...Rent-seeking, Homogeneous contest success functions, Existence of equilibrium.

JEL Classification Numbers: D72.

1 Introduction

A striking feature of the large and diverse rent-seeking literature is the frequent use of Tullock's (1980) contest success function to specify the probability that given player wins. (3) As acknowledged by Nitzan (1994), Tullock's model seems to be widely used because it is analytically tractable. Subsequently, Skaperdas (1996) showed that Tullock's contest success function is characterized by five axioms.

In this note, we focus on solving for equilibria. We retain the analytical tractability of Tullock's model while broadening the class of contest success functions beyond those that satisfy all of Skaperdas' axioms. Specifically, we consider contest success functions that are homogeneous of degree zero. This property is intuitively appealing for rent-seeking contests. The contest winner is determined by relative efforts. For example, if all players double their effort, then the probabilities of winning the contest are unchanged--the increase in effort is completely wasted.

For a symmetric n-person rent-seeking contest we provide conditions under which a unique symmetric pure-strategy Nash equilibrium exists and determine the equilibrium effort of the players. These results are determined, at least in part, by a partial derivative of a contest success function evaluated at a particular point. To determine the equilibrium effort corresponding to any specific contest success function, one simply evaluates this derivative.

2 Symmetric n-person contests

2.1 Preliminaries

Let N [equivalent to] {1,..., n} denote the index set of the n players. Player i expends effort [x.sub.i] to win the contest. Given a vector of nonnegative effort levels x = ([x.sub.1], [x.sub.2],..., [x.sub.n]), the probability that player i wins the prize is [[pi].sub.i](x). For simplicity, we assume the functions [[pi].sub.1], [[pi].sub.2],..., [[pi].sub.n] are differentiable. We refer to ([[pi].sub.1], [[pi].sub.2],..., [[pi].sub.n]) as the contest success function. All players have a common value, V, for the contest prize. The players are risk-neutral. Given the effort vector x, player i's expected payoff is [U.sub.i](x) = [[pi].sub.i](x)V - [x.sub.i], [for all]i [member of] N.

2.2 Contest success functions

Tullock's (1980) contest success function is

[[pi].sub.i](x) = [x.sub.i.sup.r]/[[[summation].sub.j[member of]N] [x.sub.j.sup.r]] [for all]i [member of] N, (1)

where r > 0. Skaperdas (1996) shows that Tullock's function is completely characterized by five axioms.

Axiom 1 (Probabilities) [[summation].sub.j[member of]N] [[pi].sub.j](x) = 1 and [[pi].sub.i](x) [greater than or equal to] [for all]i [member of] N and [for all]x; if [x.sub.i] > then [[pi].sub.i](x) > 0.

Axiom 2 (Monotonicity) [for all] i [member of] N, [[pi].sub.i](x) is increasing in [x.sub.i] and decreasing in [x.sub.j], [for all]j [not equal to] i.

Axiom 3 (Anonymity) For any permutation p : N [right arrow] N, we have

[[pi].sub.i](x) = [[pi].sub.p(i)]([x.sub.p(1)], [x.sub.p(2)],..., [x.sub.p(N)]) [for all]i [member of] N.

Axioms 1 and 2 state that the contest success function has the properties of a probability function, and increases in a player's effort increase that player's chance of winning but decrease...

NOTE: All illustrations and photos have been removed from this article.



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