Utility Function
Definition
A utility function $U: \mathcal{A} \times \mathcal{S} \to \mathbb{R}$ assigns a numerical payoff to each (action, state) pair, representing the decision-maker’s preference.
Usage in Decision Theory
The optimal action maximizes expected utility:
$$ a^* = \arg\max_{a \in \mathcal{A}} \mathbb{E}[U(a, S)] = \arg\max_{a \in \mathcal{A}} \sum_{s} \mathbb{P}(S = s)\, U(a, s). $$