Game theory perfect bayesian equilibrium

If player 1 chooses either L or M then player 2 learns that R was not chosen but not which of L or M was chosen and then chooses between two actions L' and R', after which the game ends.

Payoffs are given in the extensive form. Using the normal form representation of this game given below we see that there are two pure strategy Nash-equilibria - L,L' and R,R'. To determine which of these Nash equilibria are subgame perfect, we use the extensive form representation to define the game's subgames.

So the game above has no proper subgames and the requirement of subgame perfection is trivially satisfied, and is just the Nash equilibrium of the whole game. However, one can see that R,R' clearly depends on a noncredible threat: if player 2 gets the move, then playing L' dominates playing R', so player 1 should not be induced to play R by 2's threat to play R' given the move.

To strengthen the equilibrium concept to rule out the subgame perfect Nash equilibrium R,R' we impose the following requirements. R1: 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. For a nonsingleton information set, a belief is a probability distribution over the nodes in the information set; for a singleton information set, the player's belief puts one on the decision node.

R2: Given the beliefs, the players' strategies must be sequentially rational. That is at each information set the action taken by the player with the move and the player's subsequent strategy must be optimal given the player's belief at the information set and the other players' subsequent strategies where a "subsequent strategy" is a complete plan of action covering every contingency that might arise after the given information set has been reached.

R3: At information sets on the equilibrium path, beliefs are determined by Bayes' rule and the players' equilibrium strategies. R4: At information sets off the equilibrium path, beliefs are determined by Bayes' rule and the players' equilibrium strategies where possible.

In our example R1 implies that if the play of the game reaches player 2's non-singleton information set then player 2 must have a belief about which node has been reached or equivalently, about whether player 1 has played L or M. This belief is represented by probabilities p and 1-p attached to the relevant nodes in the tree.

Thus, simply requiring that each player have a belief and act optimally given this belief suffices to eliminate the implausible equilibrium R,R'. Sign up to join this community. The best answers are voted up and rise to the top.

Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Find a separating perfect Bayesian equilibrium Ask Question. Asked 9 months ago. Active 9 months ago. Viewed 73 times.

Now the solution manual says that there is not any separating perfect Bayesian equilibrium. What is wrong with my understanding here? Improve this question. Add a comment. Active Oldest Votes. Improve this answer. Herr K. P1 is getting a higher payoff that way.