D O C U M E N T O D E T R A B A J O

Instituto de EconomíaTESIS d

e MA

GÍSTER

I N S T I T U T O D E E C O N O M Í A

w w w . e c o n o m i a . p u c . c l

Tournaments with Homogeneous and Heterogeneous Agents.Discussion for the Chilean Policy of Bonus to Schools

Caroline Laplace.

2011

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PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE I N S T I T U T O D E E C O N O M I A MAGISTER EN ECONOMIA

TESIS DE GRADO

MAGISTER EN ECONOMIA

Laplace Caroline

Agosto 2011

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PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE I N S T I T U T O D E E C O N O M I A MAGISTER EN ECONOMIA

Tournaments with homogeneous and heterogeneous agents. Discussion

for the Chilean policy of bonus to schools.

Laplace Caroline

Comisión

Montero Juan-Pablo

Rau Tomás

Traferri Alejandra

Wagner Gert

Santiago, julio 2011

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I. Introduction

The theory of contracts establishes conditions to put in place an ideal incentive

structure. The traditional pay-per-hour cannot create efficient incentives to make people

working harder whereas the firm wants to increase her welfare. Thus, theories and practices

turned into the development of incentives based on performance according to the outputs

because the inputs are not public information. This is often possible because there is a direct

relationship between the quality of the work, the results and output that benefit the firms. So,

the performance measure wants to reveal the real effort. However, it may happen that the

output is not quantifiable like in the Public sector since the State provides “public” goods that

benefit all the society or that the output per worker is not measurable if production process is

not sufficiently separable for an unbiased estimate of each contribution to output

(Malcomson, 1986). In this case, rank-order contracts make each agent’s compensation a

function of his relative performance to the group. So, the ranking is useful when the principal

cannot determinate the exact marginal production. Moreover, a bonus or contract based on

relative performance introduces competition. Competition is useful to increase efforts

without wasting resources since only the best(s) receives a prize. Competition is well-known in

sports contests or salesmen’ salary but it can be introduced in other sectors in order to

increase efforts as in education. The idea of this paper comes from the Chilean education

policy called the SNED. The SNED is “the national subsidized school performance

evaluation system” or in Spanish “Sistema nacional de evaluación del desempeño”, created in

1996. The SNED concerns the public and private subsidized schools. It was created with two

objectives. The first is to contribute to the improvement of the education provided by the

state through a monetary reward to the 25% best schools from 1996 to 2005 and to the 35%

best schools then (100% of the bonus to the 25% best schools and 60% of the bonus to the

other 10%). The second objective is to provide the school community and parents with

information (Contreras et.al., 2003). Empirical evaluations study the effects of the SNED on

the SIMCE results1

showing that the SNED seems to be a good policy (Contreras et al., 2003;

Mizala and Romaguera, 2000 and 2004). Yet, the bonus presents weaknesses because the

evaluation is principally done on intra-cohorts’ results. (Carnoy et.al., 2007). Studying this

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the Simce is : Sistema de mediación de calidad de educación or System of measure of the educational quality.

Chilean educational government tests students’ level of fourth and eighth grade according to the minimal

objectives of education.

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policy, one may wonder how to establish an optimal rank-order bonus and how the efforts

move according to the bonus and the proportion of winners. The purpose of this work is to

propose a theoretical model to study the conditions of an optimal bonus. In particular, my

study is in line with the debate about the efficiency of a winner-take-all (WTA) scheme versus

a bonus distributed to the k first persons. The general presentation provides a framework that

may be easily used but I will underline the specificities of a public policy for teachers.

The paper is organized as follows. In section II, the related literature is presented.

The third part will present the general model before using it. Then, the fourth part will

present a benchmark with the issues the principal has to face with. The fifth part presents a

very simple case of two agents to make the reader aware of the problem and a discussion will

explain why the conclusions cannot be extended to tournament with more than two agents.

Then, the study presents a model of Nash equilibrium in which all the agents choose the

same effort and determinates the conditions of movement of efforts. In section VII, the

model is extended to the situation where the agents are heterogeneous and know their level

of ability. Thus, agents of different type do not choose the same effort. I present a general

framework, a general application and above all, a specific application with two types of effort

(high, low) and two types of ability (high, low).Conclusion reminds the main results and

implications to make optimal a rank order bonus.

II. Related literature

The theory of contracts draws a scheme that let firms and organizations understand how

to improve their workers’ productivity and how to maximize their profit. The relationship

between principal and agent (PA) is stained by asymmetric information since “by definition

the agent has been selected for his specialized knowledge and the principal can never hope to

completely check the agent’s performance” (Arrow, 1963 in Laffont and Mortimort, 2002).

Thus, the principal has to establish a scheme to make the agents do the effort he wants. To

make an efficient reward scheme, it is significant to be aware of the issues the model has to

face with. Thus, the principal may be the state, an organization or a firm. The particularity of

a state or an organization is that they may want to improve efforts and quality without

knowing exactly what the output is whereas generally a firm may measure it better. The PA

theory embraces issues of accountability and it may be particular difficult and costly in some

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cases. The PA model depends on significant conditions as the motivation of the agents, the

moral hazard, the information about the relationship between output and agents’ efforts and

whether the agents are multiple tasks (Levacic 2009). In particular, the moral hazard

increases when there is uncertainty regarding the relationship between the agents’ effort and

the output. In the case of Chile for example, it was shown that SIMCE scores are almost like

a white noise and so they do not represent well a school’s level (Mizala, Romaguera and

Urquiola 2007). The information asymmetry is all the bigger issue as the workers realize

different outputs. This asymmetry may be explained by the lack of competition in the sector.

In a competitive market, principals can easily compare the performance so they can contract

who they want according to their performance. However, in a no or low competitive sector,

this is not possible and so, the PA theory about optimal contracts can be effective when

competition is limited. However, money is not always the best solution and it is easy

imaginable to induce more efforts without monetary incentives if the workers value other

things as reputation. For example, Jurges, Richter and Schneider (2004) show through a PA

model how the creation of central tests of student achievement in Germany increased the

efforts of teachers because they value reputation.

This paper is interested in a rank-order tournament bonus. Lazear and Rosen (1979) are

the pioneers in the study of tournament. They compare a rank-order tournament to a

classical piece-rate contract and a fixed standard and demonstrate that tournament and piece-

rate are equivalent when agents are risk-neutral so tournaments dominate piece-rates when

rank is more easily observed than each individual’s level of output. This output depends on

own effort and on an additive shock that is common to all agents. They do not know the

value of the shock when they choose their level of effort but they do know its distribution.

Their model consider the case with two agents and then with N agents. On the contrary,

when they are risk-averse, tournaments can be more efficient than piece-rate when the

activities present a high degree of inherent riskiness. The relative variance of the common

shock and the stochastic component of the effort are the main variables to choose between

the three possibilities. Thus, if the variance of the common shock is large, the tournament is

the best whereas if the variance of the stochastic component of output attributable to effort is

relatively large, it is one of the other two schemes will dominate. As Green and Stokey (1983)

noted, a tournament performs well when the common shock is important because

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“competing in a tournament is like being judged against a standard that is a random variable

(the opponent’s output). This is useful if the random standard is highly correlated with the

random component of the agent’s own output and detrimental otherwise, because it

introduces additional “noise” into the relationship between effort and compensation”. Green

and Stokey offer a model of one neutral principal and many risk-averse agents. They study

the role of the common shock when the output depends on a common shock and on his

own effort as in Lazear and Rosen. They find that the tournament is better when the

distribution of the common shock is sufficiently diffuse because the tournament may reduce

the randomness of any agent’s compensation by considering all the time this shock. However,

this same tournament increases the randomness in compensation by making the reward

depend on the idiosyncratic of the peers. Thus, the preference of tournaments or contracts

depends on which effect dominates. Then, Nalebuff and Stiglitz (1982) propose a rank-order

tournament model with another form of output. The output does not depend only on one

shock but two. It depends on a common environmental variable (common to all agents) and

on an individualistic random variable. They find that a rank-order tournament may attain the

first best allocation but that a penalty to the lowest ranked individual is superior to a prize to

the highest ranked individual in motivating effort. The model proposed here won’t take into

part the idea of a penalty because it is generally impossible to put in place such a scheme but

it will follow the Nalebuff and Stiglitz’s output structure. These papers are always taken as

references in studies of tournament but the problem is that they compare piece-rate contract

to rank-order tournament. They consider that the principal may choose between the different

schemes where there are sectors in which a piece-rate scheme is obviously impossible. The

issue of this work is to study the effort and the conditions to make optimal a rank-order

tournament without considering other types of contracts.

Following the first studies of tournaments, issues have been underlined and studied.

Some of them are the problems of sabotage and positive externalities among agents

(Harbring and Irlenbush, 2005; Drago and Turnbull, 1988). When the output is the result of

a common work, the agents may try to sabotage the work of the others to win. Harbring and

Irlenbush find that the productive and destructive activities are not influenced neither by the

number of agents taking part in the tournament nor by the fraction of the winner prize. In the

case of externalities, Drago and Turnbull find that the welfare resulting from the tournament

falls if agents are risk-averse. This may explain why some firms reject the idea of

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tournaments. Closer to my study, there is a significant debate about the best choice of

number of winners. Cason et.al. (2010) compare the decision of entry between a WTA

sheme and a percentage prize (PP) in which the bonus is divided between agents according to

their share of the total achievement in an experimental study. They find that the PP

provokes more entry and more total effort than the WTA because it limits the degree to

which heterogeneity discourages weaker contests without altering the performance of stronger

entrants. Moreover, in a WTA, agents may decide to not entry if they would face with only

one much stronger opponent whereas their decision of entry in a PP is determined according

to the average performance. Finally, PP is better because it is as effective as WTA to

determinate the top agents but is more efficient to increase aggregate performance. Without

considering the problem of entry but directly the efficiency on the effort, Krishna and

Morgan (1998) show that in the case of homogeneous agents and regardless of risk

preference, the WTA principal is better in the two and three agents’ cases. On the opposite,

their study of four agents reveals that the WTA is better only if the agents are risk-neutral

whereas the prize has to be divided at least between the first and second places if they are

risk-averse. In a special work on salespeople, Kalra and Shi (2001) study the optimal design

of contest. They compare three schemes: a WTA, a multiple-winners format (MW) in which

the k best results receive the same prize and a rank-order tournament (RO) in which only the

k best agents win and the prizes are proportional to the place such as the first wins more the

second, etc. First, RO is better than MW because in a MW scheme, the last winner receives

the same prize than the other winners whereas he does less effort. Thus, salespeople may do

a more little effort than in a RO. The results are sensitive to the distributional assumption of

sales. If it is a logistic distribution, the aversion affects. The WTA is the best with risk-

neutrality whereas the number of winners has to increase and the spread between prizes

decreases with risk-aversion. On the opposite, the uniform distribution leads to conclude that

WTA is the best regardless risk-aversion. Moreover, if the WTA does not meet the

participation constraint, the optimal tournament has to provide a big prize to the top

salesperson and a small reward to many others. The objective to the small prize is to ensure

that all salespersons participate. Thus, it does not seem that there is one best scheme but

rather than the scheme has to follow the characteristics of the set. In a comparison of

multiprize vs one prize, Moldavenu and Sela (2001) find that it is the form of the cost

function that determinates the number of winners. If the players have a linear or concave cost

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functions, the optimum is a single prize whereas it is at least two prizes if they have a convex

cost function. They add ability in the model. Increasing the value of the prize generates an

increase in equilibrium efforts because a higher ability leads to a higher chance to win the first

prize. However, the probability of getting the second prize is not monotonous in ability and

the marginal effect of this prize is ambiguous. It is negative for the high abilities and positive

for middle-low abilities. Moreover, it exists a threshold such as for abilities below it, the

marginal effect of the second prize is littler than the marginal effect of the first place. Thus,

the relevant variable to maximize the average expected effort of each contestant is the average

difference between the marginal effects of the second and first prizes. If the difference is

negative, the optimum tournament is a WTA. If there is ability, agents are heterogeneous.

We may encounter two main types of heterogeneity: in the cost function and directly in the

output. Harbring and Lünser (2008) make a model of heterogeneity with a strong and a weak

agent. The strong agent has an advantage in her function of cost since it is only

whereas the

weak agent suffers a cost of

with . They work with the prize spread and find that

weak agent decides to compete against strong agent only if the prize spread is sufficiently large

and so the principal may discover the type of each agent. On the other side, Akerlof and

Holden (2008) use the heterogeneity in the output in two forms. Their objective is to

compare the winners-prize tournaments to the losers-prize tournaments so the conclusions

are not our concern but the model itself is relevant. The additive heterogeneity is when the

agent’s output is with the idiosyncratic shock, the common shock

and the agent’s type (ability). They speak about multiplicative form when the agent’s

output is . The multiplicative form acts more directly on the effort but

both forms may represent the reality. Moreover, they study three cases: when the agents learn

their types after deciding whether to participate in the tournament and after choosing an

effort level, when they learn their types before choosing the effort but after the decision to

participate and the case when they learn their types before the decisions. In my case, the

decision of participation to the tournament is not relevant since all the schools participate in

the bonus scheme without deciding it. I will study the case when agents learn their type

before choosing the level of effort because if they do not know, they will not consider it as an

advantage (or disadvantage) in the decision of effort. Thus, it would be the second one of

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their study. Then, they use the simple case that if the agents have low ability and

if they have high ability. I would present the special case of two efforts with two

abilities. In the case of schools, it would be interesting to study the case of three abilities (the

worst schools, the best ones and the middle-ones) but it complicates the study so this stays as

a possible extension of my work.

III. The general model

The paper presents different extensions of a general model. Before working with

these extensions, I present the general equations and conditions of equilibrium. Thus, as I

explained before, there are a principal and n agents. The principal can observe neither the

effort nor the shocks. It would be possible to discuss the validity of the measure of the output

yet I make the assumption that the output measures what the principal wants. Following

Nalebuff and Stiglitz (1982), this output Qi is a function of the effort ei, a common shock

and an individual shock µi:

and µ have the distribution F( . One interpretation of these shocks is an agricultural

context where the common shock is the general weather of the region where the individual

one is the weather (rainfall) on a particular farm. Another explanation may be, in the context

of a bonus to school, that the common shock is the global socio-economic level that forms a

homogeneous group of tournament whereas the individualistic variable is the particular

characteristics of a school. The principal do not observe the shocks whereas the agent

observes and decides on

Then, H is the distribution function of , h its density and we specify that:

( ( ( )

Moreover, we suppose that ( ) ( .

The agents are rational and want to maximize their welfare. Since the utility depends on the

probability of winning the bonus, the agents maximize their expected utility. Given U the

utility function, the agent i maximizes:

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( ( ( ( ( (

With: w the fixed salary that all the agents receive

b the bonus. It is a proportion of the salary.

ei the effort of the agent i

V(.) the cost function of the effort

By assumption: . So, the agents are risk-averse and the

cost is increasing at increasing rates. Actually, the results will present the case with risk-

neutrality too.

To resolve her problem, the agent chooses her effort as a function of what the agent believe

the others will play in equilibrium: (

(

( (

(

(

( ( ( (

To be sure to have an equilibrium:

(

(

(

Then, I define the function ( where k is the number of winners. It appears in the

probability of winning as I will explain in a next point:

(

(

(

Moreover, I want the effect of a variation of the bonus b or the proportion of winner k. To

determine these effects, I need to find

and

:

(

(

(

(

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Finally:

(

( (

(

)

(

(

(

)

Thus, according to the derivative of the probability, the utility function and the cost function,

I may determine the effect of a variation of b and/or k. In theory, using the system of

equations, the principal may determine the optimal par (b,k). In practice, the resolution is

difficult when agents are heterogeneous. For now, we need to study the probability.

The probability does not depend only on the i’s effort but on the others’ effort (unknown to

i) and the number of winners k in the tournament. So, we have:

(

One wins if and only if her output is among the k best outputs:

( ( )

We can interpret the action of win/not win as a binomial function with n, the number of

participants and the probability

( ) ( )

There are two ways to continue. It is possible to use the function of distribution of H or G.

The particularities of the models help to choose the form of the distribution. So, we have:

( ( ) ( ( )

( ( ( (

This is the probability that i beats j. In a tournament, the probability of winning depends on

all the other agents. So, following Green and Stockey (1983):

(

) ( (

( ( ( )

(

)∫ ( ( )

( ( ( ))

(

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However, this is only the probability of being the kth

in the tournament whereas the total

probability of winning is the sum of the probabilities of each kth

:

∑(

) ( (

( ( ( )

∫∑(

) ( (

( ( ( ))

(

Then, we need to derivate the probability to the effort i:

∑(

) ( ( ) ( (

( ( ( {(

( ( ( }

∫∑(

) ( ( ) ( ( )

(

( ( ) {( ( ( ( )

} (

Using this, we may solve the problem of optimization and conclude about the principal’s

decisions. Now, I may begin with the case where all the agents choose the same effort and in

a next part, I will try to study the behavior of heterogeneous agents.

To finish the model, note that the different extensions will use the next utility function:

( (

If agents are risk-averse.

If agents are risk-neutral.

IV. Toward a benchmark: the principal’s issues

As I underlined in the introduction, incentive theory studies how to introduce

incentives in contracts. In a career as teachers, salary does not usually depend on the effort. It

may depend on variables that would have to reflect the level of the teacher as studies or

seniority but without a clear definition of the effort. It would be wondered why the effort is

important if the teacher is a good teacher. In the case of Chile, the politics considers that the

effort does matter in different areas for the education of the pupils. It counts for the results to

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the national exams of course, but it counts for the initiatives and the integrations of the

parents too. The effort is the variable that reflects that the teacher does not only dictate a

class but that he tries to help and to really teach his pupils. Moreover, in the public sector, it

is quite difficult to check teachers’ work since their main task is in relation with children. The

real effect of the teachers can appear only years after the child leaves school. Thus, with a

salary that is not a function of the effort, the utility of one agent is just the difference between

her salary and her effort:

So she chooses her effort maximizing her utility and since the salary is not a function of the

effort:

Without incentive, an agent does not have any reason to do effort. Noticing this problem, the

principal may decide to put in place an incentive scheme. If the salary is in function of the

effort, the agent will equalize her effort to her marginal cost adjusted by her ability. So, the

principal may choose a bonus that increases the effort. The first problem is to establish the

salary as function of the effort because that means that the effort is measurable. Because it is

usually not possible to give an absolute number to the effort, the principal may decide to

distribute a bonus according to the position in a tournament. The idea is to have a measure,

which can be a composite of variables, that lets the principal order the agents. Using the

order, the principals distribute the bonus. Given this scheme, two questions appear: what is

the proportion of winners? What is the value of the bonus?

Here, it is important to stop. As we will see later but it follows the intuition, the effort

increases with the bonus. The problem is that the principal will not give an infinite bonus.

First, because he cannot. In the public sector, each area receives a budget and the superior

has to distribute this budget. Thus, the tournament will be able to supply a bonus according

to the budget. Second, to maximize the effects of the bonus and the number of winners, the

principal will equalize to zero the derivatives of the effort of equilibrium to the value of the

bonus and to the proportion of winners, such as the marginal effects are zero. He does so

because of what he is interested in. In particular, the principal does not want to use the game

to know who the bests and the worsts are (signalization). He may want to know the type of

the agents to understand what it will happen but he does not use the type to distribute the

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award. This decision may be discussed since it would be possible to consider that the State

has to help the worst agents above all (in the case of teachers, teachers who work in the worst

conditions for example). In my model, the principal has a social vision of its task as principal

of a public good. He wants to serve everyone and so, he wants every agent to increase her

effort.

The other question is the real issue of the paper. What is the optimal number of

winners? Is it better to give a prize only to the best or is it better to give several prizes? The

question is a big debate between economists. Since there is no absolute measure of the effort,

it is not possible to give a prize according to this effort, as a firm can do according to the

results’ employees. However, this prize may be thought as the case of the “month employee”:

the best wins, which is called winner-take-all in the theory. Thus, the principal may want to

put in place a WTA. The prize may be chosen according to what usually happens in the

private sector. Generally, a bonus for an executive is an extra month of pay so the principal

may want to follow this pattern. Expressing the bonus as a portion of the salary gives a

such as we can determine the effort of equilibrium.

Considering the simple case of two risk-neutral agents in the WTA scheme, the effort of

equilibrium2

is:

Thus, the principal has the idea to reach this equilibrium to improve the performance

of the agents. Thus, the objective of the paper is to “test” the effects on effort when there are

more than one agent. That is why the benchmark is the WTA model. Moreover, the paper

will try to study the differences in the effort of equilibrium when the assumptions change as

homogeneity and heterogeneity of the agents and more than two agents. We will see that the

heterogeneity and the number of agents make the model more complicated and the

comparison is not so direct.

2 I will develop the model of two agents in the next part. I just present the main result here to not repeat the

model.

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V. Introduction to the problem: a simple case of two agents

Before studying the real issue of the paper, I want to present a simple case of two agents

in order to make an introduction to the problem. Using the variables of the general model

presented in the previous part, I develop a simple model to underline the issues of such a

study. Then, a discussion explains why the results cannot be directly extended to more agents

and so why the study is relevant.

A. The model

There are two agents, i and j, a principal and one award b. The award may be distributed

between the winner and the loser such as the winner receives the proportion and the loser

receives ( with .

The agents try to maximize their expected utility with a cost function ( with

a parameter that can be interpreted as the ability. Bigger is, smaller the agent’s ability is. If

, the agents are homogeneous. Then, I study the special case of a uniform function

such as the probability of winning is: (

.

So, the agents want to maximize their expected utility:

( ( ( ( ( ( ( (

Developing the expression and using the expression of the probability and the utility and cost

functions, the expression of the expected utility with risk neutrality is:

( ( ( )

) ( (

The agent is sure to win the loser’s prize and she has a probability to win the difference

between the winner prize and the loser prize.

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Each agent maximizes her expected utility to determine her effort as a function of the effort

of the other. In the case of only two agents, the effort of the other does not appear and the

efforts of equilibrium are finally:

(

(

The efforts depend on the parameter of the agents, the proportion of the prize and the prize.

It is easily seen that if the winner and the loser receive the same prize ( , the efforts

are zero. The agents do not have any reason to make effort since they will win exactly the

same prize.

If agents are homogeneous, such as the effort of equilibrium is:

(

That means that agents do the same effort in equilibrium. It follows the next intuition. If the

agent i chooses to not do effort, j has an incentive to do an effort even little in order to be

sure to win. However, i wants to win so she wants to increase her effort above the effort of j to

increase her probability. Following this pattern, both increase their effort up to the point

when doing an effort is more costly than the gain. This point is the equalization of the

marginal cost of effort to the marginal increase in probability of winning by an increase of the

effort. That is why both do the same effort.

If they are heterogeneous, it is kind of the same reasoning. The difference is that the “point”

is not the same since the parameter of ability is different between them.

Then, it is possible to study the effects of the proportion and the value of the prize on the

effort of equilibrium for :

That is always positive with since the parameters are positive.

(

That is always positive since the parameters are positive and .

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Equalizing the expression (in this case, it is not possible to equalize to zero because it would

mean that a parameter is zero and this is not interesting) such as the marginal effects are

equal, we find:

(

One can verify that is useless since the bonus would be zero to maintain the

marginal effect. Agents would do not effort. Then, if , that means if the principal

decides to put in place a WTA scheme, the bonus has to be half of a salary (since the bonus

is expressed as a proportion of the salary as explained in the general model). We can

determine different par ( respecting the condition. Then, using the pars, it is important

to determine the par that leads to the highest effort. Calculating only the part ( of

the effort, we have:

( (

(

(

(

(

If we looked only the effort of equilibrium, it was obvious that, given a prize, the best was to

give the entire prize to the winner. If the principal wants to equalize the marginal effect, the

conclusion is the same. The difference in this case is that the principal has to be able to

provide the value of the bonus according to the condition. The conclusion leads to the results

of the literature: with two agents, the best scheme is a WTA. The intuition about this result is

quite easy to understand. If there are only two agents, the agents do effort to win the prize. If

the prize of the winner is reduced, the agent has less incentive to do effort since she will

receive a part of the prize in all the cases.

Then, it may be interesting to study what happens in the case of risk-aversion. Now, we have

the utility ( with .

Thus, the expected utility may be reduced to:

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( ( ( )

) (( ( ( ( (

The efforts of equilibrium are, for :

(( ( (

We note

.

Now, the principal equalize the marginal effects to zero:

(( ( (

(( ( ( ( (

The solution is which is not possible according to the restriction. The solution is

outside the possible values of the parameter. So, we look for a solution using the expression

of the effort of equilibrium. It is probable that the best decision be a corner solution.

Defining:

( (( ( (

We have:

( (( (

( (( (

( ((

The effort will be higher as the and the prize are higher. So the best equilibrium in which

the effort is the highest is a WTA scheme. The bonus will depend on the budget of the

principal. To illustrate the effect of the risk-aversion, I present few graphs of the

expression (( ( ( . The “A” increases as the aversion tends to the

neutrality so the real effect of the aversion is bigger than the graphs do. The objective of the

graphs is to give an idea of the effect.

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Graph 1: Value of (( ( ( in function of (

The three graphs are similar and they show that the expression (and so, the effort of

equilibrium) increases with the three variables. So, whatever the risk-aversion and the prize

are, the effort of equilibrium will be the biggest with . Then, the effort increases as the

prize is bigger and as the aversion is closer to the neutrality. The risk-aversion in the case of

two agents does not change the conclusion of giving the entire prize to the winner.

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Thus, this simple model shows that a WTA scheme leads to the highest effort of equilibrium

in the case of neutrality and risk-aversion. It is tempting to extend this conclusion to more

than two agents but I am going to expose the reasons why it is not possible to do it so easily.

B. Discussion

The conclusions of the simple two-agents model are attractive but unfortunately, it is not

reasonable to extend them to a group of N agents without studying the cases of more than

two agents. Different reasons may be exposed to justify this point of view.

First, in the perspective of a public good or more generally of a good that it is not

possible to measure perfectly the effort, the decision of giving more than one prize permits to

reduce the effect of the noise. The noise may have different origins. For example in the case

of the teachers, the results to the national exam depend a lot on the effort of the pupils. More

generally, a teacher may do the same effort for every pupil but pupils are different so the

impact of the teacher will not be the same. The noise makes the output measure be biased.

Thus, a principal may want to reduce this noise but he may be unable to reduce it in the

measure of the output because it is too costly or because he does not know how to eliminate

it. In this case, he may choose to reduce the impact of the noise in his incentive policy. One

way is the idea to provide more than one prize. Agents will be more ready to do efforts if they

know that the noise has more little impact in their probability of winning.

Then, literature supports the intuition. As I presented in the related literature,

Krishna and Morgan (1998) show that a WTA is the best scheme for two and three agents

but more prizes may be better if the group is bigger. They find that in the case of four agents,

the WTA scheme is the best scheme only if agents are risk-neutral. If they are risk-averse, the

prize has to be divided at least between the first and second places. So, the effect of risk-

aversion on the number of prizes exists but it would appear only with groups bigger than

three. On the contrary, Szymanski and Valletti (2004) demonstrate that a second prize may

be useful even in the case of only three agents. They study the case where one agent is better

than the other two so she knows she will win. The second prize creates a competition

between the other two agents because they know they can win the second prize if they

increase their effort. So, the second prize permits an increase in the effort without decreasing

21

the effort of the winner of the first prize. This point is important and underlines the problem

of heterogeneity in a tournament group.

To finish, there is a mathematical problem with the simple case. If we look at the

probability with a uniform distribution, the derivative of this probability to one effort does not

depend on the other effort since the probability is ( )

. This means it is not really a

reaction function that determines the effort according to the effort of the other. On the

contrary, the probability in the case of three agents is more complex and it really exists a

reaction function since the probability is if only the first wins and (

( if the first and the second win with ( )

the probability than i be

better than j. Thus, the three-agents case permits to create a game with reaction functions. It

is certainly closer to the reality than the simple two agents case presented in this part.

Thus, the simple case of two agents leads to the conclusion that a WTA scheme is the

best equilibrium with risk-aversion and risk-neutrality. It follows the intuition that agents will

do less effort if they win some part of the bonus when they lose. There are three principal

points that supports the importance to study more the subject: the problem of noise, the

results presented in the literature and the mathematical problem of simplicity of a two-agents

game. Moreover, the simple case divides the bonus such as the winner wins more than the

loser so it cannot explain what happens when the prize is the same for the k winners as in the

Chilean policy of bonus. For these reasons, it is significant to study more the problem to try

to understand what happens with more than two agents and to try to define the best scheme

to have an equilibrium in which every agent increases her effort.

Before turning to the rest of the paper, I want to underline what the principal wants to

do. The principal is not looking for a social efficient optimum but an equilibrium. It would

be interesting to study the case where the principal wants to induce the effort that maximizes

the society welfare but this implies a study of what it is the best for the society. In the case of

education for example, it would have to study the importance of the teacher in the pupils’

education and success and moreover, to study the importance of education in the

development and welfare of a society. My study is much more general. The principal is

22

looking for an equilibrium in which the agents increase their effort and do not have incentive

to move. So now, we can follow the study with the case of homogeneous agents.

VI. Homogeneity of the agents

Now, we consider a situation in which the agents are homogeneous and they choose

the same effort in equilibrium. We use the case of . In a Nash equilibrium where

, we have :

|

∑(

) ( ( ( ( {( ( ( }

Using this expression, it is possible to determinate the effort of equilibrium. However, it is

almost impossible to interpret general results so we now use functions of cost and distribution

of G.

(

Moreover, we need to choose a function of distribution for F and so for G. First, we use the

normal distribution since it is the most common distribution and is usually chosen to

represent the human distribution.

Since we have ( and ( , so ( ) ( So, we have:

|

∑(

)

√ (

)

(

)

And we have the e of equilibrium:

(

(

(

So:

|

(

As we want to find the par(s) (b,k) that is (are) equilibrium(s) , we would like to resolve the

system of equations:

23

However, there is no a general solution. (see appendix)

Then, to study the movements of the effort as a function of b and k, I establish

graphs. To be able to do it, I have to choose the other parameters than b and k. As the

parameter c, and only change the amplitude of the effect, I choose 1 to simplify. As in

Chile, 63% of the teachers win less than 500 mil pesos, the salary w is 500. First, I show that

the value of b does not affect so much the decision of the effort. The bonus only amplifies

the movement. That does not mean that the value of b does not have incidence but that the

principal have more power of decision on k.

Graph 2: Behavior of the effort in function of b and k, with , w=500 and

α=0,5 (n=10 ; n=100)

So, I choose a b to be able to study the movement of the effort in function of n and α. In

Chile, the bonus represents approximately the half of one month salary so b=0,04w.

Now, we have e (b,k,n,w,c, , , e (0. ,k,n, 00, , , , . We want to find the best k, that

means the k that maximizes the effort, and the k such as e=0. The next table presents few

results.

24

Table 1

Effort of equilibrium in function of n and α.

√

α n=2 n=10 n=100

0,01 1.04E-06 2.57E-06 8.31E-06

0,2 1.36E-03 3.36E-03 0.012E-03

0,6 0.149 0.366 1.18

1 5 12.3 39.80

k* 3/2 11/10 101/100

With θ=c=σ=1 ; w=500 ; b=0.04

From these results, I may make some conclusions. First, as expected, the parameter α

of risk-aversion reduces the level of effort. So, if the agents are risk-neutral, the principal may

expected a better response to its bonus policy. The most interesting is the differences of

efforts and k* with the size of the group (graph 2). The effort of equilibrium increases with

the size of the group and the possible interval of k decreases. So, the equilibrium is at the

point

with a normal distribution.

Graph 3: Efforts and k with different n.

So, in a Nash equilibrium, the size of the group of the tournament is important. The value of

the bonus is not a real variable of decision since the effort increases with the bonus all the

time. The principal has to choose the bonus in function of its welfare. The optimal k stays

25

. Thus, according to the principal’s monetary restriction, he may try to form the best

group size to optimize the effort.

These conclusions may only come from the distribution function so it is significant to study

other forms of distribution to compare. Now, I use a uniform function.

First, with two uniform functions, we have to determinate the distribution function of

( ). It is a triangular form with:

(

{

(

(

And :

( (

{

(

(

In our case, ( so if the agents choose the same effort, and so we have

( as before with a normal distribution. The difference will only be the absence of

√ because of ( so the efforts will be bigger. Thus, the conclusions do not change with a

uniform shock.

VII. Heterogeneity of the agents

A possible extension of the homogeneous-agents model is the introduction of

heterogeneity. I consider that the heterogeneity comes from differences in ability. Moreover,

the agents may know the abilities of the others. I study first the extension of the precedent

part with heterogeneity in the cost function or/and in the output. Then, I present a general

pattern in the particular case of two efforts and two abilities if N > 30. To be able to make

conclusions, I study the cases of few agents.

26

A. Study of the effort as the maximization of the expected utility

a. Heterogeneity in the cost function

Now, the agents have the same output and expected utility as before but have different cost

function. I do not consider the distribution function of because the triangular

introduces too many cases. It is easier to work with only one shock. Thus, the probability is:

( ) ( ( )

i. Two agents

First, I study the case of two agents i and j. The cost function depends on an indicator of the

ability such as higher is, higher the cost for a given effort is.

(

Thus the probability of winning depends on the realization of the shock, so it is:

∫ ( ( ) (

Considering the simple case of a uniform distribution such as ( , the

probability is:

∫ ( )

Moreover, solving the derivative of the expected utility, the result is:

((

Then, the principal wants to define the optimal bonus. To do this, we need:

if

(

)

but there are both minima. There are no maximum

because the effort increases with the bonus. Thus, with two agents, the principal has to

choose the value of the bonus according to its welfare.

27

The interesting part is the effect of the ability or “lack of ability”. We saw that the

effort depends on the ability parameter so we want to see how the ability changes the decision

of effort. The next graph shows the variation of the effort as a function of the bonus and the

ability. One point may be underlined. To have results, the aversion must be close to the

neutrality and at least, more than 0.75. Thus, we use neutrality in the graph and the other

parameter at one. The bonus has a positive effect only for little value of the . Fast, the lack

of ability and so the high cost makes the agent lose utility if she does effort. On the contrary,

with little , she may wins utility and so increases her effort.

Graph 4: The effort of equilibrium as a function of the bonus and the ability

Thus, when the ability is in the cost function in a play of two agents with a uniform

function, the agent with better ability will increase her effort whereas the other may decide to

do not effort. The bonus policy lets the principal know who the efficient agent is since it is

almost sure that the agent with higher effort will win in this case.

ii. Three agents

In the case of three agents, it is possible to study k=1 and k=2. The expressions of the

optimal effort is really large and do not indicate a lot about the decisions. The interesting part

is that the efforts of equilibrium are symmetric and they are in function of the other efforts

28

and the parameter of ability . To be able to understand the results, the graphs are more

explicit.

Graph 5: Effort in function of b with

Graph 6: Effort in function of b with

29

Thus, the graphs 5 and 6 show that there is almost no difference between the efforts

in k=1 and k=2. The agents with higher ability decide to do higher effort. To be able to

underline the difference, the bonus has to be bigger as in the graph 6. We choose an

excessive b to show what happens.

Graph 7: Effort in function of b with

Graph 8: Effort in function of b with

30

In this case, we can see the behavior of the agents. In the graph 7, the abilities are

different. With a high ability ( ), the agent does more effort when k=1. This may be

explained in the idea that the agent wants to win so does more effort when only one wins.

The other two types do the contrary. Their effort is higher when k=2. This may be explained

by the decision to try to win. When only one wins, they may think they will not be able to win

whereas when two wins, they know that one of them may win and increase their effort to be

the winner. On the other side, if the abilities are closer ( ), these agents decide

according to the same scheme. Both choose a higher effort when only one wins because they

want to win and think they can whereas the agent with the lowest ability increases her effort

only if two wins. The problem here is that one may think that the agent with the lowest ability

would not increase her effort when the other two have very closed abilities since it is almost

sure that these agents will win the bonus. This certainly comes from the form of the function

and the type of ability. It is important to study what happens when the ability is in the output

because in this case, it acts directly on the probability of winning.

b. Heterogeneity in the output

Another way to study the heterogeneity is to introduce a parameter in the output. This may

be done additively or multiplicatively: with the parameter of ability such as [ ]:

(

As before, it is impossible to study directly on a general formula with n agents. We need to

study the cases of few agents to understand what happens. I present results for the case of

additive ability.

i. Two agents

If there are only two agents, one wins and one loses, so the probability of winning is only:

( ( )

( ∫ ( ( ) (

31

with a uniform distribution: ( ∫

(

As before, we solve the problem of maximization of the agent i to determine the effort of

equilibrium. In this case, the result is:

((

The parameter of ability does not appear in the effort when there are only two agents so it is

the Nash solution. This comes from the absence of the ability in the derivative. If we want to

have it in a two-agents game, we may use the normal distribution but it complicates a lot the

resolution.

ii. Three agents

To study the case of three agents i, j, k, we are going to follow the progression of the

heterogeneity in the cost function with k = 1 and k = 2.

k=1

If only one wins, the agent i has to beat the others:

( ∫ ( ( ) ( ( (

I do this for each agent and solve the problem to determine the effort of equilibrium.

However, every effort is a function of the other efforts and on all the abilities. To solve, I

have to solve the system of three equations and three unknowns. Finally, I find:

( ((

((

And there is the same logic of effort of equilibrium for j and k.

To study the behavior of the effort of equilibrium, I present graphs of ei given ai in function

of aj and ak with the hypothesis that the ability goes from -2 to 2. Moreover, since the effort

depends on the ability of the others, the model cannot consider asymmetric information.

32

Graph 9: Effort in function of the other abilities with

. (a=-2 ; a=0 ; a=2)

Thus, if the agent knows she has a negative ability, she will do positive effort to

compensate her disability whereas if she has a zero-ability, it will depend on the other abilities

and if she has the best ability, she will not do effort. The results are negative effort but in a

realistic interpretation, they have to be seen as the decision to not do effort. This may be

explained by the output.

The output is so an agent with a high ability respect to the others

may take advantage of her position to do less effort, whereas an agent with low ability has to

compensate. This model does not show how an agent may decide to do not effort because

33

she knows she will lose whatever her effort is. This certainly comes from the form of the

distribution as earlier. Moreover, there is not reason that the behavior of the agent will follow

the same scheme if n>3 and k=1.

k=2

Actually, the case of k=2 does not change a lot from the case of k=1. The effort is higher but

follows exactly the same scheme. The problem may be that the agent with a lower ability does

not suffer an higher cost so she accepts to do more efforts. It may be interesting to underline

the behavior of the effort in the tournament when there are different costs and a parameter of

ability in the output at the same time.

Two forms of ability

Now, I consider that the ability appears in the output and the cost function and I

determinate the conditions for the sign of the effort. This may explain why some agents with

very low ability will do efforts or not. Following the same resolution as before, I find the

effort in function of the parameters of ability ai, aj, ak and ci,cj,ck. and using the expression, we

may determinate the conditions for the positivity of the effort of equilibrium. It is possible to

determinate exactly the conditions with all the parameters. However, to simplify the

presentation, I use the case I study from the beginning: .

⁄

k=1

U D

condition condition

=0

(

( )

=0

>0 (

( >0

<0 (

( <0

34

k=2

U D

condition condition

=0

( ( )

( )

>0 >0

( ( )

( )

<0

( ( )

( )

The interpretation of these conditions is not so easy because in k=1, there are five

conditions only for U and because the conditions are not easily interpretable in themselves.

The economic logic is to consider that a negative effort is a zero effort. In a general way, two

cases are interesting: U and D are positive and U positive and D negative. One may expect

that an agent with a very low ability will give up and her effort will be zero or at least, tend to

zero. In the case of a unique winner, the two D represent these phenomenons. If D is

positive, namely if the cost of i is bigger than the sum of the other two costs, the effort will

diminish as D increases so bigger the parameter of lack of ability in the cost is, smaller the

effort is. It tends to zero as we expect. On the other side, if D is negative, the effort will be

negative, actually zero. If an agent has a very low ability, she may decide to not participate by

not doing effort because her cost associated to her ability is very high. On the contrary, if an

agent has a very high ability, the most probable is that D be negative and so the effort will be

positive. However, better her ability is, smaller her cost and D are, so her effort tends to zero

too. In the case of k=2, D is always positive and increases with the value of the parameters.

The sign depends on the ability in the output. As for k=1, the difficulty is that the ability in U

depends itself on the costs so the interpretation in economic terms is not really possible. At

least, the table shows that conditions and parameters may explain the decision of an agent to

do effort or not.

35

Then, we may wonder what happens if we change the values of b and . There are a

lot of cases and present all of them will be uninteresting, above all in my study. I underline

the main conclusions in k=2 because they may help to understand the behavior of the effort

in different contexts. I may look at every condition but it is clearer to study directly the effort.

Thus, the value of the bonus does change the form of the function of decision. To keep the

form “if the agent has a very low ability and so a high cost, she prefers to do low or zero

effort”, the difference of ability between the agent and the others has to increase as the bonus

increases. If it does not, the agent increases her effort when she is disadvantaged. Then, if the

form is maintained, the bonus will change the value of the effort. The parameter of risk-

aversion is different. Given the other parameter, the risk-aversion will only diminish the value

of the effort but will not change the form of the decision. Moreover, if the agent is risk-averse,

it is possible to increase the bonus and maintain the form without changing the abilities. This

may be another way to explain some behaviors.

Thus, I developed in this part the decision of an agent i according to her parameters and the

abilities and costs of the other agents. I did it for two and three agents. When the ability is

only in the cost function or in the output, the agent with a low ability decide to compensate

her disadvantage by a higher effort. On the contrary, it is possible to underline conditions of

behavior if the ability is at the same in the output and the cost function. The conditions

themselves do not help so much to understand why and how the agent decides but show why

an agent may decide to not do effort is she has a very low or very high ability. The conditions

are more simple in the case of k=2 so it may be possible that it is more probable that more

agents will participate. The rest of the paper focuses more on this point.

Now, I turn the study into a general presentation in a perspective of theory of games with two

efforts and two abilities.

B. Restriction of compatibility of incentives for the principal if N>30

The game I am looking for may be seen from another point of view. In the traditional

theory of contracts, the principal wants to maximize her output subjected to the restrictions of

participation and compatibility of incentives. In the case of a State that wants to maximize the

output of teachers, the output is not monetarily measurable. The government may create an

indicator as the SNED in Chile but it is more an indicator of comparison between schools

36

than an absolute measure. Moreover, a school does not choose to participate or not in the

tournament. The teachers may decide to not increase their efforts but the school does not

decide to entry or not in the game. Thus, the restriction of participation and the

maximization do not interest me in the decision of the number of winners. The State has to

choose using the restriction of compatibility. With two efforts H (high) and L (low):

(

( ( (

As explained before, the total probability of winning is the sum of Bernouilli. If N is large,

the sum may be approximate by a Normal distribution.

∑

( ∑ ∑ (

The ability is in the cost function: and are the parameters of cost associated to the type

high, low. Thus, the is the parameter of the agent of type high, and the parameter is lower

than . There are persons of type h, l.

Considering the case of an agent with the ability , the restriction turns into:

√ ∫ {

(

)

(

)}

(

(

(

With:

(

(

(

)

(

)

The principal may determinate the k and b of equilibrium using this restriction and his

budget. The problem is that the resolution is complicated and it seems impossible to

simplify the expression. For this reason, I turn now into the special cases of a little number of

players.

37

C. Study in the case of two abilities and two efforts

To be able to understand the conditions the principal has to face with, I develop a

classical prisoner’s game. Each agent wants to maximize her expected utility. Comparing the

expected utilities, we may underline the choices of the agents and above all, the conditions

the principal has to respect to make the agents choose the effort he wants. Indeed, there are

two efforts, High/Low, and two abilities, high/low such as there are four cases: hH, hL, lH

and lL. The principal may want to induce the high effort to everyone or may want to induce

the high effort to the high ability and the low effort to the low ability. The next part develops

the game for two agents to understand what happens. The probability is determined using the

linear function.

a. Two agents

The agent compares her expected utility. The next table shows each expected utility for

ability h;l and effort H;L. To facilitate the equations, we note (( and

((

Ability h

Ability l

Effort H Effort L

Effort H

(

(

Effort L

(

(

Using the table, we may determinate the conditions:

All the agents choose the effort H if and only if:

√ (

38

Or if the agents are risk-neutral:

(

All the agents choose the effort L if and only if:

√ (

(

The agent of type h chooses H and the type l chooses L if and only if:

√ (

√ (

(

(

In the game of two players, the principal does not choose the number of winners. In the case

of a State, the government has a budget to the game and the bonus represents all the value of

the budget. Thus, the government may study the conditions to choose the value of bonus

before choosing the budget. If the bonus is not big enough, both players may decide to not

do effort whereas one will win the prize, so the game will be useless. If the State wants to use

the game to determinate the type of ability, he will choose the last condition. However, this is

more the role of a game in a firm. The target of the State is to increase the level of the

education of all schools, not only of good schools. To do that, he has to provide a bigger

bonus and accepts to propose a prize to the high type bigger than necessary. Thus, the ideal

is to induce high effort to each type of agent.

The case of two agents lets understand the scheme of the game but does not help to conclude

about the number of winners. I develop now the same game but for three agents.

Now, I consider the most logical case for a principal as a State. The State wants to make

every agent choose the high effort whatever the other agents choose. He wants to determinate

the bonus and the k such as a dominant strategy in the high effort exists. So now, I can

present a case where the agents do not know the decisions of the other agents. Following the

same pattern as for two agents, I determined the conditions to induce the high effort. First, I

determined for a k the bonus in every possible case of efforts such as the agent chooses the

high effort. Then, comparing these conditions, the principal has to follow the highest

39

conditions. Moreover, he has to choose the bonus according to the lowest ability. Then, I

may compare the conditions between the different k to make conclusions about the optimal

number of winners in such a tournament.

I present the results in the next part. I do not develop every case in order to simplify the

reading but present the mean results.

b. More than two agents

o Three agents

The development of the game leads to the next conditions:

k = 1 If (

(

( ( )

k = 2 If (

(

( ( )

Thus, in the case of three agents, there are two cases:

- If (

, there is no solution. The principal cannot create a dominant strategy.

- If (

both schemes are possible. Thus, the choice will depend on the

budget of the principal. Since the condition on the bonus is the same, it will be more

costly to have k=2. The State has a pre-defined budget so he may have the money to

give two bonuses. However, this would not be an efficient use of his money since he

may attain the same effect with fewer budgets. He does not know the exact value of

the parameters but may approximate them using the information he has and try to

make a WTA system.

o Four agents

In the case of four agents, the conditions are:

40

k=1 If (

Impossible

If (

(

( ( (

If (

(

( (

k=2 If (

√ Impossible

If (

√

(

( (

k=3 As k=1 As k=1

The indication “impossible” means that there are negative and positive conditions together so

the principal is not able to provide a bonus that respects all conditions (a negative conditions

means that the principal would have to create a punishment). Thus, the variance has to be big

enough to make the State able to create a tournament.

In this case, one type of tournament may be the only possible one. The bonus depends on

the variance of the output.

- If (

, the principal cannot put in place a system of tournament with bonus

to induce high effort.

- If (

but

(

√ , only WTA is possible and the value of the bonus

depends still on the variance.

- To compare a WTA with a two-prizes scheme, the condition is (

√ so it is

the bonus with (

in k=1. The case k=3 returns to the conditions of k=1. It

may remember the pyramidal form of the Nash equilibrium. Thus, the decision to

give three bonuses has to respect the same general condition than a WTA scheme.

Actually, there is a difference in the condition. The general condition is the “worst”

condition for the State and this condition does not come from the same game

(number of H versus L). Then, the comparison between prizes leads no to another

condition but a general conclusion: the WTA always needs a more little budget than

a two prizes scheme. Thus, it depends once more on the principal. If he has only the

budget for a WTA, he has no choice. On the contrary, if he has the money for a two

41

prizes scheme, he may choose between provide a WTA with a higher bonus than

necessary or provide two prizes. The results will be the same: all agents will choose to

do the high effort.

o Five agents

To confirm the previous results, I develop for the case of five agents. The conditions are:

k =1

If (

Impossible

If (

(

( ( (

(

k=2

If

√ (

Impossible

If √ (

(

( ( (

k=3 As k=2 As k=2

k=4 As k=1 As k=1

The analysis is the same as before. I found the same phenomenon of pyramidal form. It is

possible to compare a WTA with a two-prizes scheme if √ (

. Whatever the

variance is, the WTA needs a more little budget than a two-prizes scheme. So, the results are

similar to the case of four agents.

o Six agents

To finish, I develop the game for six agents to confirm the form of the conditions.

k=1

If ( Impossible

If (

(

(

If ( (

(

k=2 If ( Impossible

42

If ( (

(

k=3 (

k=4 As k=2 As k=2

k=5 As k=1 As k=1

With (

I found the same scheme as before.

o Ability in the output

It is possible to develop the same game with the ability in the cost function and in the

output. Actually, the additive ability in the output complicates the model since every possible

case has to be seen. With N agents, there may be one, two, three … N-1 agents of the same

type and if the principal does not know the number of each type, he has to choose the worst

conditions for him to induce the high effort. To give an idea of what happens, I develop the

game for two and three agents.

In the case of two agents, the ability does not influence the conditions because the

parameters vanish in the resolution. On the contrary, the parameters stay when there are

more agents than two. If only one agent wins, the “worst” condition on the bonus for the

principal appears in the game with one agent of low ability versus two agents of high ability.

The condition is (

( ( ( with ( the difference between the high

and the low abilities. So, the existence of ability in the output increases the condition of the

value of the bonus because one agent does not have the high ability. It is more costly for him

to make the high effort. If two agents win, the worst condition appears in the game of one

agent with high ability versus two agents of low ability. The condition on the bonus is the

same than in a WTA. Thus, the ability in the output introduces more differences between

agents so the principal has to increase the bonus to compensate the differences when he

wants to create a dominant strategy in the high effort. Since the case with the ability in the

cost function follows a scheme, there is no reason that the introduction of ability in the output

changes this scheme. The conditions on the variance and the bonus will be more

complicated but the existence of the ability would increase the minimal bonus.

43

Then, the results come from the hypothesis of the uniform function of probability.

The phenomenon of symmetry may exist only for this special case. I study know the case of a

normal function. I will not develop as much as before since the method is the same.

o Normal function

The idea is to develop the same game as in the previous part using not a uniform function

but the normal function. As in the homogeneous case, the probability is made with .

The problem appears in the resolution because of the form of the normal distribution. The

computation uses the approximation of the normal distribution through the function of

errors, erf. Since I am interested in the symmetry of the results, I develop the game with the

expression ( (

√ that I note E to simplify. Values are necessary to be able to develop

more.

Thus, I present the results in the next tables.

N=3

k=1

Impossible

(

(

k=2 As k=1 As k=1

N=4

k=1

Impossible

(

(

(

(

k=2

√ Impossible

√ (

(

k=3 As k=1 As k=1

44

N=5

k=1

Impossible

(

(

k=2

impossible

(

(

k=3 As k=2 As k=2

k=4 As k=1 As k=1

I do not present the results for N=6 because it is sure I will find the same symmetry. Thus,

the symmetry in the conditions on the bonus appears with a normal distribution too. As in

the case of the uniform function, the optimal decision to maximize the effort and minimize

the cost will be a WTA scheme. However, if the State has a bigger budget than the minimum

bonus, he may choose to distribute more than one prize. It is not optimal in the idea that the

money would be used for another policy but the State may consider social aspects and

problems of measure that the model does not show.

o General aspects

Since the conclusions are similar, it may be possible to extend the conclusions to a

game of N players. However, the conditions are different and the game has to be extended to

the N number of participants to find the conditions of possibility and optimality. Thus, the

WTA is the most efficient since the other ways would provide the same result but they are

more costly. However, in the special case of a State, it is easy to imagine that the State decides

to give the bonus to more than one person because of the social aspects and the

approximation of the measure of the output. In the case of schools, all schools are different

and the quality of teaching does not explain all the variability of the students’ level. Thus, the

measure of the quality, called the SNED in Chile, cannot measure exactly the work of the

teachers. The SNED tries to include the variable as the geography (urban, rural), the socio-

economic level or the type of school but the indicator cannot be perfect. Giving bonuses to

more than one person provides a kind of correction of the error of measure. Thus, the State

45

would have to think more about the quality of the indicator to choose a k such as it

represents the possible best agents more the error.

Then, an interesting conclusion is the symmetry of the conditions on the bonus. In

the homogeneous agent case, I found that the effort has a pyramidal form in function of the

number of winners. In the heterogeneous case, there are only two efforts and the conditions

make the agents choose the high effort. Here, I find that the condition on the bonus has a

pyramidal form in function of k. That leads to many possible bonuses but only one is optimal

if the principal wants to minimize his cost. The symmetry appears in the uniform and the

normal functions so the conclusions may be solid.

One critic may be about the hypothesis of neutrality in the tables. However, if the

agent is risk-averse, the condition turns into an expression as √ (

with

T in function of the parameter. Since , the condition of minimum increases with

the aversion, but given a degree of aversion, the preference for a WTA scheme will not

change. So, the aversion will only restrict more the value of the bonus. Bigger the aversion is,

more probable is that the principal chooses a WTA scheme because of his budget

restriction.

VIII. Conclusion

This study tried to participate to the debate about the number of winners in the

tournament. First, if the agents are perfectly homogeneous, namely that there is no difference

of ability and that all the agents decide according to the same pattern, I found symmetry in

the effect of the number of winners given the other parameters as the salary and the value of

the bonus. The proportion of winners that leads to the highest effort of equilibrium is

.

This is not possible because of the half but the principal may decide to establish the number

of winners according to this rule. The value of the bonus does not really play a role since the

effort increases with the value. It depends more on the budget of the principal than a

decision. Then, I tried to extend the general model to the case of heterogeneity. The ability

appears as a parameter in the output or in the cost function or both. The study is difficult

because the resolution is long and I did not perform to have results for more than three

agents. The cases of more than three agents would have to be done as extension of this

46

paper. These cases would permit to find a phenomenon to extend the conclusions to N

agents. I found that the ability in the output does not participate in the decision of effort in

the case of two agents. With three agents, the agents with the worst ability will do more effort

in the case of two winners than one winner whereas the agent with the highest ability does the

contrary. The agent with the worst ability seems to compensate its disadvantage by a higher

effort, especially if her probability of winning is higher. However, the model does not

perform to underline why some agents decide to not do efforts when their probability of

winning is very low. The case with ability in the output and in the cost function may help to

contest to this problem but the interpretation of the solution is difficult. The expressions and

conditions I present are not easily usable and it is almost sure that it would be more difficult

as the number of agents increases. At least, this part shows the existence of cases when the

agent chooses to not do effort because her disadvantage or advantage of ability. It would be

interesting to extend the work to find a way to show the decisions in these cases. Because of

the problems introduced by the heterogeneity, I study the particular case of two types of

ability and two types of effort. To do it, I consider that the principal wants to induce a

dominant strategy. Thus, the agents may not know the type of the other agents and their

decision. The principal finds the “worst” condition (the highest minimum bonus) and then

compare the conditions between a WTA scheme and more than one winner. The interesting

result is the symmetry of the conditions. Thus, the last condition (k=N-1) is as the first one,

the next to last condition as the second one etc… however, if the condition is the same on the

bonus, the last conditions represent much more money since the principal has to provide the

same minimum bonus but to more persons. Thus, the conditions with a number of winners

less than the half are better than the ones after the half. Then, the comparison between the

conditions leads to conclude that a WTA permits to minimize the expenditure. However, in

the case of a bonus for teachers as in Chile, it is easy to imagine that the State decides to give

a bonus to more than one agent. This decision may be explained by the political aspects and

the problems of measure of the output that I did not consider in my model. Indeed, since the

concept of bonus to teachers may be difficult to accept, a WTA will be certainly rejected. In

Chile, the output is measured by an indicator. The schools are separated in groups according

to parameters and the indicator tries to consider all possible variables. However, it cannot be

a perfect measure of the work of a school so the distribution of a prize to k schools permits

to decrease the problem of measure. Then, I added the ability in the output to study the

47

differences. As expected, the only difference is that the condition is more restrictive as the

difference of ability between the high and low ability increases. Then, I did the same game

but with a normal distribution of the shock to study if the symmetry only comes from the

uniform distribution. I found that the symmetry is still present and that the conclusions do

not change. The normal is more difficult to use without values of the parameters but the

games show the same pattern.

Thus, the principal conclusions are that in the case of two types of effort and two

types of ability, the best decision is a WTA scheme because it leads to the target of high

effort with the littlest cost. The results are consistent with a uniform and normal distribution,

with or without the ability in the output and with or without risk-aversion. However, the

model does not consider more realistic parameters as the political aspects of a politics of

bonus. The general case that does not establish the number of types of ability and effort is

difficult to interpret and it would be interesting to study more this case in another paper.

48

Appendix: Derivative of the effort of equilibrium.

The effort of equilibrium is determined as:

|

(

( ( ∑

( ( ((

√

Using the function of utility and the normal distribution, we may differentiate this effort to the

bonus and the k to obtain the movement of this effort.

By computation, we obtain:

( ( ∑

( ( (

√

( (

∑

( ( ((

√

With ( the gamma function.

There is no solution to the system. We can look at the second order derivative respected to

b:

( ( ( ∑

( ( (

√

Thus, the sign depends on the value of k. It may be positive or negative. Actually, we have:

Thus, as b increases, the effort will increase. The principal chooses the bonus in function of

its monetary restriction.

49

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Tanaka Business School Discussion Papers: TBS/DP04/3