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Chapter 4 - Beliefs Important concept A belief is the...

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Beliefs Important concept: A belief is the player’s assessment about the strategies of the other players (JW) For instance, if P2 in the simplistic poker game thinks that P1 will always play the strategy BB’ then that’s a belief. a N b 1 c 1 d e f 2 g h i j k H L B F B’ F’ C F C F 3,-3 1,-1 -3,3 1,-1 -1,1 -1,1
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Probabilities Prisoners’ dilemma 1 2 C D C 2,2 0,3 D 3,0 1,1 Or P1 may think that P2 will play C with probability p and D with probability 1 - p I Probabilities are always between zero and one (one=certainty that it will happen; zero=certainty that it won’t happen) I The numbers p and 1 - p reflect a probability distribution I A probability distribution describes how likely it is that each of the possible outcomes will occur I Please note that P2 may or may not randomize; the value of p just reflects P1’s uncertainty about P2’s strategy
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Beliefs for normal form games I A belief of player i is a probability distribution over the strategies of the other players (JW) I If Δ S - i is the set of all probability distributions over the strategies of all the other players then any element - i 2 Δ S - i is a belief I For purists: - i is in fact a probability (mass) function (or probability density function in the continuous case), which describes a probability distribution, but I’ll try to be consistent with JW
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