Maximum efforts in contests with asymmetric valuations

European Journal of Political Economy Vol. 20 (2004) 1059 – 1066 www.elsevier.com/locate/econbase

Maximum efforts in contests with asymmetric valuations Kofi O. Nti * University of Ghana, P.O. Box LG 78, Legon, Accra, Ghana Received 2 September 2002; received in revised form 3 November 2003; accepted 18 November 2003 Available online 12 April 2004

Abstract Efforts may be reduced when players with different valuations participate in a contest. This paper considers the problem of designing a contest to elicit maximum aggregate effort from players with asymmetric valuations. Optimal designs for different classes of contest technologies are computed and characterized. A value weighted contest is optimal in the concave case. In the unconstrained case, the optimal contest is equivalent to a first price all-pay auction with a reserve price. The optimal design discounts the effort of the high valuation player in order to induce him to compete vigorously. D 2004 Elsevier B.V. All rights reserved. JEL classification: C72; D72 Keywords: Contest design; Asymmetric contests; Rent-seeking

1. Introduction Contests are used to model many important economic and social interactions where two or more players expend efforts in hopes of winning a prize (Nitzan, 1994; Lockard and Tullock, 2001). Since an individual’s expected reward in a contest depends on his effort relative to everybody else’s, contests engender a great deal of competition among symmetric players. However, efforts may be reduced if there are asymmetries in the players’ valuations (Nti, 1999). Low valuation players tend to put in less effort because they realize that their win probabilities are less than average, which encourages high valuation players to attempt to claim the prize with reduced effort. This raises an interesting question about how to design a contest to induce players with different valuations to apply maximum efforts. * Tel.: +233-21-500591; fax: +233-21-500024. E-mail address: [email protected] (K.O. Nti). 0176-2680/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejpoleco.2003.11.003

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Although expenditures and efforts in some contests should be minimized because they are socially wasteful (for example, unproductive rent-seeking expenditures), there is no guarantee that a contest organizer would be guided primarily by social concerns. There are, in fact, certain situations where a self-interested contest organizer may want to extract maximum efforts from the players. Typical examples include: (a) athletic competition where everybody seems to enjoy and approve of intensely contested matches; (b) lobbying where a politician may desires large campaign contributions from potential rent-seekers; (c) government procurement where the sponsoring agency desires maximum cost reduction efforts from contractors; and (d) employment tournament where a company wants managers to exert their best efforts to win promotions. There is an emerging and interesting literature on contest design; namely, the construction of the rules that define the participants, the winner(s), and the prizes awarded to the players. Appelbaum and Katz (1987) study the optimal prize size. The optimal structure of prizes are discussed in Glazer and Hassin (1988) and Moldovanu and Sela (2001). Baye et al. (1993) examine the optimal selection of contestants. The optimal number of contest stages is analyzed by Gradstein (1998). Dasgupta and Nti (1998) investigate optimal choice of the contest success function. Michaels (1988) and Amegashie (1999) discuss the optimal number of contestants. Typically, the objective of the contest design problem is effort maximization. In this paper, we employ the methodology of Dasgupta and Nti (1998) to determine the effort maximizing contests for players with asymmetric valuations. The main difference between this paper and Dasgupta and Nti (1998) is that they consider players with equal valuations and restrict their analysis to concave contest technologies. In contrast, we determine the optimal design for a wider range of contest technologies, specifically concave, power functions, and the unconstrained case. Asymmetric valuations makes a difference even in the concave case. Tullock’s constant returns to scale contest is optimal in Dasgupta and Nti (1998), but we show that the optimal design in the concave case is actually a more general value weighted contest. And we show for the unconstrained case that the optimal contest is equivalent to a first price all-pay auction with a reserve price. In addition, we provide a complete analysis of the optimal design for Tullock’s increasing returns to scale technology. Because the players in our model typically supply different efforts in every plausible contest, they have to be given the differential incentives to induce them to supply maximum efforts. The optimal design discounts the effort of the high valuation player in order to induce him to compete vigorously. The rest of the paper is organized as follows. In Section 2, we describe the model and characterize the equilibrium of asymmetric valued contests with a general contest technology. In Section 3, we construct the effort maximizing contest for the concave and the unconstrained cases. Optimal design for a power function technology is computed in Section 4. Section 5 concludes the paper.

2. The model Consider a contest where two risk neutral players are competing to win a prize. Let V1 be player 1’s valuation of the prize and let V2 be player 2’s valuation, where V2 V V1.

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Suppose player i, i = 1 and 2, expends effort xi in hopes of winning the prize. Given a profile of effort levels (x1, x2), assume that the probability that player i wins the prize is Wi ¼

Pi ðxi Þ ; P1 ðx1 Þ þ P2 ðx2 Þ

ð1Þ

where the contest technology Pi (z) satisfies Pi (0) = 0, PiV> 0, and is twice differentiable. The probability that player j wins the contest equals 1Wi and we may define W1 = W2 = 1/2 if x1 = x2 = 0. The sequence of moves in the contest design problem is as follows: (a) The organizer announces the contest technology Pi(z); (b) The players expend efforts x1 and x2; (c) The organizer collects x1+x2 and awards the prize, using Pi(xi) to construct the odds. The expected payoff of player i is pi ðx1 ; x2 Þ ¼ Wi Vi  xi ¼

Pi ðxi ÞVi  xi : P1 ðx1 Þ þ P2 ðx2 Þ

ð2Þ

Assuming that V1, V2, and Pi(z) are common knowledge, we solve for the Nash equilibrium of the contest. We note that a Nash equilibrium cannot involve zero effort from either player and study the first-order conditions Vi Pi Vðxi ÞPj ðxj Þ Bpi ¼ 1¼0 Bxi ðP1 ðx1 Þ þ P2 ðx2 ÞÞ2

ð3Þ

for an interior equilibrium. Suppose we know the unique equilibrium efforts generated by a given contest technology Pi(z). A careful examination of the first-order conditions (Eq. (3)) reveals that we can employ a pair of linear contest technologies Li(z) = bi +aiz to construct another contest whose unique equilibrium outcome is the same as that obtained from Pi(z). Specifically, consider a contest where the probability that player i wins the prize, given effort profiles (x1, x2), is ˜i ¼ W

bi þ ai x i ; b1 þ b2 þ a1 x 1 þ a2 x 2

ð4Þ

The expected payoff of player i is p˜ i ðx1 ; x2 Þ ¼

Vi ðbi þ ai xi Þ  xi : b1 þ b2 þ a1 x 1 þ a2 x 2

ð5Þ

The first-order conditions for equilibrium are Vi ai ðbj þ aj xj Þ Bp˜ i ¼  1 ¼ 0; Bxi ðb1 þ b2 þ a1 x1 þ a2 x2 Þ2

ð6Þ

The second-order sufficiency conditions 2Vi ai ðbj þ aj xj Þ B2 p˜ i ¼ <0 2 Bxi ðb1 þ b2 þ a1 x1 þ a2 x2 Þ3 are satisfied provided ai > 0 and bi+ai xi > 0, for i = 1 and 2.

ð7Þ

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To ensure that Eq. (6) yields the same equilibrium solution (x1*,x2*) as Eq. (3), let us choose ai = PV(xi*) and bi = P(xi*)  xi*P V(xi*). Then the two sets of equations will be identical so the linear contest will yield the same equilibrium efforts as Pi(z). Note that ai and bi are uniquely defined through xi*. In addition, ai > 0 since xi* > 0 implies PiV(xi*) > 0. Also, bi+ai xi*=Pi (xi*) >0; therefore, the second-order sufficiency conditions are satisfied for for pi. This discussion establishes Proposition 1. Proposition 1. Suppose (x1*,x2*) is the unique equilibrium efforts for a contest that employs increasing contest technologies Pi(z), i = 1 and 2. Let ai = PiV(xi*) > 0 and bi = Pi(xi*)  xi*PiV(xi*). Then (x1*,x2*) can be induced as the unique equilibrium of a contest where the win probability of player i is given by Eq. (4). An important implication of Proposition 1 is that all equilibria of contests employing increasing contest technologies are contained in the set of equilibria of contests involving a pair of linear contest technologies Li(z) = aiz + bi, i = 1 and 2. We can therefore replicate the set of equilibria generated by all increasing contest technologies by varying the parameters ai and bi. Evidently, we must employ different linear functions to construct the players’ win probabilities when the valuations are different.

3. Maximizing aggregate effort In this section we determine the maximum aggregate effort that can be extracted from the players when the contest technologies Pi(z) are increasing. We consider the situation with concave technologies and the unconstrained case separately and exploit Proposition 1 to formulate the related optimization problems. Let (x1*, x2*) be the equilibrium efforts induced by Pi(z). According to Proposition 1, we can use a linear contest to replicate the equilibrium outcome by choosing ai and bi appropriately. Therefore, we can maximize aggregate effort by choosing (xi*, ai, bi), i = 1 and 2, to solve the following optimization problem: Max X ¼ x1* þ x2* s.t. ai Vi ðbj þ aj x*j Þ ¼ ðb1 þ b2 þ a1 x1* þ a2 x2*Þ2 ; ð8Þ and x*i z0;

ai z0:

Eq. (8) is a re-arrangement of the first-order condition given in expression (6). The nonnegativity constraint on ai is a consequence of Proposition 1, and x*i is non-negative (positive) at every equilibrium. Solving the first-order conditions for x*i yields, a1 a2 Vi2 Vj bi xi* ¼  : 2 ai ða1 V1 þ a2 V2 Þ Hence, X ¼ x1* þ x2* ¼

a1 a2 V1 V2 ðV1 þ V2 Þ ða1 V1 þ a2 V2 Þ

2



b1 b2  : a1 a2

ð9Þ

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And substituting xi* into the payoff functions produces pi ¼

a2i Vi3 ða1 V1 þ a2 V2 Þ

2

þ

bi : ai

ð10Þ

If Pi(z) is concave, then bi = P(xi*)  xi*P V(xi*) z 0. In that case, we maximize expression (9) subject to ai z 0 and bi z 0. So it is optimal to set bi = 0. Differentiating the resulting expression for X with respect to ai, we obtain the characterization result a1V1=a2V2. Hence the maximum aggregate effort that can be realized with concave technologies is 0.25(V1+V2). A substitution shows that the maximum aggregate effort can be achieved by using the linear contest technology Li(xi) = Vj xi. This produces a value weighted contest where the win probability for player i is given by Wi ðx1 ; x2 Þ ¼

Vj xi : V2 x1 þ V1 x2

ð11Þ

For the unconstrained case, the contest designer should set xi = WiVi, leaving zero payoff for each player. Therefore, aggregate effort X = x1 + x2 = W1V1 + W2V2 is maximized by setting W1=1 to obtain a maximum aggregate effort equal to V1. From Eq. (10), pi = 0 implies that bi = ai3Vi3/(a1V1 + a2V2)2. Substituting bi into expression (9) yields maximum X ¼

a1 V12 þ a2 V22 ¼ V1 ; a1 V1 þ a2 V2

which implies that a2 = 0, b2 = 0, and b1 = a1V1. The corresponding linear contest technologies are L1(x1) = a1(x1  V1) and L2(x2) = 0. Setting a1 = 1 and insisting that L1z0 yields the optimal contest, which is to award the prize to player 1 if and only if x1 z V1; the prize is awarded to player 2 if x1 < V1. (To provide a participation incentive for player 1, we may award him the prize when x1 z V1e, e >0, and come arbitrarily close to the maximum value V1.) Proposition 2 summarizes the above discussion. Proposition 2. (a) Using concave contest technologies, the maximum aggregate effort that can be elicited from the players is 0.25(V1 + V2); The optimal contest is a value weighted contest where the win probability for player i is given by Eq. (11). (b) In the unconstrained case, the maximum aggregate effort that can be elicited from the players is V1; The optimal contest awards the prize to player 1 if and only if his effort is not less than his valuation. The optimal design for concave technologies involves a value weighted contest where a player’s effective effort is the product of his own effort and valuation of his opponent. Thus the effort of the high valuation player is discounted in order to induce him to compete vigorously. The value weighted contest is a generalization of the constant returns to scale Tullock contest, and the two are equal when the players have equal valuations. In the unconstrained case the effort of the high valuation player is completely discounted if

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his effort falls below his valuation. We should point out that several different mechanisms can be used to implement the optimal design. For example, we could use an auction mechanism where the prize is awarded to the highest bidder but the contest designer has a reserve price equal V1 (see Glazer, 1993) or alternatively use a first price all-pay auction with a reservation price of V1 (see Hillman and Riley, 1989).

4. Power function technology The contest theory literature abounds with models that employ the power function technology P(z) = zr, r > 0, proposed by Tullock (1980). Michaels (1988) suggested that a contest designer could select the parameter r to maximize efforts, and observed that r = 2 is optimal for the symmetric valuations case. We analyze the asymmetric valuations situation and determine the optimal r for different profiles of player valuations. Suppose the organizer employs the power function technology Pi(z) = z r and the players exert efforts x1 and x2. Player i wins with probability xir/(x1r+x2r) and obtains an expected payoff pi ðx1 ; x2 Þ ¼

Vi xri  xi : þ xr2

xr1

As shown in Nti (1999), the unique equilibrium efforts are x1* ¼

rV1rþ1 V2r

and

ðV1r þ V2r Þ2

x2* ¼

rV2rþ1 V1r ðV1r þ V2r Þ2

;

provided V1r+V2r > rV1r z rV2r. Thus, aggregate effort is X R ¼ x1* þ x2* ¼

rV1r V2r ðV1r þ V2r Þ2

ðV1 þ V2 Þ:

Let v = V2/V1 and factor out the total valuation V1 + V2 from the expression for X R. Then given a valuation ratio v, 0 < v V 1, we may write the effort maximization problem as choosing r z 0 to Max F R ¼

rv r ð1 þ v r Þ2

s.t. rV1 þ v r : That is, given the players’ valuations, we choose an optimal value of the returns to scale parameter to induce the players to expend the maximum proportion of their total valuation. The constraint in the optimization problem is the parametric condition that ensures the existence of pure strategy equilibrium for each v.

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Table 1 Design parameters and valuation ratio v

0.1

0.2

0.285

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

R r* F*

1.08 0.67 0.10

1.16 0.96 0.14

1.217 1.217 0.178

1.23 1.23 0.19

1.30 1.30 0.23

1.38 1.38 0.28

1.47 1.47 0.32

1.57 1.57 0.36

1.69 1.69 0.41

1.83 1.83 0.45

2.00 2.00 0.50

For 0 < v V 1, let R(v) be the unique solution to the equation r = 1 + vr. Then the constraint set equals the interval [0, R(v)]. It can be shown that R(v) > 1 and that R(v) is increasing in v. Obviously R(1) = 2, and the largest feasible region, 0 V r V 2, occurs when the players have equal valuations. The objective function has BF R vr ¼ ½1 þ v r þ r log v  rv r log v: Av ð1 þ v r Þ3 The quantity 1 + vr + rlogv  rvrlogv is decreasing in r and is negative if v is small. Therefore the objective function must be decreasing at R(v) if v is small. Thus the optimal solution r*(v) must be less than R(v) if v is small. Also, rV1 + vr implies that the quantity 1 + vr + r log v  rvr log v is not less than r (1 + log vvr log v). But the latter expression is not less than rvr log v > 0 if v is large (for example, vze1 = 0.368). Therefore, the objective function is increasing on [0, R(v)] if v is large. Thus the optimal solution r*(v)=R(v) if v is large. This implies that F* = 11/R(v). The optimization problem was solved for numerous values of v. Table 1 provides the optimal design parameters R, r*, F* for different valuation ratios. The table reaffirms the qualitative properties derived earlier. In addition, we found that v=0.285 is the smallest valuation ratio for which r*(v) = R(v). Interestingly, the constant returns to scale technology is optimal only if the valuation ratio equals 0.2138. Proposition 3. Let R(v) solve the equation r = 1 + vr, where v is the valuation ratio. Then the maximum fraction of the total valuation that can be elicited within the class of power function contest technologies equals 11/R(v), if v z 0.285; The maximum fraction of the total valuation that can be elicited is greater than 11/R(v) if v < 0.285.

5. Conclusion This paper considered the problem of eliciting maximum efforts from two players who have asymmetric valuations for a prize. We determined the optimal efforts and contests for the unconstrained problem and also for concave and power function technologies. Solving the contest design problem for different classes of contest technologies revealed and rationalized several contest forms as optimal. For example, the value weighted contest and the all-pay auction with reserve price emerged naturally within our analytical framework. Typically the optimal contests discounts the effort of the high valuation player in order to induce him to compete vigorously. We can also associate an optimal contest with the exponents of the power function technology. Future research could investigate other types of contest forms that may emerge as the design objective is varied.

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Acknowledgements I thank the referees and the editor for helpful comments and suggestions. I am particularly grateful to Bouwe Dijkstra for his helpful suggestions.

References Amegashie, J.A., 1999. The design of rent-seeking competitions: committees, preliminary and final contests. Public Choice 99, 63 – 76. Appelbaum, E., Katz, E., 1987. Seeking rents by setting rents. Economic Journal 97, 685 – 699. Baye, M., Kovenock, D., de Vries, C.G., 1993. Rigging the lobbying process: an application of the all-pay auction. American Economic Review 83, 289 – 294. Dasgupta, A., Nti, K.O., 1998. Designing an optimal contest. European Journal of Political Economy 14, 769 – 781. Glazer, A., 1993. On the incentives to establish and play political rent-seeking games. Public Choice 75, 139 – 148. Glazer, A., Hassin, R., 1988. Optimal contests. Economic Inquiry 26, 133 – 143. Gradstein, M., 1998. Optimal contest design: volume and timing of rent seeking in contests. European Journal of Political Economy 14, 575 – 585. Hillman, A.L., Riley, J.C., 1989. Politically contestable rents and transfers. Economics and Politics 1, 17 – 39. Lockard, A., Tullock, G. (Eds.), 2001. Efficient Rent-seeking: Chronicle of an Intellectual Quagmire. Kluwer Academic Publishing, Boston. Michaels, R., 1988. The design of rent-seeking competitions. Public Choice 56, 17 – 29. Moldovanu, B., Sela, A., 2001. Optimal allocation of prizes in contests. American Economic Review 91, 542 – 556. Nitzan, S., 1994. Modelling rent-seeking contests. European Journal of Political Economy 10, 41 – 60. Nti, K.O., 1999. Rent-seeking with asymmetric valuations. Public Choice 98, 415 – 430. Tullock, G., 1980. Efficient rent seeking. In: Buchanan, J., Tollison, R., Tullock, G. (Eds.), Toward a Theory of the Rent-seeking Society. Texas A&M University Press, College Station, pp. 97 – 112.