designing an optimal contest

European Journal of Political EconomyŽ .Vol. 14 1998 587–603

Designing an optimal contest

Ani Dasgupta ), Kofi O. NtiSmeal College of Business Administration. Penn State UniÕersity, UniÕersity Park, PA 16802, USA

Received 1 December 1997; accepted 1 June 1998

Abstract

The paper brings a mechanism design perspective to the study of contests. We considerthe problem of selecting a contest success function when the contest designer may alsovalue the prize. We show that any equilibrium outcome that can be achieved by a concaveincreasing contest success function can be replicated by a linear contest success function.An expected utility maximizing designer should employ a linear homogeneous contestsuccess function. We explicitly derive the optimal contest for a risk-neutral designer andpresent comparative statics results. Tullock’s contest is optimal only when the designer’svaluation for the prize is low. q 1998 Elsevier Science B.V. All rights reserved.

JEL classification: D72; C72

Keywords: Contest design; Optimal contest; Rent-seeking; Mechanism design

1. Introduction

A contest is an economic or social interaction in which two or more playersexpend money or effort in hopes of winning a prize. Contests have been used tostudy a variety of economic and political phenomena, including rent-seekingŽ . Ž . Ž .Tullock, 1980 , trade protection Hillman, 1989 , patent races Loury, 1979 ,

Ž . Ž .employment tournaments Rosen, 1986 , public goods Katz et al., 1990 , andŽ .many others Dixit, 1987; Nitzan, 1994 . A common way to model a player’s

) Corresponding author. Tel.: q1-814-865-2189; Fax: q1-814-863-2381; E-mail: [email protected]

0176-2680r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0176-2680 98 00027-5

( )A. Dasgupta, K.O. NtirEuropean Journal of Political Economy 14 1998 587–603588

chances of winning the prize is to map expenditure profiles into winning probabili-ties for the players using a specific contest success function. Popular contestsuccess functions include the linear and power functions introduced in TullockŽ . Ž .1980 as well as the exponential function in Hirshleifer 1989 .

This paper is motivated by two considerations. First, we wish to extend thebasic rent-seeking paradigm to consider situations where the agent offering thecontestable rent may also value the prize; the current literature has alwaysassumed that the contest organizer has no valuation for the prize. We examine theincentives of a contest organizer to award the prize in the first place when he orshe also values the prize and must trade-off rent extraction against his or her ownvaluation for the prize. Secondly, we wish to bring a mechanism design perspec-tive to the study of contest in order to provide some decision theoretic justificationfor certain contest success functions commonly used in the literature. SkaperdasŽ .1996 has supplied a set of axioms for the power and exponential contest success

Ž .functions. Clark and Riis 1996 also employ a random utility framework to justifythe power function specification. Although both the axiomatic and random utilityapproaches provide useful insights into the logic underlying certain contest successfunctions, it still leaves open the issue of which member of a class might elicitoptimal effort levels from the contestants, where optimality is considered from thepoint of view of the contest designer. In other words, if a contest designer couldchoose the contest success function, which one should he or she employ?

There are many economic scenarios where a principal needs to solve the aboveproblem of judiciously awarding one prize among a group of potential awardeesbased on their performances. For instance, a manager in a firm may need to decideon who among a certain group of workers should be made the supervisor based ontheir work efforts. The government may need to decide on how to allocate apermit for emitting pollutants to one of several firms based on their R&Dexpenditures towards pollution control. Closer to home, a father might wish togive away a prize to one of his children based on the hours they put in for theirstudies. Finally, in the context of rent-seeking, given that it is essentially a covertactivity, although it might seem peculiar that the ‘rules of the game’ will beannounced upfront, one might point out that such announcements may also bemade covertly, inducing the rent-seeker to solve a similar problem.

In general, the problem of selecting an optimal contest success function iscomplicated by the fact that the prize may be valuable to the designer and hence,in addition to extracting rents from the contestants, he or she might also wish toretain the prize with some positive probability. If efforts or expenditures are low,the planner may not wish to award the prize to any of the contestants. Also, thisextension of the theory is relevant when the government as a contest designer islikely to consider the interests of other stakeholders who might not to able tolobby or contest for the prize directly. Suppose, the government’s objectiveconsists of both welfarist and rent-seeking motives. Now, if the ‘prize’ is thoughtof as some part of national assets, then it seems perfectly plausible to view the

( )A. Dasgupta, K.O. NtirEuropean Journal of Political Economy 14 1998 587–603 589

government’s objective as a weighted average of rents and the probability ofretaining the prize. For the situation where the designer cares only about the

Ž .contestants’ expenditures or efforts, examples discussed in Nti 1997 suggest thatcertain contest success functions induces more aggregate effort than others;

Ž .Gradstein 1995 also notes how variation in players’ abilities affect aggregateexpenditure. But the theory remains to be developed in the general case where thecontest designer cares not just about expenditures but also values the prize andmight wish to incorporate a residual probability of not granting the prize to any ofthe contestants.

In this paper, we adopt a mechanism design perspective to investigate theproblem of choosing a contest success function. We show that the class of linearcontest success functions is quite versatile. Specifically we show that any outcomethat can be achieved as an equilibrium of a contest involving any concaveincreasing contest success function can also be achieved by a linear contestsuccess function. In addition, a homogeneous linear contest success function isoptimal for a large class of utility functions that may be used to evaluate contestperformance. We then identify the specific contest success function a risk neutralcontest designer should choose and subject the results to a comparative staticsanalysis. Our results provide additional justification for the simple constant returnsto scale contest success function that has been employed in the majority of the

Ž .contest theory literature since Tullock 1980 .One byproduct of the our analysis is to show that in the risk-neutral case,

Tullock’s contest is optimal when the contest designers valuation for the prize issufficiently low. However, when the contest designer cares sufficiently about the

Ž .prize but does not necessarily value it more than the contestants do , he shoulddesign the contest so that he retains the prize with some positive probabilityregardless of the expenditure of the contestants. Thus the Tullock contest is notoptimal in this case. Surprisingly, the choice of the designer’s likelihood forretaining the prize does not matter so long as it is strictly positive. The level of theresidual probability is controlled by adjusting the slope of the optimal homoge-neous linear contest success function. In addition, the slope depends on thenumber of contestants as well as the relative valuations for the prize.

The rest of the paper is organized as follows. In Section 2, we formulate theproblem and establish results on the robustness and optimality of the homogeneouslinear contest success function. The risk neutral designer’s problem is analyzedand characterized in Section 3. This is followed by a conclusion.

2. The class of contests

We consider a contest C where n players compete to win a prize with value V,where nG2 and V)0 but we also assume that the contest designer has avaluation V for the prize. Each player must decide on the level of effort or0


expenditure it will invest in the contest. These decisions must be made simultane-ously and independently. Expenditure levels are measured in units commensuratewith the prize. Suppose player i spends x , x G0, in hopes of winning thei i

Ž .contest. Given a profile x , . . . , x of expenditures, assume that the probability1 n

that player i wins the contest is given by the logit form expression

h xŽ .iP x , . . . , x s , is1, . . . ,n , 1Ž . Ž .ni 1 n

sq h xŽ .Ý jjs1

Ž .where sG0 and h . is a twice differentiable, increasing, concave function withŽ .h 0 s0. If ss0, it will be assumed that each contestant has zero probability of

obtaining the prize if none of them puts in any effort.The probability function above is derived as follows. When player i spends xi

Ž .on the contest, his likelihood for winning the contest is given by h x . Thei

contest designer introduces a likelihood s that the prize will not be awarded to anyof the contestants. A straightforward normalization leads to the probability func-

Ž .tion specified in Eq. 1 . By introducing the parameter s into the probabilityfunction, we have described a slightly more general situation where the contestdesigner may retain the prize with positive probability if s)0; the prize will beawarded to one of the contestants for sure if ss0. 1

Given the above specification of probabilities, the expected payoff of player i,is1, . . . ,n, is

p x , . . . , x s P x , . . . , x Vyx ,Ž . Ž .i 1 n i 1 n i

Vhi2Ž .s yx ,i

sqh q hÝi jj/i

Ž . 2where h sh x . Throughout this paper we will fix the number of players nj jŽ . �and consider contests C h,s induced with different functions hg Hs h:R ™q

< Ž . X Y 4R h 0 s0, h )0, and h F0 and different values of sG0.q

1 We assume, in following the bulk of the mechanism design literature, that even though thedesigner cares about the prize himself, once the contestants have expended efforts he will not turnaround and refuse to distribute the prize to the contestants according to the preannounced probabilisticrules. The implicit presumption here, as it is in every other paper of this genre, is that the effect of anyreneging will be disastrous on the part of the principal, perhaps because of some kind of social contractor reputation considerations.

2 Ž .Baye et al. 1993 have noted that the contest organizer may extract more rent by selecting theparticipants judiciously. We however don’t consider this as a choice issue.


Ž .Although, the logit form of contests as described in Eq. 1 , with the ContestŽ . Ž .Success Function CSF h . having the properties mentioned above is ubiquitous

in the literature, it is perhaps worthwhile to discuss the choice of this class in thecontext of mechanism design. Here we adopt the perspective that besides aidinghis objectives, the planner would also like the contest to have certain desirable

Ž .fairness properties. The logit form as axiomatized by Skaperdas 1996 , is knownŽ . Ž .to be the unique possessor of the following properties: 1 It is anonymous, 2 It

rewards higher efforts by a player by higher probabilities; and similarly a player’sŽ .winning probability is lowered by higher efforts by other players, and 3 the

relative winning probabilities of players in a group are unaffected by efforts ofplayers outside of that group. However, even assuming the logit form, one may

Ž .question the use of concave h . functions. The issue is particularly relevant sinceŽ . rit is known that in the class of Tullock contests where h x sx for some r,

higher aggregate expenditures are generated for higher r. That is, a contest with aŽ .concave CSF such as Tullock contest with r-1 may be dominated, at least in

Žterms of aggregate effort, by a contest with a convex CSF such as a Tullock.contest with rs1.5, say . However, a major problem with convex CSFs is that

they are not collusion-proof. For instance, in a two-player contest, suppose players1 and 2 are expending efforts x and x , respectively. Then their total probability1 2

Ž Ž . Ž .. Ž Ž . Ž ..of winning the prize is h x qh x r sqh x qh x . Now suppose,1 2 1 2

player 2, in a collusive agreement with player 1, ‘gives away’ his expenditures tothe latter and supplies no effort himself. While the total cost of the cartel has notchanged, notice however that the total probability of winning the prize now isŽ . Ž Ž ..h x qx r sqh x qx . A simple algebraic manipulation shows that when s1 2 1 2

is strictly positive the latter expression is strictly larger than the former if and onlyŽ . Ž . Ž . Ž .if h x qx )h x qh x . But if the function h . is strictly convex with1 2 1 2

Ž .h 0 s0, this will clearly be the case. For this reason, we will be working withconcave CSFs in this paper. See Section 4 for some comments on the non-concavecase.

Another advantage of working with concave CSFs is that within this class weare assured of a pure strategy symmetric Nash equilibrium. The lemma below

Ž .characterizes such an equilibrium of any contest C h,s .

( )Lemma For hgH, any equilibrium of the contest C h,s must be symmetric. The( ) X( )unique symmetric equilibrium expenditure per player is positiÕe zero if h 0 )

( )F srV.

Proof At any equilibrium, the first order conditions for player i, is1, . . . ,n are

sq h VhXÝ j iž /Ep Epj/ii is y1F0, and x s0. 3Ž .i2nE x E xi i

sq hÝ jž /jsi


The second order sufficiency conditions

2Y Xsq h Vh 2 sq h VhŽ .Ý Ýj i j i2 ž / ž /E p j/i j/iis y F0.2 2 3n nE xi

sq h sq hÝ Ýj jž / ž /jsi jsi

are clearly satisfied since h is concave. 3

ŽFirst we rule out asymmetric equilibria. Consider any equilibrium profile x ),1.x ), . . . , x ) . Without loss of generality, suppose x ))x ). Then the first2 n 1 n

order conditions for players 1 and n imply

2n nXV sq h yh h s sq h 4Ž .Ý Ýj 1 1 jž /

js1 js1

and2n n

XV sq h yh h F sq h 5Ž .Ý Ýj n n jž /js1 js1

Ž . Ž .Note that the inequality in Eq. 5 allows x ) to be zero. And combining Eqs. 4nŽ .and 5 produces

n nX Xsq h yh h G sq h yh h . 6Ž .Ý Ýj 1 1 j n n

js1 js1

Ž . Ž . X XŽ .But x ))x ) implies that h sh x ) )h x ) sh and h sh x ) F1 n 1 1 n n 1 1XŽ . X w n x wh x ) sh since h is increasing and concave. Hence sqÝ h yh F sqn n js1 j 1

n x Ž .Ý h yh . Therefore Eq. 6 must hold as an equality. This implies thatjs1 j nŽ . Ž . XŽ . XŽ .h x ) sh x ) and h x ) sh x ) . In which case, player 1 can improve1 n 1 n

his or her payoff by deviation to x ), destroying the equilibrium. This impliesn

that there are no asymmetric equilibria.Second, if x) is a symmetric equilibrium expenditure per player then the first

Ž .order condition R x) F0 must hold, whereXsq ny1 h x h x VŽ . Ž . Ž .

R x s y1. 7Ž . Ž .2sqnh xŽ .Ž .

XŽ . Ž . XŽ .Suppose h 0 FsrV. Then R 0 sh 0 Vrsy1F0, which implies that x)s0XŽ . Ž .yields a symmetric equilibrium. But if h 0 )srV then R 0 )0. We claim that

3 The only time the first order conditions are not satisfied in equilibrium is when ss0 and everyXŽ .contestant is supplying zero effort. But this can only happen when h 0 s0, which is ruled out by our

assumption that hX)0.


Ž .lim R x -0. Note that as x™`, a concave increasing function h mustx ™`XŽ . XŽ .either have lim h x s0 or lim h x )0 when h is unbounded. In thex ™` x ™`

Ž . w Ž . x X x Ž .2first case lim R x sy1 since sq ny1 h h V r sqnh ™0 because ofx ™`X Žthe h term in the numerator the rest of the expression is easily seen to be

. Ž . w Ž . x X x Žbounded . In the second case lim R x sy1 since sq ny1 h h V r sqx ™`

.2nh ™0 because of the squared term in the denominator. Thus, in either case,Ž .R x is positive for small x and negative for large x. Hence there is an x))0

Ž .such that R x) s0. That is, x))0 yields a symmetric equilibrium. Toestablish uniqueness of x), note that

Y X Xsq ny1 h Vh Vh hŽ .XR x s y s nq1 qn ny1 h -0,Ž . Ž . Ž .2 3sqnh sqnhŽ . Ž .

8Ž .

Ž .since h is non-negative and concave. Hence R x is decreasing in x and mustŽ .have a unique x))0 that solves R x) s0.

Ž .Knowing that every equilibrium of C h,s is unique and symmetric, we canexploit the necessary conditions to construct a linear contest success function

Ž .whose unique equilibrium is the same as that of C h,s . The relationship betweenh and the linear function is given in Proposition 1. Both this proposition and the

Ž .one that follows can be thought of as characterizing the choice of an optimal h . ,once the optimal s has been chosen; they can also be thought of as characterizing

Ž .the optimal h . , if the designer had to work with a given s. We return to the issueof the optimal choice of s later in the paper.

Proposition 1 Suppose x) is the unique equilibrium expenditure per player for˜( ) ( )C h,s where n, V and s are fixed. Then there exists a linear function h x saxqb,

( )a)0 and bG0, such that i x) is also the unique equilibrium expenditure per˜ ˜( ) ( ) ( ) ( )player for C h,s and ii h x) sh x) .

Proof. Define a and b by the following two equations:

ashX x) and ax)qbsh x) . 9Ž . Ž . Ž .X XŽ . XŽ .Clearly a)0 since h )0 and bsh x) yx) h x) G0 since h is concave

Ž . XŽ .with h 0 s0. Note from the Lemma that a is well-defined even if h 0 s`

˜Ž .since x) must be positive in that situation. Now consider the contest C h,s with˜payoff functions p where the linear function h defined above replaces h in Eq.˜ i

Ž .2 . It is evident thatX

Ep Ep sq ny1 h x) Vh x)Ž . Ž . Ž .˜ i is s y1.2E x E x sqnh x)Ž .Ž .i i Ž .Ž . x ) , . . . , x )x ) , . . . , x )

10Ž .


Hence the first order condition for a maximum is satisfied at x) for p if it was˜ i

satisfied at x) for p .iTo ensure that x) is an equilibrium expenditure per player in the new game

with payoff functions p , we must show that x) is in fact a global optimum for˜ i

p given the others are choosing x). But this follows from the concavity of p in˜ ˜i i

x ; that is,i

22 a V sq ax qbŽ .Ý j2E p̃ j/iisy F0 11Ž .2 3nE xi

sq ax qbŽ .Ý jž /js1

since a)0, bG0, sG0 and x)G0. I

An important and powerful ramification of Proposition linear is that allequilibria of contests based on concave increasing contest success functions arecontained in the set of equilibria of linear contest success functions, as we vary theparameters a and b. Of course, to obtain an equilibrium outcome for a given h,we have to be choose a and b in a particular way, as noted in the proposition.

To illustrate the essence of Proposition linear, consider the well-known TullockŽ . rrent-seeking game where h x sx , with 0-rF1 to ensure concavity. The

payoff of player i in the Tullock game is

x r Vip s yx ,i ir rx q xÝi j

j/i

which yields the symmetric equilibrium expenditure

r ny1 VŽ .x sx)s .i 2n

Proposition linear suggests that we can generate the same equilibrium outcome if˜Ž . w Žwe use the linear contest success function h x saxqb, where asr r ny

. 2 xry1 Ž .w Ž . 2 xr1 Vrn and bs 1yr r ny1 Vrn . Here both a and b are positive. It isstraightforward to verify that a contest employing

rry1r ny1 V r ny1 VŽ . Ž .

h̃ x srx q 1yrŽ . Ž .2 2n n

yields the same equilibrium outcome as the Tullock game.Since every equilibrium of a concave contest success function can be induced

by a linear contest success function, involving just two non-negative parameters a


and b, it is appropriate to consider the optimal choice of a and b. Note that as wemove from the equilibrium generated by one concave contest success function tothat generated by another, both a and b also change; that is a and b arefunctionally dependent on x). Thus, to determine the optimal choice of a and b,

Ž .we must construct an appropriate utility function to evaluate every contest C h,s .Ž .Evidently, the utility function must depend on x) and h x) since this is all that

is required to determine the consequences of a given contest success function. Acontest designer would normally wish to induce more expenditure or effort fromthe contestants, therefore the utility function should be increasing in x). Weclaim that under reasonable assumptions, he should also care about a low value ofequilibrium contest success function. To see this, suppose the designer has a vonNeumann Morgenstern utility function defined over prize valuations, which weassume has units that are commensurate with expenditures. In the case when theprize is awarded, the designer gets nx), and when the prize is not awarded heobtains nx)qV where V is his valuation for the prize. His expected utility is0 0

given by

nh sDs U nx) q U nx)qVŽ . Ž .0nhqs nhqs

Straightforward differentiation then confirms the claim that expected utility isincreasing in x) and decreasing in equilibrium h. We shall thus assume that the

Ž Ž ..contest designer employs a utility function D x), h x) , which is increasing inŽ .x) but decreasing in h x) , and determine the form of the optimal contest

success function.

Proposition 2 Suppose a contest designer wishes to choose hgH for a fixed( ( ))sG0 to maximize an utility function D x) , h x) where D is increasing in its

( )first argument and decreasing in the second and x) is the equilibrium of C h,s .Then it is sufficient to employ a homogeneous linear function hsax, where a)0.

Proof. For a fixed s the solution to the utility maximization problem must yieldeither x))0 or x)s0. 4 By Proposition 1, x) can be induced with a linearˆ ˆ ˆ

˜function hsaxqb, where a)0 and bG0.If x)s0 solves the utility maximization problem, then the associated contestˆ

ˆ ˆXŽ .success function h must satisfy h 0 FsrV, as noted in the Lemma. It is evidentfrom the proof of the lemma that we can induce x)s0 by setting bs0 andˆ0-assrV.

4 Note that if ss0 then x)s0 cannot occur since hX)0. Thus x)s0 implies that s)0.ˆ ˆ


Ž .If x))0 solves the utility maximization problem, then a, b, x) must solveˆ ˆthe problem

Max DsD x),ax)qbŽ .s.t.

2sq ny1 ax)qb aVs sqn ax)qb , 12Ž . Ž . Ž . Ž .

where x)G0, aG0 and bG0. The constraints ensure that x) is non-negativeand satisfies the first order condition for each player when the linear contest

˜success function hsaxqb is employed. Let z)sax)qb. Then noting thata)0, the optimization problem may be written as

Max DsD z)yb ra, z)Ž .Ž .s.t.

2sq ny1 z) aVs sqnz) ,Ž . Ž .z)ybG0,

aG0 and bG0.

The objective function above increases when b decreases; also the feasible setincreases as b decreases. Therefore the optimal solution must have bs0. I

Proposition 2 implies that the constant returns to scale contest success functionelicits the greatest expenditure or effort from the contestants. That is, the Tullockcontest is optimal in the class of concave increasing contest success functions solong as the contest designer has no interest in retaining the prize. Furthermore,since the linearity result holds for every fixed s, clearly, there is no loss ofgenerality in confining attention to the class of linear homogeneous CSFs evenwhen s is a choice variable.

3. The risk neutral designer’s problem

In this section, we consider the situation of a risk neutral contest designer anddetermine the optimal a and corresponding x) as a function of the parameters ofˆthe contest. Suppose the risk neutral contest designer has valuation V saV for0

the prize, where the parameter a tracks the relative valuation of the prize betweenthe contestants and the designer. The contestants value the prize more than thedesigner if a-1; the situation is reversed if a)1; and the prize is equallyvaluable to both sides if as1.

Suppose the contest designer employs the homogeneous linear function hsax.This will induce an equilibrium aggregate expenditure equal to nx). The prize

Ž .will be awarded to the contestants with probability nax)r sqnax) but the


designer will keep the prize according to the complementary probability. Thepayoff D to the contest designer is

a sVDsnx)q

sqnax)

To maximize expected payoff, the contest designer must choose a), x) ands) to solve the following optimization problem.

a sVMax Dsnx)q

sqnax)

s.t.

2sq ny1 ax) aVs sqnax) 13Ž . Ž . Ž .and

x)G0 and s)G0.

Ž . Ž . 5Eq. 13 is just the first order condition Eq. 12 with bs0. As noted earlier,when ss0 every finite a essentially produces the constant returns to scale

Ž .Tullock contest. And it is clear from Eq. 13 that the corresponding expenditureŽ .x)s ny1 Vrn. Thus if ss0 the risk neutral contest designer’s will elicit a

Ž .maximum aggregate expenditure nx)s ny1 rnV.For now, we consider the risk neutral contest designer’s problem when s)0 is

fixed. We note that the contest designer can always implement x)s0 by settingassrV. This will yield an expected payoff DsaV. If the contest designerwishes to implement a positive x) then we can transform the optimizationproblem by making the following substitutions.

spsnx) and qs 14Ž .

sqnax)

Then the contest designer’s problem becomes

Max Dspqa qV

s.t.

21yqŽ .1yq Vsp. 15Ž .

n

Ž .where pG0 and 0FqF1. Using Eq. 15 to substitute for p in the objective anddropping the constant multiplicative factor V in the objective leads to the

5 Note that this is without loss of generality. For some a- sr Õ the equation holds as inequality asŽ .x)s0. But then using as sr Õ, insisting on Eq. 13 gives us the same utility.


following one dimensional constrained optimization problem. The contest designershould choose qG0 to

21yqŽ .Max Ds1yqy qa q

n

s.t.

qF1.

We note that the objective function is concave in q and form the Lagrangian

21yqŽ .Ls1yqy qa qq 1yq ,Ž .

n

where l is the Lagrangian multiplier. The optimal solution satisfies the Kuhn–Tucker conditions

E L 2 1yqŽ .sy1q qaylF0 16Ž .

E q n

E L 2 1yqŽ .q sq y1q qayl s0 17Ž .

E q n

E Ls1yqG0 18Ž .

El

E Ll sl 1yq s0, 19Ž . Ž .

El

where qG0 and lG0.The optimal solution q) depends on the magnitude of a . There are three cases

Ž .to consider: If a)1 then Eq. 16 implies that l)0, which when combined withŽ .Eq. 19 yields q)s1; If a-1y2rn then

E L 2 1yq 2 2 qŽ .-y1q q1y ylsy ylF0,

E q n n n

Ž .which when combined with Eq. 17 yields q)s0; And if 1y2rnFaF1 thenŽ .it is straightforward to verify that q)s1yn 1ya r2 and ls0 satisfy the

Kuhn–Tucker conditions.

a)1° ¶~ •1y 2rn FaF1Ž .Proposition 3 Suppose s )0. If then the contest designer¢ ß

a-1y 2rnŽ .


ssrV° ¶~ •s2 srV r 1qa 1y nr2 1yashould choose a) to induce equi-Ž . Ž . Ž . Ž .Ž .¢ ßarbitrarily large

0° ¶2~ •1r4 1ya VŽ . Ž .librium effort x)s from each contestant and obtain a˜

2¢ ß1rn ny1 VŽ .Ž .aV° ¶

2~ •aVq nr4 V 1yaŽ . Ž .maximal expected payoff D)s .¢ ß1rn ny1 VŽ . Ž .

Proof. The proposition follows directly from the solution of the optimizationŽ .problem. If aG1 we have q)s1, which implies ps0 in Eq. 15 and x)s0ˆ

Ž .in Eq. 14 . But x)s0 is induced by choosing a)ssrV, as noted earlier. IfˆŽ . Ž .a-1y2rn we have q)s0, which implies p)s 1y1rn V in Eq. 15 and

Ž . Ž . Ž .x)s ny1 Vrn2 in Eq. 14 . Also q)s0 implies ax)™` in Eq. 14 ; i.e.,Ž .a) must be arbitrarily large. If 1yn 1ya r2FaF1, we have q)s1y

Ž .Ž . Ž . Ž 2 . Ž . Ž .nr2 1ya , which implies p)s nr4 V 1ya in Eq. 15 . Using Eq. 14 ,Ž 2 .Ž . Ž . wŽ .Žwe solve to obtain x)s 1ya Vr4 and a)s2 srV r 1qa 1yˆ

Ž .Ž .xnr2 1qa . The expected payoff D) for each of the cases is calculated bydirectly substituting for q) in the objective function. I

Proposition 3 is fairly intuitive. If the contest designer values the prize morethan the contestants, he chooses a low a) to discourage participation. A low a)

means that the rate at which a contestant’s effort is converted into probability islow. If the contestants value the prize excessively compared to the designer thenthe designer should attempt to replicate the conditions of the constant returns toscale Tullock game by using an extremely large a) to neutralize the effect of s.This ensures that the prize is awarded with very high probability. In the intermedi-ate situation where the contestants value the prize moderately more than thecontest designer a) depends delicately on n, V, a and s. It is interesting toobserve that the effort induced by the optimal contract is non-increasing in a

while the expected payoff is non-decreasing. The dependence of effort and payoffŽ .on relative valuation a is illustrated below Fig. 1 .

We may also investigate the comparative statics properties of the optimalcontest as the parameters of the model changes. In particular, what happens to a)

and x) as s, n, V and a change? Again there are three cases to consider,ˆdepending on the magnitude of a .

It is obvious that for a)1, x)s0 is independent of s, n, V and a . However,ˆŽa)ssrV is increasing in s and decreasing in V. When a-1y2rn, x)s nˆ

. 2y1 Vrn is increasing in V and n but is independent of s and a . For thissituation, we do not have interesting comparative statics result for a) since it is


Fig.

1.D

epen

denc

eof

effo

rton

rela

tive

valu

atio

na

.


Ž . wŽ .Ž Ž .Žnot finite. Finally when 1y2rnFaF1, a)s2 srV r 1qa 1y nr2 1y..x Ž .a is clearly increasing in s but decreasing in n, V and a . Also x)s 1r4ˆ

Ž 2 .1ya V is increasing in V, independent of n and s, and decreasing in a . Thisyields the following result.

Proposition 4 When a) is finite, da)rds)0, da)rdV -0, da)rdnF0 andda)rdaF0; In addition, dx)rdss0, dx)rdV)0, dx)rdnG0 and dx)rˆ ˆ ˆ ˆdaF0, regardless of whether a) is finite or not.

Proposition 4 reveals the strategic dimensions of the problem. Because thecontest designer and the contestants are situated on opposing sides of the market,they have opposing desires and tendencies. The contest designer reduces theexchange rate at which a player’s effort is transformed into probabilities if aparametric variation ordinarily elicits more effort from the contestants; the ex-change rate is decreased if the parametric variation is effort reducing. In singlestage contests, effort per contestant increases with V and n but decreases with s

Ž .and a Baik, 1994; Nti, 1997, 1998 . Therefore the contest designer moves in theopposing direction by offering a lower exchange rate for transforming effort intoprobability as V or n increases, but a higher exchange rate is offered as s or a

increases. Interestingly, the effort induced at equilibrium depends only on the

Fig. 2. Dependence of payoff on relative valuation a .


number of contestants and the relative valuations but does not vary with s when sis positive.

Finally we can determine the optimal choice of both s and a. It is clear thatwhen s)s0, the choice of a) is irrelevant; all positive a) yield the same

Ž .equilibrium outcome and the payoff to the designer is 1y1rn V. This is just areplication of the constant returns to scale Tullock game. A look at Fig. 2 nowtells us that when a)1y2rn the optimal choice of s) must be strictly positivesince the optimal payoff is strictly increasing in this range. Note also that in this

Ž .range D) is independent of s). Thus for a- 1y2rn V, the designer shouldchoose s)s0 and any positive a), implying that the constant returns to scale

Ž .Tullock contest is optimal. But when aG 1y2rn V, the designer may chooseany positive s) and select the corresponding a) in accordance with the appropri-ate expression given in Proposition optimal. This is formally stated in our lastproposition.

( )Proposition 5 If the relatiÕe Õaluation a-1y 2rn then the designer shouldchoose s)s0 to implement the constant returns to scale Tullock contest.Otherwise, the designer should choose any positiÕe s) and determine thecorresponding a) in accordance with Proposition 3 to implement the optimalcontest.

4. Conclusion

In this paper, we have demonstrated that any equilibrium outcome of a contestthat employs a concave increasing contest success function can be replicated byusing a linear contest success function. We exploited this result to construct anoptimal contest for a utility maximizing contest designer who may also value theprize. We showed that the contest designer should employ a linear homogeneouscontest success function, providing a decision theoretic justification for theTullock contest. We explicitly derived the optimal contest for a risk-neutraldesigner and presented the comparative statics results. The constant returns toscale Tullock contest is optimal when the contest designers valuation for the prizeis sufficiently low but not optimal otherwise. When the designer cares sufficiently

Ž .about the prize but does not necessarily value it more than the contestants do heshould design the contest so that he retains the prize with some positive probabil-ity regardless of the effort of the contestants.

This paper dealt with a specific class of contests, namely those with concaveincreasing contest success functions, and where the contestants have equal valua-tion for the prize. A related question that suggests itself is: What happens if theclass of CSFs is extended to include nonconcave CSFs as well? This is currently

Ž .under investigation by one of the authors Dasgupta, 1998 . Preliminary results


indicate that although for all CSFs one cannot provide a linear function to replicatethe equilibrium outcome, one can always provide a piecewise linear functionŽ .consisting of two pieces to do the same. Thus, the contestants may need to crossa ‘threshold’ effort level before further efforts begin to get translated into positiveprobability of winning the prize. This result and its implications will be developedin a future paper. Further extension of this paper might consider the optimal designof contests involving asymmetric valuations of the prize. The consequences ofemploying other utility functions for the contest designer could also be investi-gated.

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