gameL5 - Knowledge and Common Knowledge Game Theory...

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Knowledge and Common Knowledge Game Theory requires us to be interested in knowledge of “parameters” like costs, valuations, and demand, but also knowledge about what other players know. Consider a f nite probability space ( ,p ) and an informa- tion partition for each player i . That is, i = { P 1 i , ..., P K i } , such that P k i P k 0 i = for all k, k 0 ,and K [ k =1 P k i = . Let P i ( ω ) denote that element of player i ’s information partition containing ω . An event is a subset of , with the interpretation that event E occurs when the state is ω if ω E .
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Player i knows E in state ω if P i ( ω ) E . De f nition: Player i ’s knowledge function is given by K i ( E )= { ω : P i ( ω ) E } . K i ( E ) is the set of states in which player i knows E . Some Facts about knowledge generated by partitions: 1. K i ( . 2. E F implies K i ( E ) K i ( F ) .I f F o c c u r s whenever E occurs, then whenever you know E you also know F. 3. K i ( E ) K i ( F K i ( E F ) f y o u k n o w E a n d knowF ,thenyouknowthatbothEandFoccur . 4. K i ( E ) E . Whenever you know that E occurs, then E in fact occurs.
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5. K i ( E ) is the union of all elements of i ’s partition that are contained in E : K i ( E )= [ j : P j i E P j i 6. K i ( E K i ( K i ( E )) .I f y o u k n o w E , t h e n y o u know that you know E.
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gameL5 - Knowledge and Common Knowledge Game Theory...

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