l'istituto italiano per gli studi filosofici e gli studi di economia

I accept for this exposition the expected-utility theory of choice.
There is a utility function, U(y), defined for all possible outcomes,
such that DM’s aim is to make its expected value as large as possible.
Most often, I will assume that,
U(y) = In y,
or sometimes, more generally, a power function. We can combine the
outcome function and the utility function into a payoff function
w (a, x) = U[Y(a, x)]
and now the DM’s aim can be expressed
Max! Ex [w (a, X)]
a
where “Max!” is the imperative to maximize, and E x means, “expectation
with respect to the random variable, X.”
Now we introduce information. Following the practices of communications
engineering and statistics, information is modeled by
another random variable, S, called a signal, which does not enter the
payoff function but is observed by DM before taking an action. Once
S is observed, the relevant probability distribution of X is that conditional
on the observed value, so that DM’s problem now is
Max! E X / S = s [w (a, X)].
a
This defines a for each s, the observed value of the signal, and so
defines a decision function, a (s). The expected payoff for the choice
of the optimal decision function depends on the realized value of S
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