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# gameL2 - Bayesian Games How do we model uncertainty about...

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Bayesian Games How do we model uncertainty about the payo ff s or (more generally) knowledge of the other players? The traditional distinction (see Fudenberg and Tirole) is that uncertainty about payo ff s is called incomplete information , and uncertainty about previous actions is called imperfect information . Harsanyi invented the trick of converting games of incom- plete information to games of imperfect information, by introducing nature as an additional player. For example, suppose we are uncertain about whether another player has low costs or high costs. This can be modelled as knowing that costs are low when nature chooses ω = L , and that costs are high when nature chooses ω = H .

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De fi nition 25.1: A Bayesian game consists of 1. A fi nite set N (players), 2. A fi nite set (states of nature), and for each player i N , 3. A set A i (actions), 4. A fi nite set T i ( signals that may be observed by player i ) and a function τ i : T i (the signal function of player i ), 5. A probability measure, p i , on (the prior belief of player i ) for which p i ( τ 1 i ( t i )) > 0 holds for all t i T i . 6. A preference relation % i on the set of probability measures over A × , where A = × j N A j .
Notes: In part (4), it is assumed that signals “partition” the set of states, so there is no noise in the process generating signals from states. This is without loss of generality, because otherwise we can simply expand the set of states to include the signal.

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