Effort levels in contests with two asymmetric players.

• Author: Baik, Kyung Hwan

1. Introduction

   A contest is a situation in which players compete with one another by expending effort to win a prize. Examples abound. Firms compete by spending R&D expenditures to win a patent which will guarantee a flow of high profits. Firms compete to obtain a monopoly or a procurement contract from a government. People in different locations compete to win the designation of the location of a government institution, a government-owned corporation, or a new highway. Political candidates compete by their campaign activities to win an election. Candidates compete for a job or to win promotion to a higher rank.

   Such contests have been studied by many economists. Loury [17], Lee and Wilde [16], Dasgupta and Stiglitz [5], Gilbert and Newbery [10], Harris and Vickers [11], Reinganum [21], and Delbono and Denicolo [6] have studied R&D competition. Tullock [25; 26], Krueger [14], Posner [20], Rogerson [23], Appelbaum and Katz [1], Hillman and Riley [12], Hirshleifer [13], Ellingsen [8], Nitzan [18], and Balk [2; 3] have studied rent-seeking contests. Lazear and Rosen [15], O'Keeffe, Viscusi, and Zeckhauser [19], and Rosen [24] have examined performance incentives associated with reward schemes. Riley [22] has analyzed the war of attrition and auctions. Dixit [7] has considered strategic commitments in contests with different applications. These are only a part of the literature on the theory of contests.

   Despite the vast literature, however, the effects of asymmetries between players have not been clarified, except by Harris and Vickers [11] who analyze, in a patent "race" model, the strategic consequences of asymmetries. The purpose of this paper is to examine the effects of asymmetries between two players on individual and total effort levels in a model with a logit-form probability-of-winning function. Effort levels expended by the players deserve attention. Effort in a rent-seeking contest is interpreted as social costs and thus total effort level is a measure of economic efficiency; effort in an R&D contest is interpreted as R&D expenditures and thus effort levels determine the expected date of invention.

   We focus on asymmetries between the players resulting from their different valuations of the prize, their different abilities to convert effort into probability of winning, or both. Such asymmetries are common in contests. For example, people in different locations may receive different economic impacts from winning the designation of the location of a new government-owned corporation. Their abilities to influence the political decision may differ. In the case where a government monopoly franchise contract is reconsidered every period, the previous contract holder may value the contract more highly than its rivals because it has invested (and sunk) resources into the particular industry and has experience in that industry. The previous contract holder may be more powerful and more efficient than its rivals in the contest for this period's contract, partly because it has been able to establish a relationship with government officials [23, 392] and partly because its accumulated knowledge and experience in that industry is respected by the authority. In the R&D context, a firm which also participates in other related industries, may put a higher value on a patent and have more ability to win the patent, compared with its rival firms which do not participate in a related industry. Finally, an incumbent monopolist may value the patent for a new substitute product more highly than potential entrants [10, 516; 11, 195]. The reason is that if the incumbent wins the patent, then it earns monopoly profits; if one of the potential entrants wins the patent and enters into the market, then the entrant earns duopoly profits. As for abilities to win the patent, the incumbent may well have more ability than potential entrants.

   In section II, we set up the basic model and derive players' reaction functions. We show that the reaction functions are nonmonotonic.

   Section III assumes that the players choose their effort levels simultaneously and employs a Nash equilibrium as the solution concept. Let the strong (weak) player be the player who has more (less) ability. Let the Nash winner (Nash loser) be the player who has a probability of winning greater (less) than 1/2 at the Nash equilibrium. Section III shows that the weak player can be the Nash winner. The weak player tries harder than the strong player and becomes the Nash winner if his relative valuation of the prize is high enough to overcome his relative weakness in ability.

   Section IV examines how individual and total effort levels at the Nash equilibrium respond when valuation and ability asymmetries between the players change. Let the even contest be a contest in which both players have the same probability of winning at the Nash equilibrium. We find the following. As a player gets hungrier for the prize, (i) he always exerts more effort; (ii) his opponent exerts more effort until the even contest is reached but after the even contest she exerts less effort; and (iii) total effort level becomes larger until the even contest is reached but after the even contest total effort level may become larger or smaller. Starting from the even contest, as a player becomes stronger relative to his opponent, both players expend less effort. This implies that individual and total effort levels are maximized in the even contest.

   Section V considers a case of endogenous timing. We model a game in which the players first announce simultaneously and independently when they will expend their effort and then, based on this timing, they choose their effort levels. Defining the subgame-perfect winner as the player who has a probability of winning greater than 1/2 in the subgame-perfect equilibrium, we show that the Nash winner is also the subgame-perfect winner. We also show that in a lopsided contest endogenous timing leads the players to expend less effort, compared with the simultaneous-move Nash equilibrium.(1) In the even contest, however, endogenous timing does not make any difference with respect to effort levels, compared with the simultaneous-move Nash equilibrium.

   Section VI provides conclusions.

2. The Basic Model

   Consider a contest in which two risk-neutral players, 1 and 2, compete with each other to win a prize. Let [x.sub.1] and [x.sub.2] represent the two players' irreversible effort levels in units commensurate with the prize and let p represent the probability that player 1 wins. We assume that the probability-of-winning function for player 1 is

   p = [Sigma]h([x.sub.i])/([Sigma]h([x.sub.1]) + h([x.sub.2])), (1)

   where [Sigma] [is greater than] 0.(2) The parameter [Sigma] represents player 1's relative ability to player 2. A value of the ability parameter greater than unity implies that player 1 has more ability than player 2. In this case, if both players exert the same level of effort, player 1's probability of winning is greater than a half. A value of [Sigma] less than unity implies the opposite and [Sigma] = 1 implies that both players have equal ability. We assume that h (0) [is greater than or equal to] 0 and h ([x.sub.i]) is increasing in [x.sub.i]. For a mathematical reason, if h(0) = 0, we define p = 0. We then obtain [Delta]p/[Delta][x.sub.i] [is greater than] 0 for [x.sub.2] [is greater than] 0 and [Delta]p/[Delta][x.sub.2] [is less than] 0 for [x.sub.1] [is greater than] 0. Each player's probability of winning is increasing in his own effort and decreasing in his opponent's effort. We also assume that

   h[double prime]([x.sub.1])[([Sigma]h([x.sub.1]) + h([x.sub.2])) [is less than] 2[Sigma][(h[prime]([x.sub.1])).sup.2] (2)

   and

   h[double prime]([x.sub.2])([Sigma]h([x.sub.1]) + h([x.sub.2])) [is less than] 2[(h[prime]([x.sub.2])).sup.2], (3)

   where h[prime] and h[double prime] denote the first and second partial derivatives of the function h. From function (1) and inequality (2), we have [Mathematical Expression Omitted] for [x.sub.2] [is greater...