Participation games: Market entry, coordination, and the beautiful blonde

We find the Nash equilibria for monotone n-player symmetric games where each player chooses whether to participate. Examples include market entry games, coordination games, and the bar-room game depicted in the movie “A Beautiful Mind”. The symmetric Nash equilibrium involves excessive participation (a common property resource problem) if participants’ payoffs are decreasing (in the number of participants), and insufficient participation if payoffs are increasing. With decreasing payoffs there can be many equilibria, but with increasing payoffs there are only 3. Some comparative static properties of changing one player's participation payoffs are counterintuitive, especially with more than two players.