Two-player asymmetric contests with ratio-form contest success functions.

Baik, Kyung Hwan

I examine players' equilibrium effort levels in two-player asymmetric contests with ratio-form contest success functions. I first characterize the Nash equilibrium of the simultaneous-move game. I show that the equilibrium effort ratio is equal to the valuation ratio, and that the prize dissipation ratios for the players are the same. I also show that the prize dissipation ratio for each player is less than or equal to the minimum of the players' probabilities of winning at the Nash equilibrium and thus never exceeds a half. Then I examine how the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels, respond when the players' valuations for the prize or their abilities change. (JEL D72, C72)

I. INTRODUCTION

A contest is a situation in which players compete with one another by expending irreversible effort to win a prize. Typical examples are various types of rent-seeking contests: competition among firms to win a monopoly rent secured under government protection or by a government procurement contract, competition between domestic and foreign firms to obtain governmental trade policies favorable to them, competition among firms to acquire a rent generated by rights of ownership to an import quota, and competition among firms to capture rents created by governmental decisions to establish tariffs or other trade barriers. Other examples of contests include auctions, patent races, research and development (R&D) competition among firms, litigation, competition for jobs among job candidates, competition among candidates to win promotion to higher ranks, election campaigns between political candidates, and competition between local governments to invite business firms, government institutions, or government-owned corporations into their districts.

Naturally, due to their prevalence and importance in economies, such contests have been studied by many economists: Loury (1979), Lee and Wilde (1980), Tullock (1980), Rogerson (1982), Rosen (1986), Appelbaum and Katz (1987), Dixit (1987), Hillman and Riley (1989), Hirshleifer (1989), Ellingsen (1991), Nitzan (1991, 1994), Krishna and Morgan (1997), Che and Gale (1998), Hurley and Shogren (1998), and Konrad (2000), to name a few. In this vast literature on the theory of contests, one of the main issues is: How much effort do the players exert in pursuit of the prize? Indeed, it is of great interest because the players' effort levels determine the profitability of the players' engaging in the contest and, in some cases, they are revenues collected by the contest organizer or bribes given to government officials. Furthermore, they account for other important outcomes of the contest. For example, in a rent-seeking contest, efforts expended by the players are viewed as social costs due to rent-seeking activities, so that total effort level is a measure of economic efficiency. In an R&D contest, effort levels expended by the players--these are R&D expenditures of the firms--determine the expected date of invention.

This article also addresses the issue: How much effort do the players exert in pursuit of the prize? But it differs from previous research by dealing with contests with ratio-form contest success functions.1 Specifically, the novelty of this article is to consider two-player asymmetric contests in which each player's probability of winning is a function of the ratio of the two players' effort levels.

Two-player contests with ratio-form contest success functions or two-player contests that can be best modeled with ratio-form contest success functions are easily observed in the real world. Examples include various types of two-player rent-seeking contests, litigation between a plaintiff and a defendant, election campaigns between two parties or candidates, and R&D competition between two firms. Consider, for example, a rent-seeking contest in which two firms, potential monopolists, compete with each other to win a government monopoly franchise contract. The firms expend outlays to influence the government officials who have the authority to grant the monopoly franchise. I believe this contest can be best modeled with ratio-form contest success functions--that is, it is natural that each firm's probability of winning the monopoly franchise should be a function of the ratio of the two firms' "investments."

Another example is a contest in which a local government has a budget for building a bridge and two communities compete against each other to win that budget. (2) The local government uses a lottery-like winner-selecting mechanism in which two participating communities purchase as many lottery tickets as they want from the local government, and the winner is selected by drawing one ticket out of the tickets sold. The collected money goes to the general fund of government revenues. In this contest, the probability that a community wins the budget is equal to the number of tickets the community holds divided by the total number of tickets sold, which means that the communities face the simplest form of ratio-form contest success functions. (3)

This article focuses on the effects of asymmetries between the two players on the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels. The asymmetries between the players arise because of their different valuations for the prize or their different abilities to convert effort into probability of winning or both. For example, in the rent-seeking contest, the firms may have different valuations for the monopoly franchise, and their abilities to influence the government officials may differ. Especially, imagine the case in which the government periodically reassigns the monopoly franchise. In a given period, the previous franchise holder may value the franchise more highly and may have more abilities to influence the government officials than its rival, because it has invested resources and has experience in that particular industry and because it has been able to establish a relationship with the government officials. In the budget-seeking contest, the communities may have different valuations for winning the budget: The benefit of one community from building its own bridge may be larger than that of the other. On the other hand, the local government may use a biased winner-selecting mechanism by charging the communities different prices for the lottery ticket.

This article considers the case where the two players choose their effort levels simultaneously and independently. I first find and characterize the Nash equilibrium of this simultaneous-move game. I show that at the Nash equilibrium, the ratio of the players' effort levels (of player 1 to player 2)--the equilibrium effort ratio is equal to the corresponding valuation ratio, and that the prize dissipation ratio for one player the proportion of his or her equilibrium effort level to his or her valuation for the prize is equal to that for the other. These results imply that a high-valuation player exerts more effort than a low-valuation player, and that a high-valuation winner--a high-valuation player as the actual winner--earns a greater net payoff than a low-valuation winner. I also show that the prize dissipation ratio for each player is less than or equal to the minimum of the players' probabilities of winning at the Nash equilibrium and thus never exceeds a half: Each player's equilibrium effort level never exceeds half his or her valuation for the prize. This means that the equilibrium total effort level never exceeds the valuation of a high-valuation player or, more tightly, the average of the players' valuations. Based on this, I may argue that over-dissipation of the prize never occurs in this contest.

Then I examine how the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels respond when the players' valuations for the prize or their abilities change. Examining the effects of changing the players' valuations for the prize, I show the following. If the players' valuations for the prize increase proportionately, then their equilibrium effort levels increase proportionately, and furthermore they increase at the same rate as the players' valuations. Therefore, in this case the equilibrium effort ratio and the prize dissipation ratios remain unchanged. Next, if the valuation ratio changes, then the equilibrium effort ratio changes in the same direction as the valuation ratio does, but the prize dissipation ratios change in either the same or the opposite direction. Finally, I show that when just one player's valuation for the prize changes, his or her equilibrium effort level changes in the same direction as his or her valuation does.

Examining the effects of changing the players' abilities on the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels, I show the following. The equilibrium effort ratio depends solely on the players' valuations for the prize--it does not depend on the players' abilities. The prize dissipation ratio for each player, each player's equilibrium effort level, and the equilibrium total effort level are maximized in the contest in which both players have equal "composite" strength.

Baik (1994) and Nti (1997, 1999) study asymmetric contests. Balk (1994) examines the effects of asymmetries between two players on individual and total effort levels in a model with general logit-form contest success functions. As in this article, the asymmetries between the players arise because of their different valuations for the prize or their different abilities or both. Nti (1997) performs comparative statics with respect to the number of contestants, the discount rate, and the value of the prize, in a model with logit-form contest success functions--more specifically, in the gametheoretic R&D model developed by Loury (1979). He looks at their effects on individual expenditures and individual profits, and on total expenditures and total profits. Nti (1999) examines the effects of asymmetric valuations on individual effort levels and individual payoffs in a model with logit-form contest success functions that were first used in Tullock (1980).

The article proceeds as follows. Section II develops the model and sets up the simultaneous-move game. In Section III, I find and characterize the Nash equilibrium of the game. In Section IV, I examine how the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels respond when the players' valuations for the prize or their abilities change. Finally, Section V presents concluding remarks.

II. THE MODEL

Consider a contest in which two risk-neutral players, 1 and 2, compete with each other by expending irreversible effort to win a prize. Let [x.sub.1] and [x.sub.2] represent the effort levels expended by players 1 and 2, respectively. Effort levels are nonnegative and are measured in units commensurate with the prize. Let [p.sub.i] denote the probability that player i wins the prize. I assume that each player's probability of winning is a function of the ratio of the two players' effort levels. More specifically, I assume the following ratio-form contest success functions for the players: [p.sub.1] = f([theta]) and [p.sub.2] = 1 - f([theta]), where the function f has the properties specified in Assumption 1 below and [theta] = [x.sub.1] /[x.sub.2]. (4)

ASSUMPTION 1. I assume that 0 <f([theta]) < 1 for all [theta] in [R.sub.+], where [R.sub.+] denotes the set of all positive real numbers. I also assume that f'([theta]) > 0, f"([theta]) < 0, and 2f'([theta]) + 0 f"([theta]) > 0, for all [theta] in [R.sub.+], where f' and f" denote the first and second derivatives of the function f.

The first part of Assumption 1 simply says that for any pair of the players' effort levels, one for each player, each player's probability of winning is between 0 and 1. This implies that when both players expend positive effort levels, the player who expends a larger effort level does not win the prize with certainty--that is, he or she may lose the prize. (5) The first assumption of the second part--the assumption that f'([theta]) > 0--says that each player's probability of winning is increasing in his or her own effort level and is decreasing in his or her opponent's effort level: In terms of the symbols, [delta][p.sub.i]/[delta][x.sub.i]> 0 and [delta][p.sub.i]/[delta][x.sub.j]< 0 for all [x.sub.i] >0 and for all [x.sub.2]>0. The other two assumptions of the second part concern the marginal effect of increasing a player's effort level on his or her own probability of winning. The assumption that f"([theta])< 0 says that the marginal effect of player 1's effort level on his or her probability of winning decreases as his or her effort level increases: [delta][p.sub.1]/[delta][x.sup.2.sub.1]<0 for all [x.sub.1] >0 and for all [x.sub.2]>0, The last assumption--the assumption that 2f'([theta]) + [theta]f"([theta]) > 0--says indirectly that the marginal effect of player 2's effort level on his or her probability of winning decreases as his or her effort level increases: [[delta].sup.2][p.sub.2]/ [delta][x.sup.2.sub.2] <0 for all [x.sub.1] >0 and for all [x.sub.]2>0. In short, the second part of Assumption 1 assumes that given the opponent's effort level, each player's probability of winning is increasing in his or her own effort level at a decreasing rate. (6) As it will be clear in Section III, this ensures that the second-order conditions for maximizing the players' expected payoffs are satisfied.

One example of the contest success functions which satisfy Assumption 1 is [p.sub.1] = [sigma][x.sub.1]/ ([sigma][x.sub.1] + [x.sub.2]) and [p.sub.2] = [x.sub.2]/ ([sigma][x.sub.1] + [x.sub.2]), where [sigma] is a positive constant. They are extensively used in the contest literature. (7) Another example is [p.sub.1] = [[alpha].sub.1][x.sup.r.sub.1]/ ([[alpha].sub.1][x.sup.r.sub.1] + [[alpha].sub.1][x.sup.r.sub.2]) and [p.sub.2] = [[alpha].sub.2][x.sup.r.sub.2]/ ([[alpha].sub.1][x.sup.r.sub.1] + [[alpha].sub.2][x.sup.r.sub.2]), where [[alpha].sub.1] > 0 and [[alpha].sub.2] > 0, which are used in Tullock (1980), Hirshleifer (1989), Skaperdas (1996), Clark and Riis (1998), and Nti (1999). The pair of contest success functions used in Kooreman and Schoonbeek (1997), [p.sub.1] = [(a[x.sub.1]).sup.r]/{[(a[x.sub.1]).sup.r] + [x.sup.r.sub.2]} and [p.sub.2] = [x.sup.r.sub.2]/{[(a[x.sub.1]).sup.r] + [x.sup.r.sub.2]}, where a>0, provides another example. (8)

The players' valuations for the prize may differ. Player 1 values the prize at [v.sub.1] and player 2 values it at [v.sub.2]. Each player's valuation for the prize is positive and publicly known. Let [[pi].sub.i] represent the expected payoff for player i. Then the payoff function for player 1 is

[[pi].sub.1] = [v.sub.1]f([theta]) - [x.sub.1],

and that for player 2 is

[[pi].sub.2] = [v.sub.2](1 - f([theta])) - [x.sub.2].

I assume that the players choose their effort levels simultaneously and independently. I also assume that all of the above is common knowledge between the players. I employ Nash equilibrium as the solution concept.

III. NASH EQUILIBRIUM

To obtain a Nash equilibrium of the game, I begin by considering the maximization problems: Maximize player i's expected payoff over his or her effort level, given player j's positive effort level. The first-order condition for maximizing [[pi].sub.1] (without the nonnegativity constraint) is

(1) [delta][pi].sub.1]/[delta][pi].sub.1] = [v.sub.1]f'([x.sub.1]/ [x.sub.2])/[x.sub.2] - 1 = 0,

and that for maximizing [[pi].sub.2] is

(2) [delta][pi].sub.2]/[delta][pi].sub.2] = [v.sub.2]f'([x.sub.1]/[x.sub.2])/[x.sup.2.sub.2] - 1 = 0,

Due to Assumption 1, player i's payoff function is strictly concave in his or her effort level, and thus the second-order condition for maximizing [[pi].sub.i] is satisfied.

Player i's reaction function shows his or her best response to every possible effort level that player j might choose. Given player j's effort level, player i's best response is defined as the effort level that maximizes his or her expected payoff. (9) Thus player 1's reaction function is derived from condition (1). It is then implicitly defined by

(3) [v.sub.1]f([x.sub.1]/[x.sub.2]) = [x.sub.2].

The implicit form of player 2's reaction function is

(4) [v.sub.2.][x.sub.1]f([x.sub.1]/[x.sub.2]) = [x.sup.2.sup.2]

which comes from condition (2).

A Nash equilibrium is a pair of effort levels, one for each player, at which each player's effort level is the best response to his or her opponent's. Thus it satisfies the players' reaction functions--equations (3) and (4)--simultaneously. (10) Let ([x*.sub.1], [x*.sub.2]) denote the interior Nash equilibrium. Then we have

(5) [v.sub.1] f'([x*.sub.1]/[x*.sub.2]) = [x*.sub.2]

and

(6) [v.sub.2][x*.sub.1]f'([x*.sub.1]/[x*.sub.2]) = [([x*.sub.2]).sup.2], which yield

(7) [x*.sub.1]/[x*.sub.2] = [v.sub.1]/[v.sub.2]

or, equivalently,

(8) [x*.sub.1]/[v.sub.1] = [[x*.sub.2]/[v.sub.2].

Expression (7) says that at the Nash equilibrium, the ratio of the players' effort levels (of player 1 to player 2) is equal to the ratio of the players' valuations (of player 1 to player 2). In expression (8), [x*.sub.i]/[v.sub.i] is the proportion of player i's equilibrium effort level--which is "dissipated" in pursuit of the prize--to his or her valuation for the prize. I call it the prize dissipation ratio for player i. From expressions (5), (6), (7), and (8), I obtain Proposition 1.

PROPOSITION 1. (a) The equilibrium effort ratio is equal to the valuation ratio. (b) The prize dissipation ratios for the players are the same. (11) (c) The equilibrium effort ratio does not depend on the function f (d) The prize dissipation ratio for each player does depend on the function f.

Parts (a) and (b) come mainly from two things. One is the assumption that the two players have the ratio-form contest success functions. The other is the solution concept of Nash equilibrium. Because each player's probability of winning is a function of the ratio of the two players' effort levels, the effort ratio should be equal to the valuation ratio to satisfy the mutual-best-responses property of Nash equilibrium. Surprisingly, the equilibrium effort ratio is equal to the valuation ratio--equivalently, the prize dissipation ratios for the players are the same--for any specific form of the function f.

Parts (a) and (b) each say that player i expends [v.sub.i]/[v.sub.j] times as much effort as his or her opponent, player j. They also say that the proportion of player i's equilibrium effort level to his or her valuation for the prize is equal to the proportion of player j's effort level to his or her valuation. Each of these implies that the players choose the same effort level if they have the same valuation for the prize; a high-valuation player exerts more effort than a low-valuation player. These make intuitive sense and are consistent with what people observe in real-world contests. Indeed, it is commonly observed that a "hungrier" player tries harder than his or her opponent. Part (b) implies that a high-valuation winner--a high-valuation player as the actual winners--earns a greater net payoff than a low-valuation winner.

Parts (c) and (d) are clearly understood with the expressions, [x*.sub.1] = ([v.sup.2.sub.1]/ [v.sub.2])f'([v.sub.1]/[v.sub.2]) and [x*.sub.2] = [v.sub.1]f'([v.sub.1]/[v.sub.2]), which are obtained from expressions (5), (6), and (7). The two expressions show that each player's equilibrium effort level depends on the players' valuations, [v.sub.1] and [v.sub.2], and the function f. Thus the prize dissipation ratio for each player should depend on the function f On the other hand, the equilibrium effort ratio does not depend on the function f--it depends solely on the players' valuations--because both expressions have the common factor f'([v.sub.1]/[v.sub.2]) and do not have any other factor related to the function f.

Consider the case where the function f takes a form, f([theta]) = [sigma][theta]/([sigma][theta] + 1) where [sigma]> 0, so that the pair of contest success functions is [p.sub.1] = [sigma][x.sub.1]/([sigma][x.sub.1] + [x.sub.2]) and [p.sub.2] = [x.sub.2]/([sigma][x.sub.1] + [x.sub.2]). In the budget-seeking contest introduced in Section I, the parameter [sigma] may reflect the difference between the ticket prices charged to the communities: Community 1 pays the local government [x.sub.1] to purchase [sigma][x.sub.1] lottery tickets, whereas community 2 pays [x.sub.2] for [x.sub.2] lottery tickets. In the rent-seeking contest mentioned in Section I, the parameter [sigma] may indicate how effective the firms' outlays are in influencing the government officials who have the authority to grant the monopoly franchise. Indeed, looking at these contest success functions, one may say that firm 1's outlay is [sigma] times as effective as firm 2's. In general, the parameter [sigma] represents player 1's abilities to convert effort into probability of winning relative to player 2's. A value of [sigma] greater than unity implies that player 1 has more abilities than player 2. In this case, if both players expend 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 abilities. Such an ability parameter is used in Tullock (1980), Rogerson (1982), Rosen (1986), Leininger (1993), and Baik (1994, 1999).

Now that parts (a) and (b) hold for any specific form of the function f, they hold for the specific form given above regardless of the value of the ability parameter [sigma]. Furthermore, according to part (c), I obtain the same equilibrium effort ratio regardless of the value of [sigma]--that is, the equilibrium effort ratio remains constant when the players' abilities change. Note, however, that different values of [sigma] lead to different pairs of the players' equilibrium effort levels and thus lead to different pairs of the prize dissipation ratios. To clarify these points,consider the rent-seeking contest in which two firms compete for themonopoly franchise. Suppose that one of the firms is replaced by another firm with different lobbying abilities. In this case, the equilibrium effort ratio is equal to the valuation ratio--equivalently, the prize dissipation ratios for the firms are the same--both before and after the change. In addition, the equilibrium effort ratio remains constant. But the equilibrium effort level and the prize dissipation ratio for each firm, obtained after the change, are different from those obtained before the change, respectively.

Proposition 2 describes another interesting finding of this article.

PROPOSITION 2. The prize dissipation ratio for each player is less than or equal to the minimum of the players' probabilities of winning at the Nash equilibrium and therefore never exceeds a half.

Proof The equilibrium expected payoff for player 1 is [[pi]*.sub.1] = [v.sub.1]f([x*.sub.1]/ [x*.sub.2]) - [x*.sub.1], and that for player 2 is [[pi]*.sub.1] = [v.sub.2](1 - f([x*.sub.1]//[x*.sub.2])) - [x*.sub.2].

Because each player's equilibrium expected payoff is nonnegative, we obtain [x*.sub.1]/[v.sub.1] [less than or equal to] f([x*.sub.1]/[x*.sub.2]) and [x*.sub.2]/[v.sub.2] [less than or greater than to] 1 - f([x*.sub.1]/[x*.sub.2]) from the two expressions. The former says that the prize dissipation ratio for player 1 is less than or equal to player 1's probability of winning at the Nash equilibrium. The latter says that the prize dissipation ratio for player 2 is less than or equal to player 2's probability of winning at the Nash equilibrium. Because the prize dissipation ratios for the players are the same, the common prize dissipation ratio must be less than or equal to the minimum of their probabilities of winning at the Nash equilibrium: [x*.sub.i]/[v.sub.i] [less than or equal to] min{f([x*.sub.1][x*.sub.2]), 1 - f([x*.sub.1][x*.sub.2])}. This implies that the common prize dissipation ratio never exceeds a half.

Given his or her opponent's effort level, each player chooses his or her best response--the effort level that maximizes his or her own expected payoff. Because each player's probability of winning is a function of the ratio of the two players' effort levels, the state of mutual best responses or, equivalently, the Nash equilibrium implies that the prize dissipation ratios for the players are the same. The Nash equilibrium also implies that each player's equilibrium expected payoff is nonnegative. Utilizing these two facts, I obtain Proposition 2.

As in Baik (1994), I define a lopsided contest as a contest in which the probability of winning for one of the players is greater than one-half at the Nash equilibrium. As in Baik (1994), let the Nash winner be the player who has a probability of winning greater than a half at the Nash equilibrium and the Nash loser the player whose probability of winning at the Nash equilibrium is less than a half. Proposition 2 then says that in a lopsided contest, the prize dissipation ratio for each player is less than or equal to the Nash loser's probability of winning.

Proposition 2 says that each player's equilibrium effort level never exceeds half his or her valuation for the prize. This implies that the equilibrium total effort level never exceeds the valuation of a high-valuation player or, more tightly, the average of the players' valuations. In this sense, I might say that over-dissipation of the prize never occurs in this contest. However, the equilibrium total effort level may exceed the valuation of a low-valuation player.

In the literature on rent seeking, efforts expended on rent-seeking activities are regarded as social costs that create economic inefficiency. Therefore, examining the extent of rent dissipation is one of the main issues in the literature. Proposition 2 establishes that less than complete dissipation of the contested rent almost always occurs in two-player asymmetric rent-seeking contests with ratio-form contest success functions. This underdissipation-of-rents result is also obtained in, for example, Tullock (1980), Hillman and Riley (1989), and Balk and Lee (2001).

IV. COMPARATIVE STATICS

As shown in Section III, each player's equilibrium effort level depends on the players' valuations for the prize, [v.sub.1] and [v.sub.2], and the function f. It is interesting, then, to examine how the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels respond when the players' valuations change or when the function f takes a different specific form. This is what I will cover in this section.

Valuations

I begin by examining the effects of changing the players' valuations for the prize on the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels. Proposition 3 is immediate from expression (7) and the fact that [x*.sub.1]/[v.sub.1] = [x*.sub.2]/[v.sub.2] = ([v.sub.1]/[v.sub.2]) f'([v.sub.1]/[v.sub.12).

PROPOSITION 3. (a) Suppose that the players" valuations for the prize change proportionately and thus the valuation ratio, [v.sub.1]/[v.sub.2], remains unchanged. Then the equilibrium effort ratio, [x*.sub.1]/[x*.sub.2], and the prize dissipation ratios, [x*.sub.1]/[v.sub.1] and [x*.sub.2]/[v.sub.2], remain unchanged; each player's equilibrium effort level changes in the same direction as the players' valuations do. (b) If the valuation ratio changes, then the equilibrium effort ratio changes in the same direction as the valuation ratio does, but the prize dissipation ratios change in either the same or the opposite direction.

Part (a) implies that if the players' valuations for the prize increase at the same rate, then their equilibrium effort levels increase at the same rate, and furthermore they increase at the very rate that the players' valuations increase. Consider, for example, the rent-seeking contest in which two firms compete for the monopoly franchise. If the firms' valuations for the monopoly franchise double, then their equilibrium effort levels double, so that the equilibrium effort ratio and the prize dissipation ratios remain constant.

Imagine the players' reaction curves drawn in the [x.sub.1][x.sub.2] space. Their intersection is the Nash equilibrium. Part (a) says that when the players' valuations rise or fall at the same rate, the old and new Nash equilibria are located on the same straight line that emanates from the origin, and the Nash equilibrium associated with higher valuations lies farther from the origin.

Part (a) implies that the equilibrium total effort level changes in the same direction as the players' valuations do. When the players' valuations for the prize increase proportionately, player i's best response to player j's positive effort level increases, so that player i's reaction curve shifts outward. (12) This leads to higher effort levels at the Nash equilibrium associated with higher valuations. To put it in familiar words, when both players' desire for the prize increases, so does the competition between them that is, the players exert more effort to win the prize, which is now more valuable than before.

Part (b) says that when the valuation ratio increases, the equilibrium effort ratio increases, but the prize dissipation ratios either increase or decrease. Geometrically speaking, the graph of the relationship between the valuation ratio and the equilibrium effort ratio is the 45[degrees] line in the corresponding two-dimensional space. Figure 1 illustrates how the prize dissipation ratios as well as the equilibrium effort ratio change as the valuation ratio increases, in the case where f([theta]) = [theta]/(1 + [theta]). As the valuation ratio, [v.sub.1]/[v.sub.2], increases from an arbitrarily small positive number, the prize dissipation ratios first increase, reach the maximum, and then decrease. The inflection point occurs at point A. The prize dissipation ratios are maximized when both players have the same valuation for the prize--that is, when [v.sub.1]/ [v.sub.2]= 1. As in Baik (1994), 1 define the even contest as a contest in which both players have the same probability of winning at the Nash equilibrium--or, equivalently, as a contest in which the probability of winning for each player is equal to one half at the Nash equilibrium. In the case where f ([theta]) = [theta]/(1 + [theta]), it is straightforward to see that the even contest occurs when both players have the same valuation for the prize. Then it follows that the prize dissipation ratios are maximized in the even contest, not in a lopsided contest. This confirms the idea that, as contestants' "strength" gets closer, competition gets fiercer. Note, however, that the equilibrium effort ratio--represented by the 45[degrees] line--keeps increasing. It happens because, as the valuation ratio increases, both players' equilibrium effort levels do not always move in the same direction. Consider, for example, the case in which player 1's valuation for the prize increases while player 2's remains constant. In this case, player 1's equilibrium effort level is increasing in the valuation ratio, but player 2's is increasing in the valuation ratio until [v.sub.1]/[v.sub.2] = 1, and is decreasing thereafter.

[FIGURE 1 OMITTED]

I now look at the effects on the players' equilibrium effort levels in the case of part (b)--where the valuation ratio changes. Using the expressions [x*.sub.1] = ([v.sup.2.sub.1]/ [v.sub.2])f'([v.sub.1]/[v.sub.2]) and [x*.sub.2] = [v.sub.1]f'([v.sub.1]/[v.sub.2]), it is straightforward to obtain the following: If player 1's valuation for the prize increases (decreases) but the valuation ratio decreases (increases), then player 2's equilibrium effort level increases (decreases). This says that if player 2's valuation for the prize increases at a higher rate than player 1's, then player 2's equilibrium effort level increases. It also says that if player 2's valuation decreases at a higher rate than player 1's, then player 2's equilibrium effort level decreases. In other words, player 2 exerts more (less) effort when his or her valuation relative to player 1's increases (decreases). In both cases, the effects on player 1's equilibrium effort level and the equilibrium total effort level are ambiguous. Finally, when replacing player 1 in the conditional statement with player 2, I cannot obtain symmetric results. This asymmetry arises because each player's probability of winning is a function of the ratio [x.sub.1]l[x.sub.2].

Another result that I obtain is as follows: If just one player's valuation for the prize changes, his or her equilibrium effort level changes in the same direction as his or her valuation does. As in the preceding case, the effects on the other player's equilibrium effort level and the equilibrium total effort level are ambiguous. Baik (1994) conducts comparative statics with respect to the valuation parameter in contests with logit-form contest success functions. He shows that, as one player's valuation for the prize increases from an arbitrarily small positive number (while the other player's remains unchanged), the other player's effort level first increases, reaches the maximum in the even contest, and decreases thereafter. He also shows that the equilibrium total effort level may not be maximized in the even contest.

Ability Parameter

I examine how the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels respond when replacing one specific form of the function f with another. I do this using f([theta]) = [sigma][sigma]/([sigma][sigma] + 1), where [sigma] > 0. As explained in Section III, the parameter o represents player 1's abilities to convert effort into probability of winning relative to player 2's. Note that different values of the ability parameter [sigma] yield different specific forms of the function f. Therefore, in this case, examining how those outcomes respond when replacing one specific form of the function f with another amounts to examining that when changing the value of the ability parameter o from one to another. Indeed, what I do in this subsection is to examine the effects of changing the value of the ability parameter [sigma] on the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels.

PROPOSITION 4. (a) The equilibrium effort ratio, [x*.sub.1]/[x*.sub.2], remains unchanged, regardless of the value of the ability parameter [sigma]. (b) The prize dissipation ratios, [x*.sub.1]/[v.sub.1] and [x*.sub.2]/[v.sub.2], are maximized in the even contest, which occurs at [sigma] = [v.sub.2]/ [v.sub.1], and decrease as the ability parameter [sigma] increases (decreases)from [v.sub.2]/[v.sub.1]. (c) Each player's equilibrium effort level and the equilibrium total effort level are also maximized at [sigma] = [v.sub.1]/[v.sub.1]--that is, in the even contest and they decrease as the ability parameter [sigma] increases (decreases) from [v.sub.2]/[v.sub.1].

Using f([theta]) = [sigma][theta]/([sigma][theta] + 1)--in this case, the players' contest success functions are [p.sub.1] = [sigma][x.sub.1]/ ([sigma][x.sub.1] + [x.sub.2]) and [p.sub.2] = [x.sub.2]([sigma][x.sub.1]+ [x.sub.2])--we obtain [x*.sub.1] = [sigma][v.sup.2.sub.1][v.sub.2]/[sigma] [v.sub.1] + [v.sub.2], [x*.sub.2] = [sigma][v.sub.1][v.sup.2.sub.2]/ [sigma][v.sub.1] + [v.sub.2]).sup.2] and [x*.sub.1] + [x*.sub.2] = [sigma][v.sub.1][v.sub.2]([v.sub.1] + [v.sub.2])/([[sigma][v.sub.1] + [v.sub.2]).sup.2]. Given these expressions, it is straightforward to obtain Proposition 4. I can also obtain part (a) directly from part (c) of Proposition 1. To understand part (b) properly, recall from Proposition 1 that the prize dissipation ratios for the players are the same whatever specific form of the function f I use, and recall that the even contest is defined as a contest in which both players have the same probability of winning at the Nash equilibrium. Baik (1994) shows that in this special case, the even contest occurs when the valuation ratio, [v.sub.1]/[v.sub.2], times the ability parameter [sigma] equals unity. This means that, given [v.sub.1] and [v.sub.2], the even contest occurs at ([sigma] = [v.sub.2]/[v.sub.1].

Part (a) implies that the equilibrium effort ratio depends solely on the players' valuations for the prize. Each player's equilibrium effort level depends not only on the players' valuations for the prize, but also on the ability parameter [sigma]--that is, their abilities. However, because the players' equilibrium effort levels are "equally" influenced by the players' abilities, their ratio specifically, the equilibrium effort ratio--does not depend on the players' abilities.

Parts (b) and (c) say that the prize dissipation ratio for each player, each player's equilibrium effort level, and the equilibrium total effort level are maximized in the even contest. An intuitive explanation for this follows. Here the winner is determined by the players' valuations for the prize and their abilities together--namely, by their composite strength. I then believe that, when both players have equal composite strength, competition for the prize is the fiercest, and therefore the players' equilibrium effort levels are maximized. Now note that the even contest is just the one in which both players have equal composite strength: The valuation ratio, [v.sub.1]/[v.sub.2], times the ability parameter [sigma] equals unity. Parts (b) and (c) also say that starting from the even contest, as player i becomes stronger relative to player j, both players exert less effort and thus the equilibrium total effort level becomes smaller. (13) An intuitive, interesting explanation for this result is that player i--the player who becomes stronger relative to the opponent--eases up facing an opponent weaker than before, whereas player j gives up, facing an opponent stronger than before.

I can also obtain Proposition 4 by using Figure 1. In the case where f([theta]) = [sigma][theta]/ ([sigma][theta] + 1), the payoff function for player 1 is [[pi].sub.1] = {[sigma][x.sub.1]/([sigma][x.sub.1] + [x.sub.2])} - [x.sub.1], and that for player 2 is [[pi].sub.2] = [v.sub.2]{[x.sub.2]/ ([sigma][x.sub.1] + [x.sub.2])} - [x.sub.2]. Because player 1's payoff function can be rewritten as [[pi].sub.1] = [[sigma][v.sub.1]/([sigma][x.sub.1] + [x.sub.2])} - [sigma][x.sub.1]/[sigma] maximizing player 1's expected payoff over his or her effort level is equivalent to maximizing [[phi].sub.1] = [sigma][v.sub.1]{[sigma][x.sub.1]/ ([sigma][x.sub.1] + [x.sub.2])} - [sigma][x.sub.1] over his or her effort level. This means that the players' equilibrium effort levels in the original contest are the same as those obtained from a new contest in which the payoff function for player 1 is [[phi].sub.1] = [sigma][v.sub.1]{[sigma].sub.1]/([sigma][x.sub.1] + x2)} - [sigma][x.sub.1], and that for player 2 is [[pi].sub.2] = [v.sub.2] {[x.sub.2]/([sigma][x.sub.1] + [x.sub.2])} - [x.sub.2]. Let [w.sub.1] = [sigma][v.sub.1] and let [y.sub.1] [equivalent to] [sigma][x.sub.1]. Then I have [[phi].sub.1] = [w.sub.1]{[y.sub.1]/([y.sub.1] + [x.sub.2])} - [y.sub.1] and [[pi].sub.2] = [v.sub.2]{[x.sub.2]/([y.sub.1] + [x.sub.2])} - [x.sub.2]. This is exactly the contest underlying Figure 1. All I need to change in the figure is to replace [v.sub.1]/[v.sub.2] and [x*.sub.1]/[x*.sub.2] with [w.sub.1]/[v.sub.2] and [y*.sub.1]/[x*.sub.2], respectively. The prize dissipation ratio for each player in the new contest is identically equal to that in the original contest: [y*.sub.1]/[w.sub.1] [equivalent to] [x*.sub.1]][v.sub.1] and [x*.sub.2]/[v.sub.2] [equivalent to] [x*.sub.2]/[v.sub.2]. Now note that a change in the valuation ratio in the new contest comes solely from a change in the value of the ability parameter [sigma] in the original contest. Therefore, I can obtain Proposition 4 from the results illustrated in Figure 1. For example, the result in part (b) that the prize dissipation ratios are maximized when [sigma = [v.sub.2]/[v.sub.1] can be obtained from the result--illustrated in the figure--that they are maximized when [w.sub.1]/[v.sub.2] = 1.

V. CONCLUSIONS

I have considered a contest in which two asymmetric players compete with each other by expending irreversible effort to win a prize, and each player's probability of winning is a function of the ratio of the two players' effort levels. Asymmetries between the players arise because of the different valuations for the prize or their different abilities to convert effort into probability of winning or both. First, in Section III, I characterized the Nash equilibrium of the simultaneous-move game and examined the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels. Then, in Section IV, I examined how the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels respond when the players' valuations change or when the function f takes a different specific form.

In Section III, I found the following. The equilibrium effort ratio is equal to the valuation ratio, and that the prize dissipation ratios for the players are the same. These results hold for any specific form of the function f. The equilibrium effort ratio does not depend on the function f, whereas the prize dissipation ratio for each player does. Finally, the prize dissipation ratio for each player is less than or equal to the minimum of the players' probabilities of winning at the Nash equilibrium and thus never exceeds a half.

In Section IV, when the players' valuations for the prize change, I found the following. If the players' valuations for the prize change proportionately and thus the valuation ratio remains unchanged, then the equilibrium effort ratio and the prize dissipation ratios remain unchanged, but each player's equilibrium effort level changes in the same direction as the players' valuations do. If the valuation ratio changes, then the equilibrium effort ratio changes in the same direction as the valuation ratio does, but the prize dissipation ratios change in either the same or the opposite direction. Examining the effects of changing the players' abilities on the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels, I found the following. The equilibrium effort ratio depends solely on the players' valuations for the prize it does not depend on the players' abilities. The prize dissipation ratio for each player, each player's equilibrium effort level, and the equilibrium total effort

level are maximized in the even contest.

Can I extend this two-player model to an n-player one? What do ratio-form contest success functions look like in n-player asymmetric contests? Are the prize dissipation ratios for the players the same in n-player asymmetric contests? Does the prize dissipation ratio for each player never exceed 1/n in n-player asymmetric contests? These are interesting questions. I leave them for future research.

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(1.) The contest success function for a player is the rule that describes the relationship between effort levels of the players and his or her probability of winning. Several forms of contest success functions have been used in the literature on the theory of contests. See Baik (1998, 686-87) for detailed explanations of different forms of contest success functions.

(2.) This example may be rather contrived, but is useful in understanding the model and results of this article.

(3.) Alternatively, the government can design and use a different winner-selecting mechanism in which the communities face more general ratio-form contest success functions.

(4.) These contest success functions hold for [x.sub.2] >0. When [x.sub.j] = 0, one may assume that [p.sub.i] = 1 and [p.sub.j] = 0 for [x.sub.i]>0, and [p.sub.1] [p.sub.2] = 1/2 for [x.sub.i] = 0. (Throughout the article, when I use i and j at the same time, I mean that i [not equal to] j.) The ratio-form contest success functions are homogeneous of degree zero in the players' effort levels. Hirshleifer (1989) coined the term ratio-form contest success function. He called the following pair of logit-form contest success functions--which was first used in Tullock (1980)--the ratio-form contest success functions: [p.sub.1] = [x.sup.r.sub.1]/([x.sup.r.sub.1] + [x.sup.r.sub.2]) and [p.sub.2] = [x.sup.r.sub.2]/([x.sup.r.sub.1] + [x.sup.r.sub.2]), where the exponent r is a positive constant. Hirshleifer (1989) also coined the term difference-form contest success function. His difference-form contest success functions are just the logistic-form ones: [p.sub.1]([x.sub.1], [x.sub.2]) = 1/[1 + exp{k([x.sub.j] - [x.sub.i])}] for i,j = 1,2 where k is a positive constant. Note that given these difference-form contest success functions, player i's probability of winning is not zero even if he or she expends zero effort. Hirshleifer (1989) shows that unlike in a model with the ratio-form contest success functions, one-sided submission or two-sided peace can occur in equilibrium in a model with the difference-form contest success functions. He argues that the ratio-form contest success functions are applicable to, for example, military struggles between two nations in which there are "idealized" conditions, such as an undifferentiated battlefield and full information, while the difference-form contest success functions are applicable to, for example, military struggles with sanctuaries, refuges, and/or imperfect information.

(5.) Hillman and Riley (1989) classify contests according to whether the player who expends the largest effort level wins the prize with certainty. Their classification divides contests into perfectly discriminating contests and imperfectly discriminating contests. The player who expends the largest effort level wins the prize with certainty in a perfectly discriminating contest, but may lose it in an imperfectly discriminating contest. According to their classification, contests under consideration in this article--that is, contests with ratio-form contest success functions--are imperfectly discriminating ones. Other examples of imperfectly discriminating contests include contests with logit-form, probit-form, or difference-form contest success functions. On the other hand, first-price all-pay and second-price all-pay contests belong to perfectly discriminating contests (see Baik, 1998).

(6.) There may exist some real-world situations in which a player's probability of winning is increasing in his or her own effort level at an increasing rate for a while and then increasing at a decreasing rate. Even in such situations, however, an equilibrium occurs only where each player's probability of winning is increasing at a decreasing rate.

(7.) See, for example, Tullock (1980), Rogerson (1982), Appelbaum and Katz (1987), Hillman and Riley (1989), Ellingsen (1991), Nitzan (1991), Leininger (1993), Baik (1994, 1999), and Baik and Lee (2001).

(8.) In the last two examples, the exponent r is a positive constant but, to satisfy Assumption 1, it must be "small"--roughly, it must be less than 2. Other specific forms of the function f are f([theta]) = 1n(1 + [theta])/{1 + 1n(1 + [theta])} and f([theta]) [{[theta]/(1 + [theta])}.sup.r], where 0<r [less than or equal to] 1.

(9.) If I adopt the assumption mentioned in footnote 4, then, when player j expends zero effort, player i's best response is to expend infinitesimally small effort.

(10.) Let J([x.sub.1], [x.sub.2]) be the Jacobian of equations (3) and

(4). Then

[MATHEMATICAL CONVERSION NOT REPRODUCIBLE IN ASCII.]

If J([x.sub.1], [x.sub.2]) is negative quasi-definite for all strategy profiles, then the game has a unique Nash equilibrium (see Friedman, 1990, 86). (Note that this uniqueness theorem gives sufficient, but not necessary, conditions for uniqueness.) It is straightforward to show that J([x.sub.1], [x.sub.2]) is negative quasi-definite for all strategy profiles if [1 + 2[square root of [v.sub.1] [v.sub.2]/([v.sub.1] + [v.sub.2])] f'([theta]) + [theta]f"([theta]) >0 and [1 - 2[square root of [v.sub.1] [v.sub.2]/([v.sub.1] + [v.sub.2])] f'([theta]) + [theta]f ([theta]) <0 for all [theta] in [R.sub.+]. Szidarovszky and Okuguchi (1997) prove the existence and uniqueness of a pure-strategy Nash equilibrium in rent-seeking contests with logit-form contest success functions.

(11.) Nti (1999) also obtains this result in a rent-seeking contest with Tullock's pair of contest success functions.

(12.) Player i's best response to player j's "sufficiently high" effort level may be to exert zero effort and may remain unchanged at the zero effort levels, when the players' valuations for the prize change.

(13.) An increase in the ability parameter [sigma] means that player 1 becomes stronger relative to player 2. Naturally, it also means that player 2 becomes weaker relative to player 1.

KYUNG HWAN BAIK, I am grateful to Dennis W. Jansen, In-Gyu Kim, Alain Marciano, Jay Shogren, Cheng-Chen Yang, and two anonymous referees for their helpful comments and suggestions. An earlier version of this article was presented at the 76th Annual Conference of the Western Economic Association International, San Francisco, CA, July 2001. This work was supported by the Brain Korea 21 Project in 2001.

Baik: Professor, Department of Economics, Sungkyunkwan University, Seoul 110-745, South Korea. Phone 82-2-760-0432, Fax 82-2-744-5717, E-mail khbaik@skku.edu