Beliefs and Sequential Rationality

Transcription

1 Beliefs and Sequential Rationality A system of beliefs µ in extensive form game Γ E is a specification of a probability µ(x) [0,1] for each decision node x in Γ E such that x H µ(x) = 1 for all information sets H. E[u i H, µ,σ i,σ i ] denotes player i s expected utility starting at her information set H if her beliefs regarding the conditional probabilities of being at the various nodes in H are given by µ, if she follows strategy σ i, and if her rivals use strategies σ i. A strategy profile σ in extensive form game Γ E is sequentially rational at information set H given a system of beliefs µ if we have E[u ι(h) H, µ,σ ι(h),σ ι(h) ] E[u ι(h) H, µ, σ ι(h),σ ι(h) ] for all σ ι(h) (S ι(h) ). If σ satisfies this condition for all information sets H, then we say that σ is sequentially rational given belief system µ. 1

2 Weak PBE A profile of strategies and system of beliefs (σ, µ) is a weak perfect Bayesian equilibrium (weak PBE) in extensive form game Γ E if it has the following properties: (i) σ is sequentially rational given µ. (ii) µ is derived from σ through Bayes rule whenever possible. That is, µ(x) = Prob(x σ) Prob(H σ) for any information set H with Prob(H σ) > 0. No restrictions at all are placed on beliefs off the equilibrium path. It may not be structurally consistent. A weak PBE may not be subgame perfect. In the literature, it is not uncommon that a set of strategies σ will be referred to as an equilibrium with the meaning that there is at least one associated set of beliefs µ such that (σ, µ) satisfies this definition. 2

3 σ is a Nash equilibrium of Γ E µ such that (i) σ is sequentially rational given µ at all information sets H such that Prob(H σ) > 0. (ii) µ is derived from σ through Bayes rule whenever possible. 3

4 Sequential Equilibrium A strategy profile and system of beliefs (σ, µ) is a sequential equilibrium of extensive form game Γ E, if it has the following properties: (i) σ is sequentially rational given µ. (ii) There exists a sequence of completely mixed strategies {σ k } k=1 with lim k σ k = σ, such that µ = lim k µ k. where µ k denotes the beliefs derived from strategy profile σ k using Bayes rule. In every sequential equilibrium (σ, µ) of an extensive form game Γ E, σ constitutes a subgame perfect Nash equilibrium of Γ E. 4

5 Extensive Form Trembling-Hand Perfect Nash Equilibrium [Selten (1975)] A strategy profile in extensive form game Γ E is an extensive form tremblinghand perfect Nash equilibrium def it is a normal form trembling-hand perfect Nash equilibrium of the agent normal form derived from Γ E. The agent normal form is the normal form that we would derive if we pretended that the player had a set of agents in charge of moving for her at each of her information sets (a different one for each), each acting independently to try to maximize the player s payoff. A normal form trembling-hand perfect equilibrium may not be subgame perfect (without having the trembles occurring at each information set rather than over strategies). It insures that equilibrium strategies are sequentially rational. We can use the sequence of equilibrium strategies in the perturbed 5

6 games for deriving sequential equilibrium beliefs. It can also eliminate some sequential equilibria in which weakly dominated strategies are played. In general, however, the concepts are quite close [see Kreps and Wilson (1982) for a formal comparison]; and because it is much easier to check that strategies are best responses at the limiting beliefs than it is to check that they are best responses for a sequence of strategies, sequential equilibrium is much more commonly used. 6

7 Exercises What are the sequential equilibria in the games in Exercises 9.C.3. 9.C.7 B Consider the extensive form game depicted in Figure 9.Ex.2. (a) Find a subgame perfect Nash equilibrium of this game. Is it unique? Are there any other Nash equilibria? (b) Now suppose that player 2 cannot observe player 1 s move. Write down the new extensive form. What is the set of Nash equilibria? (c) Now suppose that player 2 observes player 1 s move correctly with probability p (0,1) and incorrectly with probability 1 p (e.g., if player 1 plays T, player 2 observes T with probability p and observes B with probability 1 p. Suppose that player 2 s propensity to observe incorrectly (i.e., given by the value of p) is common knowledge to the two players. What is the extensive form now? Show that there is a unique weak perfect Bayesian equilibrium. What is it? 7

8 References Kreps, O. M. and R. Wilson (1982) Sequential equilibrium, Econometrica, Vol. 50, pp Selten, R. (1975) Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory, Vol. 4, No. 1, pp