MIT博弈论课件GT_fall2003_lecture_1.pdf

Knowledge and Common Knowledge Copyright 2003 by Drew Fudenberg. Do not post or redistribute We will represent player i’s knowledge using a partition H of a “state space” Ω: i When the true state is ω, player i knows that is in the element of his partition that contains ω; the elements of the partition are the states i considers possible. Call this set h (ω) . i Implicit : the state space Ω all relevant uncertainty: the player’s information/uncertainty about the state of nature, his information about others information etc. Note also that since by definition ω∈h (ω) , player i i always thinks that the true state is possible. Assume: Ω is finite, there is a common prior p on Ω, all states have positive probability. (drop zero- probability states.) Assuming finiteness makes the math a lot easier, but later we will need to deal with larger state spaces if only to understand how restrictive the finiteness assumption is. Definition: “Player i knows E at ω” if h (ω) ⊆E . i K (E ) ≡{ω| h (ω) ⊆E }: i i this is the set of states where i knows E. This definition satisfies the following properties. (proof is HW) Necessitation: K (Ω) Ω : i Player i always knows the state space. As a consequence, player i knows all statements that are true for every point in the state space, i.e. all tautologies. K (E ) K K (E ) i i i (i knows E if and only if she knows that she knows it. Implicitly players know their own information structure.) Introspection: −K (−K (E )) ⊆K (E ) i i i If you don’t know that you don’t know E, you know E. So players can’t be unaware of any possibilities. Now define the event “everyone knows E”