Static Games of Incomplete Information

• The idea is largely similar to Static Games of Complete Information except now each player has to respond to each type of player, not unlike a Game of imperfect information.

  • We can still use the usual game matrix, except each row and column denotes a combination of prescribed actions against each type.

    More formally if are the set of types, then each row (for player ) corresponds to where is the action taken when the player type is .

Bayesian Games §

• The normal-form representation of an -player static Bayesian Game of Incomplete Information is specified with a tuple

  Where is the set of players. is the set of actions. is the action of the -th player. Let - the set of type spaces - the type space of player - the set of value functions of the players
  - is the -th value function. - is the set of belief. Each function is of the form

• The Bayesian game as one that proceeds as follows:

  • Nature chooses a profile of types
  • Each player learns his own type which is his private information and then uses his prior to form posterior beliefs over the other types of players.
  • Players simultaneously choose actions (as in the static game).
  • The payoffs are realized using the action profile .

• The private values case will assume that depends only on the private information. In the common values case, we will assume wherein the payoff depends on the other player types.

• A pure strategy for player is a function that prescribes a pure action that player will choose when his type is . A mixed strategy is a probability distribution over a player’s pure strategies.

  • Each player chooses his type-contingent strategy before he learns his type.
  • In effect, each player views the others as always choosing a mixed strategy.

• A strategy profile is a pure strategy Bayesian Nash Equilibrium if, for every player , for each of player ’s types and for each , solves

\mathbb{E}{\theta{-i}} \ \bigg[ \phi_i (\theta_{-i}\mid \theta_i) \ v_i(s_i^\ast (\theta_i ) \ , s_{-i}^\ast (\theta_i) \ ; \theta_i) \mid \theta_i \bigg] \ge \mathbb{E}{\theta{-i}} \bigg[v_i(a_i,s_{-i}^\ast (\theta_{-i}); \theta_i) \mid \theta_i\bigg]