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363

Effort levels in contests The public-good

prize case

Kyung Hwan Baik Department

of Economics,

Appalachian

State University,

Boone,

NC 28608, USA

Received 8 February 1993 Accepted 5 April 1993

Abstract We examine the equilibrium individual players’, groups’, and total effort levels expended in a contest where groups compete with one another to win a group-specific public-good prize and the players choose their effort levels simultaneously and independently.

1. Introduction

A contest is a situation in which individuals agents or groups compete with one another by expending effort to win a prize. Examples include R&D competition, rent-seeking and rentdefending contests, competition for promotion, election campaigns, litigation, and sports events. Contests have been studied by many economists [see, for example, Loury (1979), Tullock (1980), Rosen (1986, 1988), Dixit (1987), Hillman and Riley (1989), Reinganum (1989), and Ellingsen (1991)]. Despite the vast literature, however, contests in which groups compete to win group-specific public-good prizes have not been explicitly studied. The purpose of this paper is to examine the individual players’, groups’, and total effort levels expended in such contests. To do so, we consider a contest in which groups compete with one another to win a group-specific public-good prize, the players choose their effort levels simultaneously and independently, and the players value the prize differently. Contests involving group-specific public-good prizes are easily observed. For example, consider a contest in which people in different locations compete to win the designation of location of a government institution, a government-owned corporation, or a new highway. All people living in the winning location benefit from the prize won. Competition between consumers and firms over a monopoly position provides another example. If consumers win the contest, all consumers enjoy the rent defended.

2. The model Consider a contest in which n groups compete with one another by expending effort to win a prize. The number of the groups is greater than unity: n > 1. Group i consists of m, risk-neutral

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K.H.

Baik I Economics

Letters 41 (1993) 363-367

players. Let xik be the irreversible effort level expended by player k in group i and let X, represent the effort level expended by all the players in group i: Xi = cl!1 xii. Effort levels are non-negative and are measured in units commensurate with the prize. Let pi(X, , ...,X,j be the probability that group i wins when the effort levels of the groups are (X1, . . . , X,). Then the probability-ofwinning function for group i can be written as

where Cy=, pi = I. Assumption 1. We assume that ap,lax, 2 0,d2pildXf s 0, dpilaXj 5 0, and a2pilaX? 2 0 for i #i. we further assume that ap,lax, > 0 and d2pildX~ < 0 when at least one Xj > 0, and aPi/aXj < 0

and a2pil ax: > 0 when Xi > 0. In Assumption 1, we assume that each group’s probability of winning is increasing in its own effort level and is decreasing in the opponent’s effort level. The marginal effect of each group’s effort level on its own probability of winning decreases as its effort level increases. Finally, the marginal effect of the opponents’s effort level on each firm’s probability of winning decreases as the opponent’s effort level increases. The prize is a public good within each group. Valuations of the prize may differ across players in the contest. Let uik represent the valuation of the prize of player in k in group i. Assumption

2. Without

loss of generality,

we assume that uih_, 2 uih > 0 for h = 2,...,mi.

Although players in each group have the same goal of winning the group-specific public-good prize, they choose their effort levels independently. We assume that all of the above is common knowledge. The expected payoff of player k in group i is 7Tin,, = U&(X,,

. . . ) X,) - Xik .

Given the effort levels of all the other players in the contest, let Zik denote the best response of player k in group i. Then Ziikis obtained by Tiik=rnrx U&(X,) . . . ) X,) - Xik such that xik 2 0 . The first-order

condition for this problem is

~,(ap,/ax,) - 1 = 0 for giik> 0

(1)

Uik(a&axi)-150

(2)

or forx”,, = 0.

If player k in group i expends a positive effort level, his marginal gross payoff, u,(dp,laX,), must be equal to his marginal cost, 1, at that effort level. If he chooses zero effort, his marginal gross payoff must not exceed his marginal cost at that zero effort. His marginal gross payoff, uik(dpildXi), decreases in his effort level, xik, due to Assumption 1. This implies that the second-order condition for this maximization problem is satisfied and giik is unique.

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K.H. Baik I Economics Letters 41 (1993) 363-367

We assume that the players in the contest choose their effort levels simultaneously and we employ a Nash equilibrium as the solution concept. A Nash equilibrium occurs when each player’s effort level is the best response to the effort levels of the other players. Let xz and XT denote the equilibrium effort level of player k in group i and the equilibrium effort level of group i, respectively. Let (X*)_i = (XT, . . . , XZ!_l, Xi*, 1, . . . , Xz). Definition

response,

‘. Given the other group’s equilibrium Xi(k), is defined as &k)

effort levels, (X*)_i, group i’s player-k-best

=m;x uikpi(Xi, (X*)_i) -Xi such that Xi 2 0

Group i’s player-k-response, &k), represents the best response of group i as a whole to the other group’s equilibrium effort levels when player k’s valuation of the prize is taken into account. The first-order condition for this maximization problem is uik(apil~Xi) - 1 = 0

if *i(k) > 0

(3)

uik(8pilaXi) - 15 0

if gi(k) = 0.

(4)

or

The term uik(dpildXi) decreases in Xi due to Assumpti_on 1. Therefore, condition is satisfied and group i’s player-k-best response, Xi(k), is unique, Lemma

the second-order

1. Zi(h - 1) 2 Zi(h) for h = 2,. . . , mi .

Proof Since the term uik(apili3Xi) in the first-order conditions (3) and (4) decreases in Xi, we obtain that for any h and k, X,(h) > &i(h) holds if uih > uik and Xi(h) = zi(k) holds if uih = uik. This and Assumption 2 yield Lemma 1. Cl Lemma 2. Given the other groups’ equilibrium effort levels, (x*)_i, level is neither greater nor less than group i’s player-l-best response,

group i’s equilibrium

effort

X,(l).

Proof Let Xi be-the effort level of group i. First, suppose that Xi > ki( 1) holds. Then by Lemma 1, Xi > Xi(l) 2 Xj(k) holds for any k, where 1 I k I mi. This and Assumption 1 imply that uik(apildXi) - 1 c 0 for any k [see first-order condition (3)]. Looking at first-order condition (l), then, we know that for any player expending a positive effort level, his effort level is not the best response to the effort levels of all the other players in the contest since his marginal gross payoff is less than his marginal cost. Therefore, group i’s effort level, Xi, is not an equilibrium one. Next, suppose that Xi < zi(l) holds. Then looking at first-order condition (3) and using Assumption 1, we know that uiI(dpildXi) - 1 > 0 holds at group i’s effort level Xi. This implies that first-order condition (1) or (2) for player 1 does not hold. His marginal gross payoff is greater than his marginal cost at his effort level. Thus, his effort level is not the best response to the effort levels of all the other players in the contest. Therefore, group i’s effort level, Xi, is not an equilibrium one. 0 Lemma 2 implies that given the other groups’ equilibrium

effort levels, group i’s equilibrium

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K.H. Baik I Economics Letters 41 (1993) 363-367

effort level equals group i’s player-l-best response: Xy = Xi(l). equilibrium effort levels of the players in group i in Lemma 3.

Using this,

we construct

Lemma 3. Given the other groups’ equilibrium effort levels, (X*)_i, (a) if uil > ui2, then x; = Xi(l) and .x,t = 0 for h = 2,. . . , mi, and (b) if uil = uir> uil+, for Some t, then cfzl x~; = X,(l) and x,t=Oforh=t+l,..., mi, where 2atImi. Proof. (a) Consider first the case where Xi(l) = 0. By Lemma 1, Xi(k) = 0 holds for any k, where 1 I k 5 mi. Then it follows from first order-condition (4) that if group i’s effort level is zero, then uik(~pilaXi) - 15 0 holds for any k. This implies that if all players in group i expend zero effort, first-order condition (2) is satisfied for each player in group i. Therefore, given the other groups’ equilibrium effort levels, (X*)_i, and zero effort of the other players in group i, zero effort is the best response of each player in gr_oup i.

Next, consider the case where Xi( 1) > 0. Using first-order condition (3) and Assumption 1, we obtain: If group i’s effort equals Xi(l), then uiI(apilaX,) - 1 = 0 and uih(api18Xi) - 1 < 0 for h=2,..., mi hold since uil > ui2. Hence, if player 1 expends Xii<1) and the rest of the players in group i choose zero effort, first-order condition (1) is satisfied for player 1 and first-order condition (2) is satisfied for each of the rest. Therefore, the proposed effort level of each player in group i is his best response to the other groups’ equilibrium effort levels, (X*)_i, and the proposed effort levels of the other players in group i. Finally, we show that. there are no other equilibrium effort levels of the players in group i. Suppose on the contrary that (xi,, . . . , xi,,.) are the equilibrium effort levels of the players in group i such that Xi = Cl21 xij = Xi(l) and xi, # 0 for some t, where 25 t 5 m,. Then since first-order condition (1) must be satisfied for player t, at these effort levels u,(dp,18Xi) - 1 = 0 must hold. Since vi1 > uir, it follows from first-order condition (3) and Assumption 1 that at these effort levels u,,(~p,l~X,) - 1 < 0 must also hold. This leads to a contradiction. (b) The proof of part (b) is similar to that of part (a) and is omitted. q If one and only one player in a group has the highest valuation, the equilibrium effort levels of the players in the group are unique. Only the hungriest player expends positive effort. If more than one player in a group has the highest valuation, the equilibrium effort levels of the players in the group are not unique. It is possible, in this case, that some hungriest players expend zero effort. Note, however, that in both cases, the group’s equilibrium effort level is equal to its player-l-best response to the other groups’ equilibrium effort levels. We describe the equilibrium individual players’, groups’, and total effort levels in Proposition 1 which follows immediately from Lemmas 2 and 3. Proposition 1, If a player in a group expends positive effort at a Nash equilibrium, his valuation of the prize must be highest one in his group. The equilibrium groups’ and total effort levels equal those resulting when only one of the hungriest players in each group competes for the prize.

The first part of Proposition 1 implies that the equilibrium effort level of a player whose valuation of the prize is less than somebody else’s in his group expends zero effort and thus is a free rider. The second part implies that the number of players and the distribution of valuations in each group do not affect groups’ and total effort levels as far as the highest valuation of the group does not change. To obtain the equilibrium groups’s or total effort levels, one only needs to consider a reduced contest with n hungriest players, one for each group.

K.H. Baik I Economics Letters 41 (1993) 363-367

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3. Conclusion

We have examined the equilibrium individual players’, groups’, and total effort levels expended in a contest where groups compete with one another to win a group-specific public-good prize, the players choose their effort levels simultaneously and independently, and the players value the prize differently. We have demonstrated the following. If a player’s equilibrium effort level is positive, he must be one of the hungriest players in his group. The equilibrium effort level of a player whose valuation of the prize is less than somebody else’s in his group expends zero effort. The equilibrium group’s and total effort levels depend neither on the number of players in each group nor the distribution of valuations in each group. It depends on the valuation of the hungriest player in each group and the number of groups in the contest.

References Dixit, A., 1987, Strategic behavior in contests, American Economic Review 77, 891-898. Ellingsen, T., 1991, Strategic buyers and the social cost of monopoly, American Economic Review 81, 648-657. Hillman, A.L. and J.G. Riley, 1989, Politically contestable rents and transfers, Economics and Politics 1, 17-39. Loury, G.C., 1979, Market structure and innovation, Quarterly Journal of Economics 93, 395-410. Reinganum, J.F., 1989, The timing of innovation: Research, development, and diffusion, in: R. Schmalensee and R.D. Willig, eds., Handbook of industrial organization (North-Holland, Amsterdam) 849-908. Rosen, S., 1986, Prizes and incentives in elimination tournaments, American Economic Review 76, 701-715. Rosen, S., 1988, Promotions, elections and other contests, Journal of Institutional and Theoretical Economic 144, 73-90. Tullock, G., 1980, Efficient rent seeking, in: J.M. Buchanan, R.D. Tollison and G. Tullock, eds., Toward a theory of the rent-seeking society (Texas A&M University Press, College Station) 97-112.