On Existence and Properties of Pure-strategy Equilibria under Contests

The use of ‘ratio form’ probability of success function dominates the existing literature on contests. Very few works have focused on the ’difference form’functions, notwithstanding their robust theoretical foundations and intuitive appeal in several contexts. Assuming the cost of efforts to be linear, Hirshleifer (1989) and Baik (1998) have argued that under the difference form contests, there is no interior pure strategy Nash equilibrium. In contrast, existence of interior pure strategy Nash equilibrium is well known for the ratio form contest functions. In this paper we use strictly convex cost functions and demonstrate existence of pure strategy Nash equilibrium for the difference form. Moreover, we show that several properties of equilibria and the comparative statics for the difference form closely resemble those for the ratio form. However, unlike the ratio form, under a difference form contest the existence of pure strategy Nash equilibrium is sensitive to the value of the prize.