Contests with a stochastic number of players. Game Econ

We study Tullock’s (1980) n-player contest when each player has an independent prob-ability 0 < p ≤ 1 of participating. A unique symmetric equilibrium is found for any n and p and its properties are analyzed. In particular, we show that for a fixed n> 2 individ-ual equilibrium spending as a function of p is single-peaked and satisfies a single-crossing property for any two different numbers of potential players. However, total equilibrium spending is monotonically increasing in p and n. We also demonstrate that ex-post over-dissipation is a feature of the pure-strategy equilibrium in our model. It turns out that if the contest designer can strategically decide whether to reveal the actual number of participating players or not, then the actual number of participants is always revealed