Lecture 2. Sequential Equilibrium

Transcription

1 ECON601 Spring, 2015 UBC Li, Hao Lecture 2. Sequential Equilibrium Strategies and beliefs Applying idea of subgame perfection of extensive games with perfect information to extensive games with imperfect information: each player s strategy has to be optimal at each of his information sets. An example of what this requirement buys us: Figure 219.1, O&R. An example of the deficiency of this requirement: Figure 220.1, O&R. The need to include beliefs as part of solution concept: unlike in extensive games with perfect information, belief about history of the play is important in determining optimal strategy at an information set. 1

2 Ideas behind sequential equilibrium Sequential equilibrium consists of a behavioral strategy profile, as in extensive games with perfect information, and a new component, called a belief system that assigns a probability distribution over each information set. Two requirements: sequential rationality and belief consistency. Sequential rationality is an extension of subgame perfection in extensive games with perfect information. At each information set, given the belief assigned by the belief system, the player chooses optimally among his available actions. As in extensive games of perfect information, this requirement applies to all information sets, not just those reached according to the strategy profile under consideration ( on the path ). 2

3 Belief consistency has the following components. Consistency with strategies, as in Nash equilibrium: for each information set on the path, the belief at the information set must be derived using Bayes rule Figure 220.1, O&R. Consistency with the extensive game form: For an information set not reached according to the strategy profile under consideration ( off the path ), the belief at the information set can be generated by some alternative strategy profile using Bayes rule. Consistency with each other: The beliefs for information sets off the path do not contradict each other. Remarks: Consistency with strategies is minimal, but the other two are less natural. Often sequential equilibrium does not do enough for beliefs off the path. 3

4 Sequential equilibrium Definition. An assessment in extensive game is a pair (β, µ), where β is a behavioral strategy profile, and µ is a belief system that assigns to every information set a probability measure on the set of histories in the information set. Let (β, µ) be an assessment in N, H, P, f c, (I i ), ( i ). µ(i)(h) denotes the probability that player P(I) assigns to the history h I, conditional on I being reached, where P(I) is the player to whom I is assigned. Sequential rationality. Let outcome O(β, µ I) be the distribution over the set Z of terminal histories by assessment (β, µ) conditional on information set I having been reached: for any terminal history h = (a 1,..., a K ), if a subhistory h = (a 1,..., a L ) of h is in I then O(β, µ I)(h ) = µ(i)(h) K 1 k=l β P(a 1,...,a k ) (a1,..., a k )(a k+1 ); and if there is no subhistory of h in I, then O(β, µ I)(h ) = 0. 4

5 Definition. Fix an extensive game N, H, P, f c, (I i ), ( i ). An assessment (β, µ) is sequentially rational if for every player i N and every information set I i I i, we have O(β, µ I i ) i O((β i, β i ), µ I i) for every strategy β i of player i. Sequential rationality is minimum requirement to refine Nash equilibrium in extensive games with imperfect information. Belief consistency. We say that a behavioral strategy profile is completely mixed if it assigns strictly positive probability to every available action at every information set. Definition. Fix an extensive game N, H, P, f c, (I i ), ( i ). An assessment (β, µ) is consistent if there is a sequence ((β n, µ n )) n=1 of assessments that converges to (β, µ) and has the properties that each strategy profile β n is completely mixed and that each belief system µ n is derived from β n using Bayes rule. 5

6 Remarks: The idea is to use convergence and consistency of belief system with strategies to assign beliefs conditional on unexpected events: this way of imposing belief consistency is a very specific way of meeting consistency with extensive game form and consistency with each other. An alternative way of imposing belief consistency is structural consistency: belief system µ in an extensive game is structurally consistent if for every information set I there is a strategy profile β with the properties that I is reached with positive probability under β and µ(i) is derived from β using Bayes rule. Since belief system under consistency uses the same perturbation to generate beliefs at all information sets, and since different strategy profiles may be used to justify beliefs at different information sets, belief systems that are struturally consistent may not be consistent. More surprisingly, a consistent belief system may not be structurally consistent. 6

7 Definition. An assessment is a sequential equilibrium of an extensive game if it is sequentially rational and consistent. Remarks: A sequential equilibrium strategy profile is a Nash equilibrium. Sequential equilibrium is equivalent to subgame perfect equilibrium in games with perfect information. Verifying sequential equilibrium Figure 220.1, O&R Example 225.2, O&R Example 226.2, O&R An example of sequential equilibrium not being structurally consistent: Example 228.2, O&R 7