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 DecisionMaking 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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 Spring '10
 AsuOzdaglar
 Game Theory

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