ECO290E: Game Theory

Lecture 13

Dynamic Games of Incomplete Information

Perfect Bayesian NE

• In dynamic games of incomplete information, there is no subgame other than the entire game. So, SPNE coincides with NE and thereby cannot help excluding implausible equilibria.

• The new concept, perfect Bayesian NE, strengthens the requirements of SPNE by explicitly analyzing the players’ beliefs, as in Bayesian NE.

Belief

• At each information set, the player with the move must have a belief about which node in the information set has been reached by the play of the game. More precisely, for an information set contains multiple nodes, a belief is a conditional probability distribution over the nodes in the information set; for a singleton information set, the player’s belief puts probability one on the single decision node.

Sequential Rationality

• Given their beliefs, the players’ strategies must be sequentially rational. That is, at each information set the action taken by the player with the move must be optimal given the player’s belief at that information set and the other players’ subsequent strategies, i.e., complete action plans of other players covering every contingency that might arise after the given information set has been reached.

Consistency of Beliefs

• At information sets on the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies.

• At information sets off the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies where possible.

Perfect Bayesian NE

• Perfect Bayesian NE is a solution concept that incorporates sequential rationality and consistency of beliefs.

• A perfect Bayesian NE consists of strategies (complete plan of actions for all players) and beliefs (defined over every information set) satisfying that (1) each player’s strategy specifies optimal actions given her beliefs and the strategies of the other players, and (2) the beliefs are consistent with Bayes’ rule wherever possible.