In decision theory, a decision rule is said to dominate another if the performance of the former is sometimes better, and never worse, than that of the latter. Formally, let δ 1 {\displaystyle \delta _{1}} and δ 2 {\displaystyle \delta _{2}} be two decision rules, and let R ( θ , δ ) {\displaystyle R(\theta ,\delta )} be the risk of rule δ {\displaystyle \delta } for parameter θ {\displaystyle \theta } . The decision rule δ 1 {\displaystyle \delta _{1}} is said to dominate the rule δ 2 {\displaystyle \delta _{2}} if R ( θ , δ 1 ) ≤ R ( θ , δ 2 ) {\displaystyle R(\theta ,\delta _{1})\leq R(\theta ,\delta _{2})} for all θ {\displaystyle \theta } , and the inequality is strict for some θ {\displaystyle \theta } . This defines a partial order on decision rules; the maximal elements with respect to this order are called admissible decision rules.
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