Bayesian Game - Specification of Games

Specification of Games

The normal form representation of a non-Bayesian game with perfect information is a specification of the strategy spaces and payoff functions of players. A strategy for a player is a complete plan of action that covers every contingency of the game, even if that contingency can never arise. The strategy space of a player is thus the set of all strategies available to a player. A payoff function is a function from the set of strategy profiles to the set of payoffs (normally the set of real numbers), where a strategy profile is a vector specifying a strategy for every player.

In a Bayesian game, it is necessary to specify the strategy spaces, type spaces, payoff functions and beliefs for every player. A strategy for a player is a complete plan of action that covers every contingency that might arise for every type that player might be. A strategy must not only specify the actions of the player given the type that he is, but must specify the actions that he would take if he were of another type. Strategy spaces are defined as above. A type space for a player is just the set of all possible types of that player. 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 (i.e. the belief is . A payoff function is a 2-place function of strategy profiles and types. If a player has payoff function and he has type t, the payoff he receives is, where is the strategy profile played in the game (i.e. the vector of strategies played).

One of the formal definitions of such game looks like the following:

The game is defined as:, where

1. is the set of players.

2. is the set of the states of the nature. For instance, in a card game, it can be any order of the cards.

3. is the set of actions for player i. Let .

4. is the types of player i, decided by the function . So for each state of the nature, the game will have different types of players. The outcome of the players is what determines its type. Players with the same outcome belong to the same type.

5. defines the available actions for player i of some type in .

6. is the payoff function for player i. More formally, let, and .

7. is the probability distribution over for each player i, that is to say, each player has different views of the probability distribution over the states of the nature. In the game, they never know the exact state of the nature.

The pure strategy should satisfy for all . So the strategy for each player only depends on his type, since he may not have any knowledge about other players' types. And the expected payoff to player for such strategy profile is .

Let be the set of pure strategies,

A Bayesian Equilibrium of the game G is defined to be a (possibly mixed strategy) Nash equilibrium of the game . So for any finite game G, Bayesian Equilibria always exists.