The optimal entry fee-prize ratio in Tullock contests

We analyze a three-stage game where an organizer sets an entry fee for a Tullock contest event, and a finite population of homogeneous agents simultaneously decide whether to participate or not. We show that in the unique symmetric subgame perfect Nash equilibrium, the larger the population size, the lower the probability the agents enter the contest, but the organizer's optimal entry fee-prize ratio could either increase or decrease with the population size. When the population size approaches infinity, the number of contestants converges to a Poisson distributed random variable.