Cooperative dilemmas with binary actions and multiple players

The prisoners’ dilemma, the snowdrift game, and the stag hunt are simple two-player games that are often considered as prototypical examples of cooperative dilemmas across disciplines. However, surprisingly little consensus exists about the precise mathematical meaning of the words “cooperation” and “cooperative dilemma” for these and other binary-action games, in particular when considering interactions among more than two players. Here, we propose new definitions of these terms and explore their consequences on the equilibrium structure of cooperative dilemmas in relation to social optimality. We find that a large class of multi-player prisoners’ dilemmas and snowdrift games behave as their two-player counterparts, namely, they are characterized by a unique equilibrium where cooperation is always underprovided, regardless of the number of players. Multi-player stag hunts allow for the peculiarity of excessive cooperation at equilibrium, unless cooperation is such that it induces positive individual externalities. Our framework and results unify, simplify, and extend previous work on the structure and properties of binary-action multi-player cooperative dilemmas.