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Unformatted text preview: Lecture VII: Common Knowledge Markus M. M¨obius March 4, 2004 This is the one of the two advanced topics (the other is learning) which is not discussed in the two main texts. I tried to make the lecture notes selfcontained. • Osborne and Rubinstein, sections 5.1,5.2,5.4 Today we formally introduce the notion of common knowledge and discuss the assumptions underlying players’ knowledge in the two solution concepts we discussed so far  IDSDS and Nash equilibrium. 1 A Model of Knowledge There is a set of states of nature Ω = { ω 1 ,ω 2 ,..,ω n } which represent the uncertainty which an agent faces when making a decision. Example 1 Agents 1 , 2 have a prior over the states of nature Ω = { ω 1 = It will rain today ,ω 2 = It will be cloudy today , ω 3 = It will be sunny today } where each of the three events is equally likely ex ante. The knowledge of every agent i is represented by an information partition H i of the set Ω. Definition 1 An information partition H i is a collection { h i ( ω )  ω ∈ Ω } of disjoint subsets of Ω such that 1 • (P1) ω ∈ h i ( ω ) , • (P2) If ω ∈ h i ( ω ) then h i ( ω ) = h i ( ω ) . Note, that the subsets h i ( ω ) span Ω. We can think of h i ( ω ) as the knowledge of agent i if the state of nature is in fact ω . Property P1 ensures that the true state of nature ω is an element of an agent’s information set (or knowledge)  this is called the axiom of knowledge. Property P2 is a consistency criterion. Assume for example, that ω ∈ h i ( ω ) and that there is a state ω 00 ∈ h i ( ω ) but ω 00 6∈ h i ( ω ). Then in the state of nature is ω the decisionmaker could argue that because ω 00 is inconsistent with his information the true state can not be ω . Example 1 (cont.) Agent 1 has the information partition H 1 = {{ ω 1 ,ω 2 } , { ω 3 }} So the agent has good information if the weather is going to be sunny but cannot distinguish between bad weather. We next define a knowledge function K . Definition 2 For any event E (a subset of Ω ) we have K ( E ) = { ω ∈ Ω  h i ( ω ) ⊆ E } . So the set K ( E ) is the collection of all states in which the decision maker knows E ....
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This note was uploaded on 05/19/2010 for the course 412 002 taught by Professor Dingli during the Spring '10 term at École Normale Supérieure.
 Spring '10
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