Competitive Problem Solving and the Optimal Prize Schemes Toru Suzukiy July 2, 2011

Abstract Agents compete to solve a problem. Each agent is endowed with technology which is private information. Given a prize scheme, each agent simultaneously decides whether to make the e¤ort. In the same time, if the agent makes the e¤ort, he chooses whether he uses his endowed technology or tries a new technology. This paper analyzes prize schemes as a tool to induce "optimal risk-taking." The stake of the prize promotes e¤ort while the relative prize size between ranks controls the risk taking level. It is shown that when the winner-take-all scheme induces excessive risk-taking given the optimal level, there exists a prize scheme which induces the optimal risk-taking in equilibrium. Moreover, the existence of such optimal prize scheme is guaranteed whenever the stake and the number of agents are su¢ ciently large. Keywords: Optimal prize scheme, Risk taking, Rank dependent prize, Experimentation, Problem solving JEL codes. D82

I thank two anonymous referees for valuable suggestions. I also thank Jacob Glazer, Hsueh-Ling Huynh, Stephen Morris, Barton Lipman, participants of Boston University theory workshop and Max Planck Institute ESI Spring workshop for helpful comments on earlier version of this paper. y Max Planck Institute of Economics. E-mail address: [email protected]

1

Introduction

In economics, contest is usually studied as an incentive scheme to promote investment/e¤ort. However, for some practitioners, contest is a tool to provide the incentive for risk-taking. For example, the X-prize foundation, one of leading innovation prize organizers, emphasizes the importance of risk-taking in innovation and claims that innovation prize is an e¤ective incentive scheme to promote risk-taking. This paper investigates prize scheme as an incentive to induce "optimal risk-taking." We provide a condition which guarantees the existence of a prize scheme which induces "optimal risk-taking." In some contests, whether to take a risk can be more important decision than how much to invest. For example, in the Human Genome Project, National Institute of Health (NIH) and Celera genomics competed for determining the sequence of chemical based pairs for human DNA. NIH employed a well known safe method and Celera genomics employed a risky method whose e¤ectiveness was unknown. The Celera genomics won the race and the risky method became a popular sequencing method after the competition. The bene…t of risk-taking is not only breakthrough. Even if a risk-taking brings failure, it can be valuable as experimentation. Especially, if the failures are public information, the gain from experimentation becomes even larger. On the other hand, risk-taking can be costly. If all agents take a risk to outperform others, the average performance can be lower than the case where agents use a safe method. Hence, for a contest designer, the "optimal risk-taking level" can be di¤erent depending on the situation and "optimal prize scheme" has to be designed for each situation. In the …rst part of this paper, we analyze the equilibrium of "problem solving contest" given a prize scheme. The basic setting is introduced in Section 2. The model consists of a …nite number of agents who face a common problem. Each agent is endowed with technology which is private information. Then, each agent simultaneously decides whether to make the e¤ort. In the same time, if he makes the e¤ort, he also decides whether he uses the endowed technology “safe method” or tries a new technology “risky method.” Concretely, if he uses the safe method, the time to …nd a solution is known. On the other hand, if he uses the risky method, the time to …nd a solution is determined by a random draw. Then, the prize scheme determines the payo¤ of each agent based on his rank in the contest. In Section 3, it is shown that all equilibria in any prize scheme have a monotonic property 1

such that, if one type uses the safe method, all types with higher technology always use the safe method. Moreover, we show that, whenever the stake is su¢ ciently large given a prize scheme, all agents make the e¤ort in equilibrium. Then, we provide the analysis of the winner-take-all scheme, which is one of most popular prize schemes. First, it is shown that there exists a unique equilibrium where all agents make the e¤ort if the stake is su¢ ciently large. Moreover, we show that the winner-take-all scheme induces the risk-taking even if the expected time to …nd a solution with the risky method is much longer than the expected time with the safe method. Second, we show that if the stake and the number of agents are su¢ ciently large, all types use the risky method in the equilibrium. Section 4 analyzes the optimal prize scheme. We de…ne "optimal prize scheme" to be a prize scheme which provides incentive to make the e¤ort while it induces the optimal risk-taking in an equilibrium. We provide a condition which guarantees the existence of an optimal prize scheme. Concretely, it is shown that whenever the winner-take-all scheme induces "excessive risk-taking" given an optimal risk-taking level, the existence of an optimal prize scheme is guaranteed. Moreover, we also show that whenever the stake and the number of agents are su¢ ciently large, the existence of an optimal prize scheme is always guaranteed. That is, any level of risk-taking can be induced by some prize schemes as long as the stake and the number of agents are su¢ ciently large. The idea of the main result is as follows. Note that, by allocating the prize from a higher rank to lower ranks, the designer can reduce "competitive pressure" of the prize scheme. For instance, when the prize is equally allocated among all ranks, the prize is rank independent and generates no competitive pressure. Then, if the winner-take-all scheme induces excessive risk-taking, the designer can induce the optimal risk-taking level by reducing the level of competitive pressure. Related literature. Since agents compete to …nd a solution, this model is related to strategic search models. In most of strategic search models, the intensity of search is a key choice variable, e.g., Fershtman and Rubinstein (1997), R&D race models such as Dasgupta and Stiglitz (1980). However, in our model, each agent decides not only whether to make an e¤ort but also whether to take a risk. As we mentioned before, this paper analyzes contest as incentive for risk-taking or experimentation. One of standard models to investigate experimentation is multi-armed bandit 2

models, e.g. Bergemann and Valimaki (2008). In the multi-armed bandit model, the decision maker has incentive to try a risky choice because the costly experimentation can improve the future payo¤. On the other hand, in our paper, we focus on the situation where the game is one-shot and, without any prize scheme, no agent has incentive to try a new technology. Then, we provide a prize scheme which induces the optimal level of experimentation. The game we analyze in this paper is a class of contest game. Unlike most of contest models where each agent chooses the e¤ort or investment level, e.g., Lazear and Rosen (1981), each agent chooses not only whether to make the e¤ort but also whether to take the risk in our game. There are not many models which take into account risk-taking in contest. Hvide (2002), Hvide and Kristiansen (2003), and Krakel (2008) analyze risk-taking in contest and provide some interesting economic implications, e.g., explanation of the relative performance evaluation puzzle. Unlike their papers, the main interest of our paper is in the design of the prize scheme. Thus, this paper is also related to contest design literatures, e.g., Taylor (1995), Moldovanu and Sela (2001, 2006), Che and Gale (2003). The main di¤erence is that they design contest/tournament to induce the optimal investment/e¤ort level. On the other hand, our prize scheme is designed to induce the optimal risk-taking. 2

Model

There are I agents, i = 1; 2; ::; I who compete to solve a problem. When agent i …nds a solution at ti 2 [0; 1) and he is the n-th agent to …nd a solution, he receives prize zn 2 [0; 1) at ti : Then, we de…ne prize scheme to be 2 where 8 < n+1 = ( 1 ; 2 ; ::; n ; ::; I ) 2 [0; 1]I : P :

n

n

Then, given prize budget x > 0; n-th prize zn is determined by

9 for any n = ; =1

n

n x:

1

That is,

the relative prize size for n-th ranked agent. We focus on the case where n. Note that, if

n+1

>

n;

n+1

n

determines n

for all

then the agent who …nds a solution at t0 has incentive to report

the solution at some t00 > t0 to get higher prize. Thus, such prize scheme cannot be e¤ective. 1

Another interpretation of x is a bonus pool. Rajan and Reichelstein (2006, 2009) study the e¤ectiveness of such bonus pool system from a di¤erent perspective.

3

To solve the problem, each agent has to make the e¤ort which costs c > 0: On the other hand, if the agent chooses "no e¤ort" ?, then he never …nds any solution and his payo¤ is 0. When the agent …nds a solution at t and receives prize zn ; his payo¤ is u(zn ; t)

c: u(zn ; t)

satis…es the following assumptions. Assumption 1 (Monotonicity). Given any t 2 [0; 1); u(zn ; t) is continuous and

strictly increasing in zn :

Assumption 2 (Impatience). Given any zn 2 [0; 1); u(zn ; t) is continuous and strictly

decreasing in t:

In short, the agent prefers larger prize if the delivery timing is the same. Moreover, if the prize size is the same, the agent prefers earlier delivery timing. The continuity guarantees that a "small" change of the timing or the prize size brings also "small" change in the payo¤. When agent i makes the e¤ort, he has to specify his problem solving method which determines the time to …nd a solution ti . There are two possible problem solving methods: one is a safe method S and the other is a risky method R: More concretely, agent i is endowed with technology

i

2

=[

min ; max ]

which is private

information. Then, when the agent chooses the safe method, it means that the agent uses the endowed technology to solve the problem. With the safe method, agent i can …nd a solution at

i

2

for sure, i.e., ti =

i

with probability 1. It is assumed that

i

is independently

drawn from absolutely continuous distribution function G( ) with supp(g) =

:

When the agent chooses the risky method, it means that he tries a new technology with which the time to …nd a solution ti is uncertain. Formally, when agent i chooses the risky method, ti is independently drawn from absolutely continuous distribution

(t)

2

with supp( ) = [0; T ]: This formulation follows search models where "experimentation" is modeled as a random draw from a distribution. We assume that the performance of search mainly depends on "luck" and

(t) is independent of endowed technology :3

The following two assumptions clarify the economic environment we are interested in. Assumption 3.

min

> 0 and

max

< T:

2

The result of this paper is preserved even if we allow to have T = 1: Suppose the risky method also depends on ; e.g., lower tends to have lower t: As long as the distribution of t is su¢ ciently "close" between types, all results are preserved. 3

4

This assumption provides the key characteristic of the risky method. With the risky method, any type has a chance to outperform the best type who uses the safe method. On the other hand, any type who employs the risky method can be outperformed by the worst type who uses the safe method. Assumption 4.

R

(t)u(z; t)dt < u(z;

max )

for any z > 0:

This assumption states that, without any incentive scheme, every type prefers to stay with his current technology. Thus, in this paper, whenever some type uses the risky method, their decisions are induced by a prize scheme. The timing of the game is as follows. First, Nature independently draws the type of each agent from distribution G( ): Then, each agent simultaneously decides whether to make the e¤ort. When he makes the e¤ort, he also chooses a problem solving method in the same time. When a solution is found, the agent receives a prize based on his ranking and a prize scheme. That is, if he is n-th person who solves the problem, he receives zn . To de…ne agent’s strategy, let A = fS; R; ?g be the set of actions. Then, problem

solving strategy of agent i is a mapping

i

! A.4 We analyze the game with Bayesian

:

Nash equilibrium. Moreover, we focus on symmetric equilibria, i.e., i

=

i( i)

=

j( j)

if

j:

The rest of this paper consists of two parts. In the …rst part, we provide the equilibrium analysis given a prize scheme. In the second part, we analyze the optimal prize scheme which induces the optimal risk-taking from the perspective of a prize designer. 3

Equilibrium

A strategy is monotonic if, whenever ( 0 ) = S; then ( 00 ) = S for any 00 < 0 : Moreover, a strategy is full e¤ort cuto¤ (FEC) if there exists ^ 2 such that ( ) = S whenever < ^ and ( ) = R whenever > ^: The cuto¤ ^ of FEC strategy does not need to be interior of the type space. That is, ^ = ( ) = R for all

2

max

if

( ) = S for all

2

and ^ =

min

if

: An equilibrium is full e¤ort cuto¤ (FEC) equilibrium if it

consists of a FEC strategy pro…le. 4

We focus on pure strategies since the set of types who play a mixed strategy in equilibrium is always measure zero.

5

The following lemma provides the property of equilibria: Lemma 1. Given any

2 ; all equilibria are monotonic.

Proof. Suppose there is an equilibrium which is not monotonic. Then, for some

0

>

00

;

( 0 ) = S; and ( 00 ) = R: Note that, when agent i chooses the safe method, given any t i ; the rank of agent i with

i

=

00

is higher than the rank of agent i with

zn is decreasing in n; the expected payo¤ from the safe method for that for

0

00

i

= 0 . Then, since

is strictly higher than

: Then, since the expected payo¤ from the risky method is the same for any type,

it is pro…table for

00

to use the safe method whenever

0

prefers to use the safe method, a

contradiction. Q.E.D. The next proposition states that all equilibria are FEC equilibrium if the budget size is su¢ ciently large. Proposition 1. Given any prize scheme , all equilibria are FEC equilibrium for su¢ ciently large x. Proof. See appendix. The idea of Proposition 1 is as follows. Note that, given any

i;

the probability that

agent i gets a positive prize from the risky method is strictly positive for any . Then, if x is su¢ ciently large, the expected payo¤ from the risky method is always positive. Hence, there is no equilibrium where some types choose "no e¤ort" which yields 0. Then, by Lemma 1, all equilibria are FEC for su¢ ciently large x: In the rest of this section, we analyze a popular prize scheme in practice. A prize scheme is winner-take-all if

1

= 1 >

n

= 0 for all n 6= 1: Patent race is a good example of

the winner-take-all scheme. The next proposition claims that the winner-take-all promotes the risk-taking even if the expected time to …nd a solution with the risky method is much longer than that with the safe method. Moreover, under su¢ ciently large stake and number of agents, the winner-take-all scheme makes all types to take the risk. Proposition 2. Suppose the prize scheme is winner-take-all. For su¢ ciently large x; there exists a unique FEC equilibrium. Moreover, the equilibrium cuto¤ is strictly smaller than

max :

Furthermore, for su¢ ciently large I and x; the equilibrium cut o¤ type is 6

min :

Proof. See appendix. To provide an intuition of the risk-taking, observe that there is no equilibrium where all types use the safe method in the winner-take-all scheme. This is because the worst type has no chance to win the race when all types use the safe method. Hence, even if the expected time to …nd a solution with the safe method is much longer, the worst type has incentive to use the risky method. When the number of agents becomes larger given a FEC strategy pro…le, the winner tends to be an agent who uses the risky method and …nds a solution before

min :

Thus, when the number of agents becomes su¢ ciently large, even

min

has no

incentive to use the safe method.5 4

Optimal prize scheme

This section analyzes the optimal prize scheme. Consider a designer who chooses a prize scheme to maximize his interest. As usual contest design models, the designer prefers that all agents make the e¤ort. Moreover, the designer has preference over the level of risk-taking. Then, suppose the prize designer has preference over the set of FEC strategy pro…les. We de…ne optimal problem solving (OPS) to be a FEC strategy pro…le which maximizes the designer’s interest and denote the cuto¤ type of the OPS by ^OP S . Moreover, we say a prize scheme

is optimal if the OPS is an equilibrium strategy pro…le given .

In this paper, we do not specify the objective function of the designer since the objective function can be various depending on the situation of the designer. For instance, consider the case where a prize designer wishes to minimize the expected time to …nd a solution. Then, if the expected time to …nd a solution with the risky method is much longer than that with the safe method, the designer may prefer a FEC strategy in which all agents use the safe method, i.e., ^OP S = max . On the other hand, when the risk-taking has an aspect of experimentation which has a positive externality to the society, the designer may wish to promote the risk-taking as much as possible, i.e., ^OP S = min : Then, our question is the following: Given an OPS, what is a su¢ cient condition for the existence of an optimal prize scheme? 5

For risk taking in other kinds of winner-take-all games, see, Tirole (1988), Dekel and Scotchmer (1999), Hvide (2002).

7

Before investigating the general case, consider the case where ^OP S = one of optimal prize schemes is

n

=

1 I

max :

In this case,

for all n: Note that, this prize scheme is rank

independent and the prize only depends on the absolute performance. Since all agents prefer the safe method to the risky method without any competitive prize, this is optimal given ^OP S = max : Note that, any prize schemes which are "close to" the rank independent scheme are also optimal in this case. Turning to the general case, the next proposition states that whenever the winner-take-all scheme induces "excessive risk-taking" given su¢ ciently large stake, there exists an optimal prize scheme. Let ^W T A (I; x) be the equilibrium cuto¤ in the winner-take-all scheme given I and x:. Proposition 3. There exists an optimal prize scheme if ^OP S

^W T A (I; x) for su¢ -

ciently large x.

Proof. See appendix. The idea of the proof is as follows. Suppose we parameterize a prize scheme by as

so that

goes to 0, the prize scheme becomes the winner-take-all while the prize scheme becomes

rank independent as goes to 1. In other words, controls "competitive pressure" of the ^W T A (I; x); we can show that the expected payo¤ for ^OP S from prize scheme. If ^OP S the risky method is higher than that from the safe method as goes to 0: On the other hand, as goes to 1, the expected payo¤ for ^OP S from the risky method is lower than that from the safe method by Assumption 4: Then, if the expected payo¤ given the OPS is continuous in , there exists 0 such that, for type ^OP S ; the expected payo¤ from the risky method and the expected payo¤ from the safe method becomes the same given

0

:

Intuitively, when the winner-take-all scheme induces excessive risk-taking, the designer can induce the optimal risk-taking by reducing the competitive pressure. It is not obvious whether ^OP S ^W T A (I; x) is a necessary condition for the existence of an optimal prize scheme. If there is a scheme which can promote more risk-taking than the winner-take-all, an optimal scheme can exist when ^OP S < ^W T A (I; x): On the other hand, it is not trivial that the winner-take-all scheme always promotes more risk-taking than any 8

other schemes. Suppose, instead of the winner-take-all scheme, consider a prize scheme which rewards both …rst and second ranked agents. Then, the probability of getting some prize becomes higher for both risky and safe methods. However, since the risky method and safe method are not ordered in term of the …rst order dominance, it is hard to see which method gains more from the second prize in general. The next proposition states that, whenever the stake and the number of agents are su¢ ciently large, an optimal prize scheme always exists. Proposition 4. Given any ^OP S 2

; there exists an optimal prize scheme for su¢ -

ciently large I and x.

Proof. The proof is immediate from Proposition 2 and 3. Recall that, by Proposition 2, the equilibrium cuto¤ type in the winner-take-all scheme is

min

for su¢ ciently large I and

x: Then, the existence of an optimal prize scheme is guaranteed by Proposition 3. Q.E.D. Intuitively, Proposition 4 says that, if there is an OPS which cannot be induced by any 2

given x and I; the designer can induce the OPS by increasing the stake and the number

of competitors. When the stake and the number of competitors are large, the winner-takeall scheme induces excessive risk-taking. Then, by choosing a prize scheme which optimally reduces the competitive pressure, the designer can induce the OPS. 5

Concluding remarks

This paper showed that competitive prize can be an useful incentive scheme to induce the optimal risk-taking. The stake promotes e¤ort and the relative prize size between ranks controls the risk-taking behavior. It is shown that, whenever the stake and the number of competitors are su¢ ciently large, the existence of an optimal prize scheme is guaranteed given any optimal risk taking level. Our results seem to be consistent with prize schemes in the real world. For example, in innovation contests, the prize scheme is often winner-take-all and lower ranked agents are rarely rewarded. Since the designer’s interest is mainly in innovation in this case, such prize scheme, which promotes risk-taking, may be optimal. In motor sports, the point system

9

is usually based on the rank rather than the lap record. Since fans enjoy watching the risk-taking of drivers, such point system seems to be an e¤ective incentive scheme. In academics, the prize scheme is less competitive than the winner-take-all. The …rst solution of an open question tends to be published in a top journal. On the other hand, when another researcher …nds a simpler solution later, it can be published in a …eld journal. If the purpose of research is not only "breakthrough" but also re…nement of existing solutions, such prize scheme is optimal since it reduces the amount of "speculative" research. Finally, suppose a manager has to get a solution of a problem before a deadline and it can be done by a safe method. In this case, the manager may simply reward the worker as long as he …nds a solution before the deadline. When the manager uses a competitive prize scheme, workers may take a risk to get higher prize and the manager may fail to get the solution before the deadline. 6 6.1

Appendix Proof of Proposition 1

First, we establish the existence of FEC equilibrium. Let qn (ti jI; ^) be the probability that

agent i becomes n-th ranked agent conditional on ti given FEC strategy pro…le with cuto¤ ^. Suppose i = ^: The expected payo¤ from the safe method for agent i given z is US ( i jI; ^) =

X n

qn ( i jI; ^)u(zn ; i )

c:

One the other hand, the expected payo¤ from the risky method for agent i given z is XZ ^ UR ( i jI; ) = qn (ti jI; ^) (ti )u(zn ; ti )dti c: n

ti

Note that both US ( i jI; ^) and UR ( i jI; ^) continuous in ^: Then, let 8 > ^ > if there exists ^ s.t. UR ( i jI; ^) = US ( i jI; ^) > < ^ (z) = if UR ( i jI; ^) < US ( i jI; ^) for any ^ max > > > : if U ( jI; ^) > U ( jI; ^) for any ^ min

R

i

S

10

i

We claim that we can construct a FEC equilibrium with ^ (z) whenever x is su¢ ciently large. First, note that, since US ( jI; ^ (z)) is strictly decreasing in and UR ( jI; ^ (z)) is constant in ; it is not pro…table for any type to use the other problem solving method. Second, suppose ^ (z) 6= max : Then, if x is such that XZ

qn (tjI; ^ ) (t)u(

n x; t)dt

c;

t

n

no type has incentive to choose "no e¤ort." On the other hand, if ^ = that

X

qn (

n

max jI; max )u( n x; max )

max ;

then if x is such

c;

there is no incentive for any types to choose "no e¤ort." Now, we claim that all equilibria are FEC for large x: Observe that the probability of being n -th ranked agent is strictly positive for any n with the risky method. Hence, the expected payo¤ from the risky method is strictly increasing in x given any strategy pro…le. Hence, for su¢ ciently large x; the expected payo¤ from the risky method is strictly positive. Thus, there is no equilibrium where some types choose "no e¤ort" when x is su¢ ciently large: Then, by Lemma 1, all equilibria are FEC equilibrium for su¢ ciently large x: Q.E.D. 6.2

Proof of Proposition 2

First, we establish that any equilibrium cuto¤ type is strictly smaller than max : Suppose ^ i = : Note that, whenever agent i uses the safe method and there are some agents who also use the safe method, the payo¤ from the safe method for agent i is zero. Then, the expected payo¤ from the safe method for agent i given FEC strategy pro…le with ^ is US ( i jI; ^) = q1 ( i jI; ^)u(x; i ) G(^))I 1 (1

= (1

c (^))I 1 u(x; i )

c:

On the other hand, the expected payo¤ from the risky method for agent i given FEC strategy pro…le with ^ is UR ( i jI; ^) =

Z

ti

q1 (ti jI; ^) (ti )u(x; ti )dti

11

c

Observe that as ^ !

max

all agents use the safe method. Hence, lim^!

max

On the other hand, given ti ;

Thus, lim^! lower than

max

max :

8 < [1 ^ lim q1 (ti jI; ) = ^! max :

G(ti )]I

1

if ti <

0 if ti >

max

q1 ( i jI; ^) = 0:

:

max

[UR ( i jI; ^) US ( i jI; ^)] > 0: Hence, the equilibrium cuto¤ type is strictly

To prove the uniqueness of the equilibrium cuto¤, suppose agent i is always cuto¤ type. 0 0 0 Claim 1. If UR ( i jI; ^ )j i =^0 US ( i jI; ^ )j i =^0 0 given i such that j > ^ for all 00 00 0 0 j 6= i; then UR ( i jI; ^ )j i =^00 US ( i jI; ^ )j i =^00 > UR ( i jI; ^ )j i =^0 US ( i jI; ^ )j i =^0 for ^00 > ^0 :

Case 1. Suppose

i

00 is such that ^ >

j

0

>^ >

for some j and j 0 : Then, if

j0

i

0

=^;

the probability that agent i wins the contest with the risky method is Pr(ti < minf

min

j2fj 0 6=ij

Note that, given such minj2fj 0 6=ij

j

j:

00 0 <^ g

Pr(ti < minf

Thus,

min

j2fj 0 6=ij 00

Thus, UR ( i jI; ^ )j is zero for both

00

#fj 0 j

i;

j

0 0 <^ g

j;

min j2fj 0 j

min j2fj 0 6=ij

0 > ^ g > #fj 0 j

0

0 0 >^ g

j0

j

0 0 >^ g

tj g):

00 > ^ g and minj2fj 0 6=ij

tj g) < Pr(ti < minf

min j2fj 0 6=ij

j

j;

00 0 <^ g

j

0 0 <^ g

min j2fj 0 j

j

00 0 >^ g

j

=

tj g)

0 > UR ( i jI; ^ )j i =^0 : Since the expected payo¤ from the safe method 0 and ^ given such i ; ^00

i=

00

UR ( i jI; ^ )j 0

> UR ( i jI; ^ )j

j0

j

j

j

j;

0 0 <^ g

^00

i=

^0

i=

00

US ( i jI; ^ )j 0

US ( i jI; ^ )j

^00

i=

^0

i=

00

= UR ( i jI; ^ )j 0

= UR ( i jI; ^ )j

^00

i=

^0 :

i=

0 Case 2. Suppose i is such that j > ^ for any j 6= i but there exists some j 0 such that 00 00 < ^ : Note that the expected payo¤ from the safe method for i = ^ is 0. Then, since 0

UR ( i jI; ^ )j

^0 i=

0

US ( i jI; ^ )j

0 given

^0 i= 00

UR ( i jI; ^ )j 0

> UR ( i jI; ^ )j

^00 i= ^0

i=

such that

i

00

US ( i jI; ^ )j 0

US ( i jI; ^ )j 12

^00 i= ^0

i=

j

0

> ^ for all j; we have 00

= UR ( i jI; ^ )j

^00

i=

given such

i:

00 0 00 Case 3. Suppose i is such that j > ^ for all i 6= j: Then, UR ( i jI; ^ ) = UR ( i jI; ^ ) 00 0 00 00 and US ( i jI; ^ ) < US ( i jI; ^ ) given such i : Hence, UR ( i jI; ^ )j i =^00 US ( i jI; ^ )j i =^00 > 0 0 UR ( i jI; ^ )j ^0 US ( i jI; ^ )j ^0 given such i : i=

i=

i=

i=

0 0 00 Case 4. Suppose i is such that j < ^ for all i 6= j: Then, UR ( i jI; ^ ) = UR ( i jI; ^ ) and 00 0 00 00 US ( i jI; ^ ) = US ( i jI; ^ ) = 0 given such i : Hence, UR ( i jI; ^ )j i =^00 US ( i jI; ^ )j i =^00 = 0 0 UR ( i jI; ^ )j ^0 US ( i jI; ^ )j ^0 given such i :

Note that

i

is determined by i.i.d. draw from G( ): Hence, we established Claim 1. 0

< ^ for some j; we ^0 ^0 > 0: Thus, if UR ( i jI; )j i =^0 i= i= i= 0 0 0 US ( i jI; ^ )j i =^0 < 0 given US ( i jI; ^ )j i =^0 = 0; we always have UR ( i jI; ^ )j i =^0 i 0 00 00 such that j > ^ for all j 6= i: Then, by Claim 1, UR ( i jI; ^ ) US ( i jI; ^ ) > 0 when0 0 ever UR ( i jI; ^ )j ^0 US ( i jI; ^ )j ^0 = 0. Then, whenever there exists ^ such that Now we establish uniqueness. Observe that, given any 0 0 0 have UR ( i jI; ^ )j ^0 US ( i jI; ^ )j ^0 = UR ( i jI; ^ )j

i=

i

such that

j

i=

UR ( i jI; ^ ) = US ( i jI; ^ ), ^ is the unique solution.

Finally, we prove that the equilibrium cuto¤ type is

min

First, to establish the existence of FEC equilibrium with

for large x and I: min ;

suppose all types use the

risky method. Then, since the probability of being the winner is the same for all agents, the expected payo¤ of each type is at least I1 u(x; T ): Then, suppose an agent with type min

(1

deviates and uses the safe method. In this case, the expected payo¤ for this agent is (

I 1 u(x; min ): min ))

Observe that lim

I(1

(

I 1 u(x; min ) min ))

u(x; T )

I!1

Hence, for su¢ ciently large I; US (

min jI; min )

= 0:

< I1 u(x; T ): Then, if x is su¢ ciently large,

no type has incentive to deviate from the risky method. Second, to establish that

is the only FEC equilibrium cuto¤ type for large I and x; suppose there exists a FEC equilibrium with ^ > min : Suppose i = min : Then, the min

expected payo¤ from the safe method for agent i is US ( i jI; ^) =

I X

Pr(kjI; ^)(1

k=0

13

(

k min )) u(x; min )

c

where

0

I

Pr(kjI; ^) = @

1

1

A (1

k

Then, limI!1 US ( i jI; ^) = 0:

G(^))k G(^)I

k 1

:

On the other hand, if agent i deviates to the risky method, the expected payo¤ is at least 1 u(x; min ) I

(

min ):

Then, note that IUS ( i jI; ^) = 0: I!1 u(x; min ) ( min ) lim

Hence, for su¢ ciently large I; US ( 6.3

min jI;

^) < 1 u(x; I

min )

(

min );

a contradiction. Q.E.D.

Proof of Proposition 3

The proof is constructive. Consider any ( (i) (ii) (iii)

8n;

))In=1 such that

n(

: [0; 1] ! [0; 1] is continuous 8 < 1 if n = 1 lim !0 n ( ) = : 0 if n 6= 1 n

lim

(iv) (v)

) = 1=I for any n

n(

!1

PI

n=1

n(

)

An example of such function is 8 < 1 PN 1 n=1 n( ) = 1 : [1 I

n(

n+1 (

1 [1 I

(1

)=1 ) for any n

(1

)wn ] if n = 1

:

)wn ] if n > 1

with appropriate (w1 ; w2 ; ::; wI 1 ): Then, let zn =

n(

)x and suppose

= ^OP S : Given an OPS, the expected payo¤ from

i

the safe method for agent i is US ( i j ; I; ^OP S ) =

X n

qn ( i jI; ^OP S )u(

n(

The expected payo¤ from the risky method for agent i is XZ ^ UR ( i j ; I; OP S ) = qn (ti jI; ^OP S ) (ti )u( n

ti

14

)x; i ):

n(

)x; ti )dti :

Since

n(

) is continuous in ; both US ( i j ; I) and UR ( i j ; I) are also continuous in :

Now, by property (iii) of

n(

); we have

x lim US ( i j ; I; ^OP S ) = u( ; ^i ); !1 Z I x lim UR ( i j ; I; ^OP S ) = (t)u( ; t)dt: !1 I t Then, by Assumption 4, lim [UR ( i j ; I; ^OP S )

US ( i j ; I; ^OP S )] < 0:

!1

On the other hand, by property (ii) of

); we have

n(

lim US ( i j ; I; ^OP S ) = q1 ( i jI; ^OP S )u(x; i ); !0 Z ^ lim UR ( i j ; I; OP S ) = q1 (ti jI; ^OP S ) (ti )u(x;i t)dti : !0

ti

Note that as

! 0; the prize scheme coverages to the winner-take-all scheme. Then, as we showed in the proof of Proposition 2, if i = ^; then lim !0 [UR ( i j ; I; ^) US ( i j ; I; ^)] ^ 0 for any ^ W T A (I; x): Thus, if OP S W T A (I; x); then lim [UR ( i j ; I; ^OP S ) !0

US ( i j ; I; ^OP S )]

0

Then, since UR ( i j ; I; ^OP S )

exists

US ( i j ; I; ^OP S ) is continuous in given the OPS, there 2 [0; 1] such that UR ( i j ; I; ^OP S ) US ( i j ; I; ^OP S ) = 0:Then, no type has

incentive to use the other method.

Finally, suppose x is such that XZ qn (ti jI; ^OP S ) (ti )u( n

n(

)x; ti )dti

c:

ti

Then, no type has incentive to choose "no e¤ort." Q.E.D. References [1] Che, Y., Gale, I. "Optimal design of research contests." American Economic Review, 2003

15

[2] Bergemann, D., Valimaki, J., "Bandit problems," The New Palgrave Dictionary of Economics, Second Edition, 2008 [3] Dasgupta, P., Stiglitz, J. "Uncertainty, Industrial Structure, and the Speed of R&D" The Bell Journal of Economics, 1980 [4] Dekel, E., Scotchmer, S., "On the Evolution of Attitudes towards Risk in Winner-TakeAll Games", Journal of Economic Theory, 1999 [5] Fershtman,C., Rubinstein, A., "A Simple Model of Equilibrium in Search Procedures", Journal of Economic Theory, 1997 [6] Hvide, H., "Tournament rewards and risk taking," Journal of Labor Economics, 2002 [7] Hvide, H., Kristiansen, E., "Risk taking in selection contests," Games and Economic Behavior, 2003 [8] Krakel, M., "Optimal risk taking in an uneven tournament game with risk averse players," Journal of Mathematical Economics, 2008 [9] Lazear, E., Rosen, S., "Rank-Order Tournaments as Optimum Labor Contracts," Journal of Political Economy, 1981 [10] Moldovanu, B., Sela, A., "The optimal allocation of prizes in contests," American Economic Review, 2001 [11] Moldovanu, B., Sela, A., "Contest architecture," Journal of Economic Theory, 2006 [12] Rajan, M. V. and Reichelstein, S., "Subjective Performance Indicators and Discretionary Bonus Pools," Journal of Accounting Research, 2006 [13] Rajan, M. V. and Reichelstein, S., "Objective versus Subjective Indicators of Managerial Performance", The Accounting Review, 2009 [14] Taylor, C., "Digging for Golden Carrots: An analysis of research tournaments," American Economics Review, 1995 [15] Tirole, J., The Theory of Industrial Organization, MIT Press (1988)

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