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122 CHAPTER 4 Bayesian Games · · · is a measurable application from the set of channel states, i = i 1 , . . . , i m i , m i N , to the set of observation of transmitter i which is S i , with |S i | < +; q(s -i |s i ) is the probability that the transmitters other than i observe s -i given that transmitter i observes s i ; a strategy for a player i is a function i : S i P i where i i and i is the set of possible mappings; u B is the expected utility for transmitter i which is defined by: i u B ( 1 , . . . , K , s i ) = E ,s -i u i 1 (s 1 ), . . . , (s K ), |s i . i (4.11) For each transmitter i, denote by i (s i ) the ambiguity set associated with the observation s i . For instance, in the case of perfect global CSI, this set boils down to a singleton: i (s i ) = { -1 (s i )} = {}. Using this notation the expected utility can be rewritten as: P() q(s -i |s i )u i 1 (s 1 ), . . . , (s K ), . u B ( 1 , . . . , K , s i ) = i Pr[ i (s i )] s i (s i ) -i (4.12)