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In particular, the belief a player holds about another player's type might change according to his own type.
In a Bayesian game, one has to specify type spaces, strategy spaces, payoff functions and prior beliefs.
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.
In addition to the actual players in the game, there is a special player called Nature.
Nature randomly chooses a type for each player according to a probability distribution across the players' type spaces.
A Bayesian Nash equilibrium is defined as a strategy profile that maximizes the expected payoff for each player given their beliefs and given the strategies played by the other players.
That is, a strategy profile Bayesian Nash equilibrium can result in implausible equilibria in dynamic games, where players move sequentially rather than simultaneously.
For example, Alice and Bob may sometimes optimize as individuals and sometimes collude as a team, depending on the state of nature, but other players may not know which of these is the case. Both must simultaneously decide whether to shoot the other or not.
Incompleteness of information means that at least one player is unsure of the type (and therefore the payoff function) of another player.
Such games are called Bayesian because players are typically assumed to update their beliefs according to Bayes' rule.
A strategy profile determines expected payoffs for each player, where the expectation is taken over both the set of states of nature (and hence profiles of types) with respect to beliefs .
In a non-Bayesian game, a strategy profile is a Nash equilibrium if every strategy in that profile is a best response to every other strategy in the profile; i.e., there is no strategy that a player could play that would yield a higher payoff, given all the strategies played by the other players.