Dominating decision rule

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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 $δ 1$ and $δ 2$ be two decision rules, and let $R (θ,δ)$ be the risk of rule $δ$ for parameter $θ$ . The decision rule $δ 1$ is said to dominate the rule $δ 2$ if $R(\theta,\delta_1)\le R(\theta,\delta_2)$ for all $θ$ , and the inequality is strict for some $θ$ .