Information Policy in Contests with Little Noise

doi:10.4236/tel.2011.13012

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Theoretical Economics Letters, 2011, 1, 53-56

doi:10.4236/tel.2011.13012 Published Online November 2011 (http://www.SciRP.org/journal/tel)

Information Policy in Contests with Little Noise

Pouya Azimi Garakani, Oliver Gürtler

Department of Economics, University of Cologne, Albertus-Magnus-Platz, Cologne, Germany

E-mail: [email protected], [email protected]

Received August 15, 2011; revised September 12, 2011; accepted September 22, 2011

Abstract

In this paper, we analyze a contest organizer’s decision to release intermediate information to the contestants.

Contrary to the existing literature, we assume that there is little noise affecting the contest outcome so that

there may be only mixed-strategy equilibria at the contest stage. Optimal information policy is found to de-

pend on the prize spread and information may be revealed if contestants are very heterogeneous. This is dif-

ferent from results in the previous literature, which finds that the optimal information policy is independent

of the prize structure and that information should be concealed from the contestants in case they are hetero-

geneous.

Keywords: Information Policy, Contest, Mixed-Strategy Equilibrium

1. Introduction

For a long time economists have recognized that people

can affect other peoples’ behavior by strategically re-

leasing information. Spence [1], for example, has shown

that job applicants can induce employers to make better

wage offers by providing (favorable) information about

their own education. Crawford and Sobel [2] have dem-

onstrated that a decision-maker takes (unverifiable) in-

formation provided by another person into account if

preferences of the two persons are not too dissimilar.

Recently, this idea has found its way into the study of

contests or tournaments. A contest describes a situation,

in which several contestants compete for prizes that have

been fixed in advance. Examples abound. Rent seeking,

promotions in internal labor markets, R & D races, liti-

gation and many more economic settings can be modeled

as a contest. The contest organizer often receives infor-

mation that the contestants do not have. Take the exam-

ple of a contest for promotion to a higher layer in the

hierarchy. The task of the contest organizer is to evaluate

the contestants and to select a winner that gets promoted.

If the contest lasts for a considerable period of time, the

organizer collects intermediate performance information

during the contest. Moreover, he may just like some of

the contestants, while he dislikes others, generating ine-

quality between the contestants. Finally, he may be in a

relatively better position to assess whether the skills of

the contestants fit well to the tasks they have to conduct

(in particular, if the contestants are young and have just

entered the labor market). No matter what interpretation

one chooses, the contest organizer has information that

the contestants do not have. She may thus decide to stra-

tegically reveal or conceal this information to affect con-

testants’ actions. A couple of economic papers have ana-

lyzed how the contest organizer’s optimal information

policy should look like.1 One finding of these papers is

that the contest organizer tends to conceal information

when one of the contestants is heavily favored over the

other (e.g., because of different intermediate perform-

ance ratings). If information was released to the contest-

ants in this case, competition would be low because the

trailing player understands that he has little chance to

overcome the advantage of his opponent. Hence, the or-

ganizer prefers concealing information in this case. An-

other finding is that information policy may depend on

the specific form of the contestants’ effort cost function,

but not on the structure of prizes.

In the contest or tournament literature it is well known

that contestants’ behavior is structurally different de-

pending on whether the contest outcome is affected by

lots of (random) noise or whether noise is relatively un-

important. In the former case, contestants choose pure

strategies in the contest (see Lazear and Rosen [6] and

Gürtler [7]), whereas there are only (nondegenerate)

mixed-strategy equilibria in the latter case (see Baye et al.

[8]). The literature on information policy in contests has

been restrictive in so far that only the first type of contest

1See, for example, Ederer [3], Gürtler and Harbring [4] and Goltsman

and Mukher

[

]

P. A. GARAKANI ET AL.

has been considered. The aim of the current note is to

analyze whether the findings of the literature carry over

to the second type of contest. Considering a simple con-

test with discrete effort choices, we find that this is not

true. For instance, we find that the optimal information

policy may well depend on the prize spread. If the con-

test organizer leaves the contestants uncertain about their

initial heterogeneity, there is typically a pure-strategy

equilibrium where the contestants either choose low or

high effort for sure, depending on the prize spread.2 If

information is revealed to the contestants and they get to

know that they are heterogeneous, there is a mixed-

strategy equilibrium where each contestant chooses low

and high effort with positive probability. Hence, depend-

ing on whether or not contestants are expected to choose

high effort in the first situation (which in turn depends on

the prize spread), concealing or revealing information is

the preferred option. Related to this result we find that

the contest organizer sometimes prefers to reveal infor-

mation to the contestants if they are very heterogeneous.

As indicated before, this is because they may choose low

effort for sure when they do not receive any information.

2. Description of the Model and Notation

A principal organizes a contest between two agents

and . All parties are risk-neutral. The winner of

the contest receives the winner prize 1, while the loser

gets . Prizes are exogenously given. Denote by

the prize spread. The agent with the

higher performance wins the contest. If both perform

equally well, a fair coin determines the winner. Agent

’s performance

<ww

w:=ww

i(i =,)

B is given by

=eyi





The variable denotes the agent’s effort.

Effort is costly and costs are defined as



i0,1e



e=0=0c

and



e=1= 0,2

w













can be interpreted in

different ways. For instance, it can describe an agent’s

previous performances in the contest. Alternatively, it

may be used to describe a situation where the principal

favors one agent over the other (if AB



). Finally, it

may account for ability differences.

Under either interpretation, the principal should have

superior information concerning the realization of i



This is because she typically gathers the performance in-

formation, knows her preferences for the agents and may

be in a better position to judge, whether the agents are able

to handle certain tasks. To account for this, we assume



to be a random variable and the principal the only

one to observe its realization (before agents choose their

efforts). More concretely, we assume three states of na-

ture. We either have



 

1=,=1,0



,





or =0





3=0,1

. This means that each agent can be in the

leading position, but also that they may tie before choos-

ing their efforts3. The states occur with probability 1,

2 and 3, respectively. At the outset, the agents are

homogeneo us and, thus, .

While the principal privately observes the state of na-

ture, she can reveal her informatio n to the agen ts.4 In th is

context, we assume that she cannot misrepresent the in-

formation, i.e., she cannot lie about the state of nature.

For instance, if performance information is recorded

within a firm, the principal may withhold the records

from the agents, but she cannot forge them. Moreover,

the principal is restricted in that she can either inform

both agents about the state of nature or none of them.5

=pp

Summarizing, a strategy for the principal consists of a

triple





123

=,,

III, where

() is a vari-

able indicating whether the principal reveals her infor-

mation to the ag ents () in state or not ().

In total, the principal has different (pure) strate-

gies some of which lead to the same outcome (see the

analysis in Section 3). We assume that the principal

cannot commit to a strategy, but chooses the revelation

policy that is optimal for her ex post, i.e., after the state

of nature has been realized. The agents’ strategies can be

written as a quadruple 123An

=1,2,3j



I=0



qqqq, with

the agent’s effort choice if he is told to be in state

and n as his effort if the principal withholds her in-

formation from the agents.

2Hence, uncertainty imposed by the contest organizer has a simila

effect as noise in that both may induce the contestants to play a pure

strategy.

3Note that only the difference between A



and



has an effect on

the agents’ behavior. Hence, in the second state, we may also assume



1,1,=



The principal chooses her strategy so as to maximize

the sum of expected effor ts. The agents choose their strate-

gies in order to maximize the expected payment minus

costs entailed by effort.

4Note that the contest outcome is not affected by another random vari-

able in addition to A



and



. Hence, if the agents get to know

these two variables, there is no uncertainty. The model results would

continue to hold if the contest outcome was affected by another noise

term, as long as the influence of this noise term on the outcome was

sufficiently low.

5A possible reason could be that

cannot prevent the agents fro

sharing information.

3. Model Solution

When agents decide about their effort, they trade-off the

higher effort costs from a positive effort with the in-

P. A. GARAKANI ET AL.

creased probability of receiving the winner prize. De-

pending on which is higher, they choose an effort of ,

an effort of or they mix between these two alterna-

tives. If agents know which state they are in, they are

able to calculate the increase in winning-probability that

a high effort entails. This, however, is no longer ensu red

if the principal keeps information secret and agents do

not know what the relevant state is. Here, agents use

Bayes’ rule to update their belief about the state. The

updating process, in turn, depends on the strategy that the

agents expect the principal to play. This means that we

cannot separate the optimal choice of the agents from

that of the principal. Hence, in the following lemmas we

characterize agents’ optimal efforts for different strate-

gies of the prin ci pal .6

Lemma 1 If the principal plays





1,1,1 ,





1,1,0

e=1

or , the agents always correctly infer

what the relevant state is. If agent believes to be in

state 2, he chooses . Otherwise, he chooses



1,0,1

 

0,1,1 i

e=1

with probability 2

lwc



 (if he believes to be

leading by 1) or with probability

e=0 2



(if

he believes to be trailing by 1).

Lemma 1 characterizes the agents’ behavior if they

know which state they are in. If there is a tie, both have

realistic chances of winning and, therefore, both choose

high effort for sure. If one agent is ahead of the other, in

contrast, there is no longer a pure-strategy equilibrium.

This is because the leading agent can secure himself a

win by choosing high effort. This would discourage the

trailing agent who would choose zero effort and drop out

of the contest. Then, however, th e leading agent need no

longer bear the cost of a positive effort since zero effort

would suffice to win the contest Yet, if the leading agent

chooses zero effort, th e trailing agent might find it bene-

ficial to choose high effort to catch up with his opponent

and so on. Summarizing, uneq ual starting positions lower

competition. Agents choose low and high effort with

strictly positive probability.

Lemma 2 If the principal plays , the agents’

only information is whether or not they are in state 2. If

agent is told to be in state 2, he chooses i. Oth-

erwise, he chooses if and i if

(for both effort levels lead to the

same payoff).



0,1,0

>4wc



e=1

=4c0

ww

Lemma 2 shows that the strategy of the principal ( i.e.,

the information policy) can have significant effects on

the agents’ efforts. More concretely, we see that there

exists no longer a (non-generic) mixed-strategy equilib-

rium once the principal reveals information only in state

2. The intuition is as follows: The mixed-strategy equi-

librium resulted from the fact that, in case the state is

commonly known and agents do not tie, the leading

agent could ensure himself a win by choosing positive

effort, which would completely discourage the trailing

agent. If the principal plays , however, agents do

not know whether they are leading or trailing behind. In

this case, choosing a positive effort increases the win-

ning-probab ility in at least one state of nature (reg ardless

of the opponent’s effort) and, hence, the expected win-

ning-probability. As a result, we have an equilibrium in

pure strategies. If the prize spread is relatively hig h, both

agents choose high effort. This means that concealing

information may increase expected efforts by leaving the

contestants uncertain on whether they are in the leading

position or trailing behind. To get unambiguous results,

we assume in the following that both agents choose the

high effort in case



0,1,0

=4wc



Lemma 3 If the principal plays , the agents

do not get any information about the realized state.



0,0,0



Agent chooses if

e=1

 and

e=0

 (for



 both effort levels

lead to the same payoff).

Lemma 3 is in spirit similar to Lemma 2. It implies

that a mixed-strategy equilibrium does not exist if the

agents do not know which state they are in. Again, con-

cealing information may lead the agents to choose the

high effort for sure and, hence, may increase expected

effort. Again, we assume both agents to choose the high

effort in case

.

Lemma 4 If the principal plays , the agents’

only information is whether or not they are in state 1. If

agent



1,0,0







B is told to be in state 1, he chooses e=1

( ) with probability e=1

Awc



 (2



). If

not, optimal efforts depend on the prize spread. If >w











c, both agents choose the high effort for sure.









21 ,

B chooses the high effort for sure, If

while

mixes between the two effort levels. If <w









21 c







, Be=1

Ae=1

chooses ( ) with proba-

6All formal proofs a re available from the authors upon request.

P. A. GARAKANI ET AL.

bility

 

211 2

pcp w

rpw



 (

 

pwc

rpw





Lemma 4 combines the ideas from the previous lem-

mas. The principal informs the agents in state 1, in which

case they play a mixed-strategy equilibriu m. In the other

states, she conceals her information. By doing this, she

may induce both agents to choose the high effort for sure.

Note that the analysis is completely analogous for the

principal’s strategy .



0,0,1

Having analyzed the consequences of the principal’s

strategies, we are now able to characterize her behavior.

This is done in the following proposition:7

Proposition 1 If , the principal either plays 4w

(0,0,0) or . If (0,1,0)

2,4



 





, she plays

. Finally, if (0,0,0)

c1,1,1)



, she plays (,

or .



1,1,0



0,1,1

Proposition 1 states that often wishes to conceal

the information from the agents. This is intuitive since

we have seen before that agents often choose higher ef-

forts in case they do not learn whether they are in the

leading position or not. Only for a very low prize spread

it is optimal for the principal to always reveal the state of

nature (either directly or ind irectly). If the prize spread is

low, concealing information indu ces the agents to choose

zero effort. Then, of course, prefers to reveal her

information. Contrary to the previous literature on feed-

back in contests we therefore find that optimal informa-

tion policy and optimal prize structure may be interde-

pendent and should not be determined in isolation. More-

over, we see that sometimes prefers to reveal infor-

mation even if agents are heterogeneous. As indicated in

the introduction, this finding again differs from findings

in the existing literature.

4. Acknowledgements

We would like to thank an anonymous referee as well as

Bernd Irlenbusch, Dirk Sliwka and participants at the

research seminar in Cologne for helpful comments.

5. References

[1] A. M. Spence, “Job Market Signaling,” Quarterly Jour-

nal of Economics, Vol. 87, No. 3, 1981, pp. 355-374.

doi:10.2307/1882010

[2] V. Crawford and J. Sobel, “Strategic Information Trans-

mission,” Econometrica, Vol. 50, No. 6, 1982, pp. 1431-

1451. doi:10.2307/1913390

[3] F. Ederer, “Feedback and Motivation in Dynamic Tour-

naments,” Journal of Economics and Management Strat-

egy, Vol. 19, No. 3, 2010, pp. 733-769.

doi:10.1111/j.1530-9134.2010.00268.x

[4] O. Gürtler and C. Harbring, “Feedback in Tournaments

under Commitment Problems: Experimental Evidence,”

Journal of Economics and Management Strategy, Vol. 19,

No. 3, 2010, pp. 771-810.

doi:10.1111/j.1530-9134.2010.00269.x

[5] M. Goltsman and A. Mukherjee, “Interim Performance

Feedback in Multistage Tournaments: The Optimality of

Partial Disclosure,” Journal of Labor Economics, Vol. 29,

No. 2, 2011, pp. 229-265. doi:10.1086/656669

[6] E.P. Lazear and S. Rosen, “Rank-Order Tournaments as

Optimum Labor Contracts,” Journal of Political Econ-

omy, Vol. 89, No. 5, 1981, pp. 841-864.

doi:10.1086/261010

[7] O. Gürtler, “The First-Order Approach in Rank-Order

Tournaments,” Economics Letters, Vol. 111, No.3, 2011,

pp. 185-187. doi:10.1016/j.econlet.2011.02.013

[8] M. R. Baye, D. Kovenock and C. G. de Vries, “The All-

Pay Auction with Compl ete Informa tion, ” Economic The-

ory, Vol. 8, No. 2, 1996, pp. 291-305.

7The proof of the proposition proceeds in two steps. First, we analyze which strategies lead to the highest payoff for the principal, i.e., the highest

expected value for the aggregate efforts. Second, we analyze whether the principal has an incentive to deviate from these strategies. Not all strategies

are robust to deviations. Suppose P plays in which case the agents infer to be in state 2 if they do not get informed about the state of nature.

Then, in states 1 and 3 P would also want to conceal information in order to make the agents think that they are in state 2. A similar problem arises

with the strategy , at least for certain specifications of the off-equilibrium beliefs. Still, there exist off-equilibrium beliefs sustaining

as an equilibrium, for example if agents believe that they are in state 1 in case information is withheld.

(1,0,1)

(1,1 ,1)