MIT博弈论课件GT_fall2003_lecture_2.pdf

Last Time: Defined knowledge, common knowledge, meet (of partitions), and reachability. Reminders: • E is common knowledge at ω if ω∈K ∞(E ) . I • “Reachability Lemma” :ω ∈M (ω) if there is a chain of states ω ω ,ω,...ω ω such that for 0 1 m each ω there is a player i(k) s.t. k h (ω ) h (ω ) : i ( k ) k i (k ) k +1 • Theorem: Event E is common knowledge at ω iff M (ω) ⊆E . How does set of NE change with information structure? Suppose there is a finite number of payoff matrices 1 L u ,...,u for finite strategy sets S ,...,S 1 I State space Ω, common prior p, partitions H , and a i map λso that payoff functions in state ω are uλ(ω) (.) ; the strategy spaces are maps from H into S . i i When the state space is finite, this is a finite game, and we know that NE is u.h.c. and generically l.h.c. in p. In particular, it will be l.h.c. at strict NE. The “coordinated attack” game A B A B A 8, 8 - 10, 1 A 0, 0 - 10, 1 B 1,- 10 0, 0 B 1,- 10 8, 8 a b u u Ω= 0,1,2,…. In state 0: payoff functions are given by matrix b u ; In all other states payoff functions are given by a