Arms races and negotiations

We consider an “arms race ” game, where two players simultaneously decide whether or not to aquire new weapons. Each player has a type, which is his private information. Types are independent. With probability close to one, the best outcome for each player is for neither to acquire new weapons (although each prefers to aquire new weapons if he thinks the opponent will). There is a small probability that a player is a dominant strategy type who always prefers to acquire new weapons. We find conditions under which the unique Bayesian Nash equilibrium involves an arms race with probability one. However, if the probability that a player is a dominant strategy type is sufficiently small, then there is an equilibrium of the cheap-talk extension of the game where the prob-ability of an arms race is close to zero.