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The suspect would rather shoot if he is a criminal, even if the sheriff does not shoot, but would rather not shoot if he is a civilian, even if the sheriff shoots.
Thus, there is incomplete information (because the suspect has private information), making it a Bayesian game.
The resulting "stochastic Bayesian game" model is solved via a recursive combination of the Bayesian Nash equilibrium (see below) and the Bellman optimality equation.
The definition of Bayesian games and Bayesian equilibrium has been extended to deal with collective agency.
The beliefs of a player describe the uncertainty of that player about the types of the other players.
Each belief is the probability of the other players having particular types, given the type of the player with that belief.
As in games of complete information, these can arise via non-credible strategies off the equilibrium path.