Dilemma of the Equality: an All-pay Contest with Individual Differences in Resource Holding Potential

Abstract

An all-pay contest in which many players compete for an indivisible resource and each player continuously maintains a different resource holding potential (RHP) is analysed. There exists the unique pure ESS function, which is common sense; that is, a higher RHP induces a higher level of investment, and, as a consequence, a player with a greater RHP always wins. As the variance in distribution of RHP converges to zero, the ESS becomes equal to the symmetric mixed Nash-equilibrium reported by Rose, which does not satisfy the condition of ESS. This suggests that some unstable symmetric Nash equilibria change to ESS functions in some games when we extend the games by assuming one more random continuous parameter of the player's condition. Moreover, the smaller the individual differences in RHP, the more intense the competition becomes, and the smaller becomes the expected payoff for almost all individuals as well as the average payoff of the population. This negative correlation between equality in RHP and the payoff in the population was first found by Kura & Kura in a war-of-attrition game and named the “dilemma of equality”. Its biological implications are also discussed.