A decision rule selects an action from a utility, loss, or outcome criterion. In causal decision-making, it evaluates the utility distribution under each intervention, written conceptually as P(U(Yₓ)), then chooses an action according to the selected rule. Automated systems often call a decision rule a policy; public-health applications may call it a treatment regime. section 12.2.2
Common decision rules translate different attitudes toward utility, risk, and imperfect behavior:
- Maximize expected utility: choose the action with the greatest expected reward, such as the investment expected to make the most money. section 12.2.2
- Minimax: choose the action that minimizes the largest possible loss, prioritizing protection during bad outcomes. section 12.2.2
- Softmax: sample actions with probabilities proportional to their utilities; the noise parameter moves from uniform choice toward utility maximization. section 12.2.2
Admissibility compares rules rather than actions. A rule is admissible when no other rule performs at least as well everywhere and better somewhere under the utility function. Softmax can be inadmissible when the true outcome distribution is known, because expected-utility maximization dominates it. With unknown parameters or latent variables, Bayesian decision theory uses posterior expected utility; Bayes rules have an admissibility guarantee under mild regularity conditions. section 12.2.3
