Stag Hunt with unknown outside options

We study the Stag Hunt game where two players simultaneously decide whether to cooperate or to choose their outside options (defect). A player’s gain from defection is his private information (the type). The two players’ types are independently drawn from the same cumulative distribution. We focus on the case where only a small proportion of types are dominant (higher than the value from cooperation). It is shown that for a wide family of distribution functions, if the players interact only once, the unique equilibrium outcome is defection by all types of player. Whereas if a second interaction is possible, the players will cooperate with positive probability and already in the first period. Further restricting the family of distributions to those that are sufficiently close to the uniform distribution, cooperation in both period with probability close to 1 is achieved, and this is true even if the probability of a second interaction is very small.