# Game theory bayesian updating

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.

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.

These beliefs are represented by a probability distribution over the possible payoff functions. Harsanyi describes a Bayesian game in the following way.

Each player in the game is associated with a set of types, with each type in the set corresponding to a possible payoff function for that player.

In game theory, a Bayesian game is a game in which players have incomplete information about the other players.

For example, a player may not know the exact payoff functions of the other players, but instead have beliefs about these payoff functions.

A type space for a player is just the set of all possible types of that player.

The definition of Bayesian games has been combined with stochastic games to allow for environment states (e.g.

physical world states) and stochastic transitions between states.

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.