122 Review

Ecn 122 -- GAME THEORY

REVIEW of MAIN CONCEPTS

1. NORMAL-FORM GAMES

2. EXTENSIVE GAMES WITH PERFECT INFORMATION

3. EXTENSIVE GAMES WITH IMPERFECT INFORMATION

· Subgames and subgame-perfect equilibrium. Behavioral strategies instead of mixed strategies.

4. KNOWLEDGE AND BELIEFS

· Information partitions. Definition: an individual knows event E at state w if the cell of her information partition that contains w is entirely contained in E (i.e. is a subset of E).

· Interpersonal knowledge and common knowledge. The event that individual 1 knows event E is the set of states at which individual 1 knows E (see previous definition). Since it is an event (a set of states) we can easily determine at which states individual 2 knows that individual 1 knows E, etc.

Definition

· Beliefs: probability distribution over the set of states that the individual considers possible (at state w an individual considers possible all and only those states that belong to the information set containing w).

· Bayes’ rule for updating probabilistic beliefs: for any two events E and F, the probability assigned to event E upon receiving information represented by the event F is

5. GAMES WITH INCOMPLETE INFORMATION

· Use Nature, states and probabilities to convert the incomplete information situation into an extensive game with imperfect information.

· A Bayesian equilibrium of an incomplete information game is just a Nash equilibrium of the corresponding imperfect information game.

· Weak Sequential equilibrium. Ingredients: (1) beliefs (= probability distribution) at each information set, (2) optimal choice given beliefs at each information set, (3) beliefs must be obtained using Bayes' rule whenever possible (i.e. at all those information sets that are reached with positive probability if the equilibrium strategies are played).

6. CORE and SHAPLEY VALUE
· Cooperative games: characteristic function and imputations. The strategic/rationality approach: the core. The axiomatic approach: the Shapley value.