lecture7 - Lecture VII Common Knowledge Markus M Mbius o...

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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 self-contained. 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
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(P1) ω h i ( ω ) , (P2) If ω 0 h i ( ω ) then h i ( ω 0 ) = 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 0 h i ( ω ) and that there is a state ω 00 h i ( ω 0 ) but ω 00 6∈ h i ( ω ). Then in the state of nature is ω the decision-maker could argue that because ω 00 is inconsistent with his information the true state can not be ω 0 . 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 . We are now ready to define common knowledge (for simplicity we only consider two players).
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