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# Eg one choice would be whether to accept a gamble

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Unformatted text preview: distribution over diﬀerent outcomes. E.g., one choice would be whether to accept a gamble which pays \$10 with probability 1/2 and makes you lose \$10 with probability 1/2. von Neumann and Morgenstern’s expected utility theory shows that (under their axioms) there exists a utility function (also referred to as Bernoulli utility function) u (c ), which gives the utility of consequence (outcome) c . Then imagine that choice a induces a probability distribution F a (c ) over consequences. 5 Game Theory: Lecture 2 Introduction Decision-Making under Uncertainty (continued) Then the utility of this choice is given by the expected utility according to the probability distribution F a (c ): U (a ) = � u (c ) dF a (c ) . In other words, this is the expectation of the utility u (c ), evaluated according to the probability distribution F a (c ). More simply, if F a (c ) is a continuous distribution with density f a (c ), then � U (a) = u (c ) f a (c ) dc , or if it is a discrete distribution where outcome outcome ci has probability pia (naturally with ∑i pi...
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