

�1� 2x1� x

��h0(u)� h0(

�U � xu

1� x)

�Recall that we are assuming 1 > x � 1

2and u > �U: By property (*), it

34

follows that h00; h000 > 0 (to see the argument, see the proof of Proposition

1). It therefore follows that the above expression is less than zero. Thus, if

�0(x) = 1; 0(x) < 0: From the de�nition of �; it follows that

�0(x) = 1 + 0(x)

(x)(1� x)

Since, (x) = nR x0�(x)dx; 0(x) = n�(x) = n�dnxe: Thus,

�0(x) = 1 +n�dnxe (x)

(1� x)

Suppose that F is a symmetric and uniform distribution. It is easily shown

that: �i = 0 for 1 < i < n: This implies that �0(x) = 1 for x 2 [ 1n; n�1n):

From this, it follows that when F is a symmetric uniform, the j0 tournament

dominates the j tournament where j0 > j � n=2: Hence, for F a symmetric

uniform, the optimal j tournament is the strict loser prize tournament. This

proves Proposition 2.

If we consider a distribution other than the uniform distribution, when will

the optimal two-prize tournament be the strict loser-prize tournament? From

the expression for �0(u); we see that there is an interaction between �0(x)

and h when �0(x) 6= 1: �0(u) will be negative even with such an interactionunless �0(x) is considerably less than 1 and h00 is large relative to h000. These

conditions are equivalent to (i)��� �iPi

j=1 �j� i��� is large for 1 < i < n

2and (ii)

3u0(u00)2�(u0)2u000(u0)6 is very small relative to �u00

(u0)2 : In fact, it is not even clear that

when (i) and (ii) hold that �0(u) will be positive, and even if �0(u) is positive,

it does not follow that the optimal tournament is not the strict loser-prize

tournament.

Proof of Proposition 3. Consider a �-Additive strict loser-prize tourna-

ment and a �-Additive strict winner-prize tournament, both of which pay out

prizes w1 and w2: Let ui = u(wi): We will assume that F is symmetric and

�su¢ ciently dispersed� and that c(e) = e2

2: Let us denote the e¤ort of the

high and low types in the winner-prize case by eW� and eW0 respectively and

35

let us denote the e¤orts in the loser-prize case by eL� and eL0 : Let us denote

the sum of e¤orts by eW and eL in the winner- and loser-prize cases respec-

tively We will now proceed to show that eW � eL. First consider e¤ort in

the winner-prize tournament:

Pr(winj�; e) = E"i�F (e� eW� + "i)

n=2�1F (e� eW0 + � + "i)n=2�

eW� = argmaxePr(winj�; e)(u1 � u2)�

e2

2

eW� = E"i

" �n2� 1�F ("i)

n=2�2F (�W + � + "i)n=2f("i)

+n2F ("i)

n=2�1F (�W + � + "i)n=2�1f(�W + � + "i)

#(u1 � u2)

where �W = eW� � eW0

eW� = (�W + �)(u1 � u2)

where (x) = E"i

" �n2� 1�F ("i)

n=2�2F (x+ "i)n=2f("i)

+n2F ("i)

n=2�1F (x+ "i)n=2�1f(x+ "i)

#

Pr(winj0; e) = E"i�F (e� eW� � � + "i)

n=2F (e� eW0 + "i)n=2�1�

eW0 = argmaxePr(winj0; e)(u1 � u2)�

e2

2

eW0 = E"i

" �n2� 1�F ("i)

n=2�2F (��W � � + "i)n=2f("i)

+n2F ("i)

n=2�1F (��W � � + "i)n=2�1f(��W � � + "i)

#(u1 � u2)

= (��W � �)(u1 � u2)

Now let us consider e¤ort exertion in the loser-prize tournament:

Pr(notlosej�; e) = 1� Pr(losej�; e)

= 1� E"i�(1� F (e� eL� + "i))

n=2�1(1� F (e� eL0 + � + "i))n=2�

= E"i

" �n2� 1�F ("i)

n=2�2F (��L � � + "i)n=2f("i)

+n2F ("i)

n=2�1F (��L � � + "i)n=2�1f(��L � � + "i)

#(u1 � u2)

36

= (��L � �)(u1 � u2)

eL� = argmaxePr(notlosej�; e)(u1 � u2)�

e2

2

eL� = E"i

" �n2� 1�(1� F ("i))

n=2�2(1� F (�L + � + "i))n=2f("i)

+n2(1� F ("i))

n=2�1(1� F (�L + � + "i))n=2�1f(�L + � + "i)

#(u1�u2)

= E"i

" �n2� 1�F (�"i)n=2�2F (��L � � � "i)

n=2f("i)

+n2F (�"i)n=2�1F (��L � � � "i)

n=2�1f(�L + � + "i)

#(u1 � u2)

= (u1�u2)ZR

" �n2� 1�F (�"i)n=2�2F (��L � � � "i)

n=2f("i)

+n2F (�"i)n=2�1F (��L � � � "i)

n=2�1f(�L + � + "i)

#f("i)d"i

= (u1�u2)ZR

" �n2� 1�F (�"i)n=2�2F (��L � � � "i)

n=2f(�"i)+n2F (�"i)n=2�1F (��L � � � "i)

n=2�1f(��L � � � "i)

#f(�"i)d"i

Let z = �"i

eL� = (u1�u2)ZR

" �n2� 1�F (z)n=2�2F (��L � � + z)n=2f(z)

+n2F (z)n=2�1F (��L � � + z)n=2�1f(��L � � + z)

#f(z)dz

= (u1 � u2)

ZR

" �n2� 1�F (z)n=2�2F (��L � � + z)n=2f(z)

+n2F (z)n=2�1F (��L � � + z)n=2�1f(��L � � + z)

#f(z)dz

= E"i

" �n2� 1�F ("i)

n=2�2F (��L � � + "i)n=2f("i)

+n2F ("i)

n=2�1F (��L � � + "i)n=2�1f(��L � � + "i)

#(u1 � u2)

eL� = (��L � �)(u1 � u2)

Pr(notlosej0; e) = 1� Pr(losej�; e)= 1� E"i

�(1� F (e� eL� � � + "i))

n=2(1� F (e� eL0 + "i))n=2�1�

eL0 = argmaxePr(notlosej0; e)(u1 � u2)�

e2

2

37

eL0 = E"i

" �n2� 1�(1� F ("i))

n=2�2(1� F (��L � � + "i))n=2f("i)

+n2(1� F ("i))

n=2�1(1� F (��L � � + "i))n=2�1f(��L � � + "i)

#�(u1 � u2)

= E"i

" �n2� 1�F (�"i)n=2�2F (�L + � � "i)

n=2f("i)

+n2F (�"i)n=2�1F (�L + � � "i)

n=2�1f(��L � � + "i)

#(u1 � u2)

= (u1�u2)ZR

" �n2� 1�F (�"i)n=2�2F (�L + � � "i)

n=2f(�"i)+n2F (�"i)n=2�1F (�L + � � "i)

n=2�1f(�L + � � "i)

#f("i)d"i

Let z = �"i

= (u1 � u2)

ZR

" �n2� 1�F (z)n=2�2F (�L + � + z)n=2f(z)

+n2F (z)n=2�1F (�L + � + z)n=2�1f(�L + � + z)

#f(z)dz

= E"i

" �n2� 1�F ("i)

n=2�2F (�L + � + "i)n=2f("i)

+n2F ("i)

n=2�1F (�L + � + "i)n=2�1f(�L + � + "i)

#(u1 � u2)

= (�L + �)(u1 � u2)

We see then that

eW = eW� + eW0

=�(�W + �) + (��W � �)

�(u1 � u2)

= h(�W + �)(u1 � u2)

where h(x) = (x) + (�x)eL = eL� + eL0

=�(�L + �) + (��L � �)

�(u1 � u2)

= h(�L + �)(u1 � u2)

There are two things that we need to show to complete the proof. (1)

�W � �L, and (2) h(x) is decreasing. It immediately follows from (1) and

(2) that eW � eL:

Proof of (1):

38

�W =�(�W + �)�(��W � �)

�(u1 � u2)

�L =�(��L � �)�(�L + �)

�(u1 � u2)

Let g(x) = [(x)�(�x)] (u1 � u2): Then �W = g(�W + �) and �L =

�g(�L + �): Let us now �nd a simpler expression for g:

(x) = E"i

" �n2� 1�F ("i)

n=2�2F (x+ "i)n=2f("i)

+n2F ("i)

n=2�1F (x+ "i)n=2�1f(x+ "i)

#

= E"i

h�n2� 1�F ("i)

n=2�2F (x+ "i)n=2f("i)

i+E"i

hn2F ("i)

n=2�1F (x+ "i)n=2�1f(x+ "i)

i

(�x) = E"i

h�n2� 1�F ("i)

n=2�2F (�x+ "i)n=2f("i)

i+E"i

hn2F ("i)

n=2�1F (�x+ "i)n=2�1f(�x+ "i)

i

= E"i

h�n2� 1�F ("i)

n=2�2F (�x+ "i)n=2f("i)

i+

ZR

n

2F ("i)

n=2�1F (�x+ "i)n=2�1f(�x+ "i)f("i)d"i

Let z = �x+ "i

= E"i

h�n2� 1�F ("i)

n=2�2F (�x+ "i)n=2f("i)

i+

ZR

n

2F (x+ z)n=2�1F (z)n=2�1f(z)f(x+ z)dz

39

= E"i

h�n2� 1�F ("i)

n=2�2F (�x+ "i)n=2f("i)

i+E"i

hn2F ("i)

n=2�1F (x+ "i)n=2�1f(x+ "i)

i

g(x) = (x)�(�x)

= E"i

h�n2� 1�F ("i)

n2�2F (x+ "i)

n2 f("i)

i�E"i

h�n2� 1�F (�x+ "i)

n2F ("i)

n2�2f("i)

i= E"i

h�n2� 1�F ("i)

n2�2f("i)

�F (x+ "i)

n2 � F (�x+ "i)

n2

�ig0(x) = E"i

"n

2

�n2� 1�F ("i)

n2�2f("i)

F (x+ "i)

n2�1f(x+ "i)

+F (�x+ "i)n2�1f(�x+ "i)

!#� 0

So, g0(x) � 0: Observe that

limx�!1

jg(x)j = limx�!1

�����E"i"�n2� 1�F ("i)

n2�2f("i)

F (x+ "i)

n2

�F (�x+ "i)n2

!#�����=

���E"i h�n2 � 1�F ("i)n2�2f("i) (1� 0)i���=

���E"i h�n2 � 1�F ("i)n2�2f("i)i��� <1Since this limit is �nite, it follows that for x su¢ ciently large, g(x + �) < x:

It is obvious that g(0) = 0. Since g is increasing, it follows that g(�) � 0:

Thus, for x = 0, g(x + �) � x: By continuity, there is a value of x, x �0, with g(x + �) = x: Hence, �W � 0: Now, suppose �W < �L: �L

solves �L = �g(�L + �): Since �W + � � 0 and g(0) = 0; we know that

�g(�W + �) � 0. Thus, �W � �g(�W + �): Since g is increasing, �W < �L

implies that �L > �g(�L + �), which is a contradiction. It therefore follows

that, if a solution for �L exists, �W � �L: For x = ��; x < �g(x + �) and

for x = �W ; x � �g(x + �): Thus, by continuity, there exists a solution for

40

�L with �� � �L � �W : This proves (1).

Proof of (2): Since �L = �g(�L + �); it follows that

d(�L + �)

d�=

1

1 + g0(�L + �)� 0

Since, �L(�) + � � �W (�) + �; it follows that there exists �0 � � such that

�L(�0) + �0 = �W (�) + �

Since eW (�) = h(�W (�) + �)(u1 � u2) and eL(�) = h(�L(�) + �)(u1 � u2); we

see that eW (�) = eL(�0) where �0 � �: We will now argue that eL(�0) � eL(�)

for �0 � �: From this, it immediately follows that eW (�) � eL(�); which proves

the claim.

Consider �0 � �: High ability types are more certain to perform well relative

to low ability types than they were before. When all high types exert the same

level of e¤ort, as they will in equilibrium, their prospects do not change at all

relative to one another. Therefore, for a given level of e¤ort by the low types,

eL0 , high types choose to reduce their level of e¤ort exertion. Low ability

types, on the other hand, are more certain to perform poorly relative to high

ability types than they were before. When all low ability types exert the same

level of e¤ort, as they will in equilibrium, their prospects do not change at all

relative to one another. Therefore, low ability types also have less incentive

to exert e¤ort for a given eL� than they did before. As a result, both types

reduce their e¤ort levels in equilibrium. Hence, eL(�0) � eL(�), which proves

the proposition.

Proof of Proposition 4. Let us denote the e¤ort of the high and low types

in the winner-prize case by eW� and eW0 respectively and let us denote the e¤orts

in the loser-prize case by eL� and eL0 : Let us denote the sum of e¤orts by eW

and eL in the winner- and loser-prize cases respectively In the homogeneous

case, holding u1 � u2 constant produced the same e¤ort in the winner-prize

and loser-prize tournaments, and this allowed us to conclude that the loser-

prize tournament dominated. We will show that, holding u1�u2 constant, as

41

� !1 eW

eL!1: This suggests that, for � su¢ ciently large, the winner-prize

tournament produces higher pro�ts than the loser-prize tournament. Hence,eW

eL! 1 is a su¢ cient condition for the result. First consider e¤ort in the

winner-prize tournament:

Pr(winj�; e) = E"i�F (e� � eW� � + "i)

n=2�1F (e� � eW0 + "i)n=2�

eW� = argmaxePr(winj�; e)(u1 � u2)� c(e)

c0(eW� ) = �E"i

" �n2� 1�F ("i)

n=2�2F (�W + "i)n=2f("i)

+n2F ("i�)

n=2�1F (�W + "i)n=2�1f(�W + "i)

#�(u1 � u2)

where �W = eW� � � eW0

c0(eW� ) = �(�W )(u1 � u2)

where (x) = E"i

" �n2� 1�F ("i)

n=2�2F (x+ "i)n=2f("i)

+n2F ("i)

n=2�1F (x+ "i)n=2�1f(x+ "i)

#

Pr(winj0; e) = E"i�F (e� eW� � + "i)

n=2F (e� eW0 + "i)n=2�1�

c0(eW0 ) = argmaxePr(winj0; e)(u1 � u2)� c(e)

c0(eW0 ) = E"i

" �n2� 1�F ("i)

n=2�2F (��W + "i)n=2f("i)

+n2F ("i)

n=2�1F (��W + "i)n=2�1f(��W + "i)

#�(u1 � u2)

= (��W )(u1 � u2)

Now let us consider e¤ort exertion in the loser-prize tournament:

Pr(notlosej�; e) = 1� Pr(losej�; e)= 1� E"i

�(1� F (e� � eL� � + "i))

n=2�1(1� F (e� � eL0 + "i))n=2�

eL� = argmaxePr(notlosej�; e)(u1 � u2)� c(e)

42

c0(eL� ) = �E"i

" �n2� 1�(1� F ("i))

n=2�2(1� F (�L + "i))n=2f("i)

+n2(1� F ("i))

n=2�1(1� F (�L + "i))n=2�1f(�L + "i)

#�(u1 � u2)

= E"i

" �n2� 1�F (�"i)n=2�2F (��L � "i)

n=2f("i)

+n2F (�"i)n=2�1F (��L � "i)

n=2�1f(�L + "i)

#(u1 � u2)

= (u1 � u2)

ZR

" �n2� 1�F (�"i)n=2�2F (��L � "i)

n=2f("i)

+n2F (�"i)n=2�1F (��L � "i)

n=2�1f(�L + "i)

#�f("i)d"i

= (u1 � u2)�

ZR

" �n2� 1�F (�"i)n=2�2F (��L � "i)

n=2f(�"i)+n2F (�"i)n=2�1F (��L � "i)

n=2�1f(��L � "i)

#�f(�"i)d"i

Let z = �"i

= (u1 � u2)�

ZR

" �n2� 1�F (z)n=2�2F (��L + z)n=2f(z)

+n2F (z)n=2�1F (��L + z)n=2�1f(��L + z)

#f(z)dz

= �E"i

" �n2� 1�F ("i)

n=2�2F (��L + "i)n=2f("i)

+n2F ("i)

n=2�1F (��L + "i)n=2�1f(��L + "i)

#(u1 � u2)

= �(��L)(u1 � u2)

43

Pr(notlosej0; e) = 1� Pr(losej�; e)= 1� E"i

�(1� F (e� eL� � + "i))

n=2(1� F (e� eL0 + "i))n=2�1�

eL0 = argmaxePr(notlosej0; e)(u1 � u2)� c(e)

c0(eL0 ) = E"i

�" �

n2� 1�(1� F ("i))

n=2�2(1� F (��L + "i))n=2f("i)

+n2(1� F ("i))

n=2�1(1� F (��L + "i))n=2�1f(��L + "i)

#�(u1 � u2)

= E"i

" �n2� 1�F (�"i)n=2�2F (�L � "i)

n=2f("i)

+n2F (�"i)n=2�1F (�L � "i)

n=2�1f(��L + "i)

#�(u1 � u2)

= (u1 � u2)

�ZR

" �n2� 1�F (�"i)n=2�2F (�L � "i)

n=2f(�"i)+n2F (�"i)n=2�1F (�L � "i)

n=2�1f(�L � "i)

#f("i)d"i

Let z = �"i= (u1 � u2)

�ZR

" �n2� 1�F (z)n=2�2F (�L + z)n=2f(z)

+n2F (z)n=2�1F (�L + z)n=2�1f(�L + z)

#f(z)dz

= E"i

" �n2� 1�F ("i)

n=2�2F (�L + "i)n=2f("i)

+n2F ("i)

n=2�1F (�L + "i)n=2�1f(�L + "i)

#(u1 � u2)

= (�L)(u1 � u2)

Let = (c0)�1: Since c(x) = dx�; (x) = xd�

1��1 : Since � > 1; limx!1 (x) =

1: We see then that

eW = eW� � + eW0

= ((�W )�2(u1 � u2)) + ((��W )(u1 � u2))

eL = eL� + eL0

= ((�L)(u1 � u2)) + ((��L)�2(u1 � u2))

44

As � ! 1; we see that eW

eL! lim

�!1 ((�W )�2(u1�u2)) ((��L)�2(u1�u2))

= lim�!1

�(�W )(��L)

� 1��1

:

We know that

�W = ((�W )�2(u1 � u2))� ((��W )(u1 � u2))

�L = ((��L)�2(u1 � u2))� ((�L)(u1 � u2))

It is easily shown that limx!1

(x) = c > 0 and limx!�1

(x) = 0 and that (x)

is �nite for all x. It therefore follows that lim�!1

�W = 1 and lim�!1

�L = 1:

From this, we conclude that lim�!1

(�W )(��L) =1, and that

eW

eL!1 as � !1:

We have shown, therefore, that for the same prizes w1 and w2 in the strict

winner- and loser-prize tournaments, the ratio of induced e¤orts, eW

eL; goes to

in�nity as � ! 1: It immediately follows that, for � su¢ ciently large, the

principal will prefer the optimal strict winner-prize tournament to the optimal

strict loser-prize tournament.

45