# 9 Subjective expected utility theory

Subjective expected utility theory is used in situations of uncertainty, where the probabilities are not known. People maximize expected utility using a subjective estimate of probability.

The agent maximises subjective expected utility, E[U(x)], using a subjective probability \pi(x_i) for each outcome x_i.

While people may have different beliefs about the probabilities of different outcomes, as the word “subjective” indicates, these are typically assumed to comply with Bayesian probability theory. Decision makers are Bayesian, in that, given the evidence, they have beliefs that are consistent with Bayes’ rule.

The equation for subjective utility theory is as that for expected utility theory, except that the probabilities are subjective.

\begin{align*} E[U(x)]&=\pi(x_1)U(x_1)+\pi(x_2)U(x_2)+...+\pi(x_n)U(x_n)\\[6pt] &=\sum_{i=1}^n \pi(x_i)U(x_i) \end{align*}

You can think of this formula as comprising the following steps:

Define utility U(x_i) over final outcomes x_1,...,x_n.

Define subjective probability \pi(x_i) over final outcomes x_1,...,x_n.

Weight the utility of each outcome U(x_i) by the subjective probability \pi(x_i).

Add the weighted utilities.