A universally optimal action has zero partial-monitoring minimax regret

Let $G$ be a finite partial-monitoring game. Suppose there is an action $a$ whose loss is no larger than the loss of any action $b$ on every outcome $i$: $L_{a i} \le L_{b i}$. Then, for every horizon $n$,

Indeed, always playing $a$ achieves zero worst-case regret, while comparison with $a$ makes every policy's worst-case regret nonnegative. This lemma isolates the decision-theoretic part of the zero-regret case and is reusable independently of the cell geometry.

Formalization Note The statement includes the discrete-signal measurability assumptions needed to construct and evaluate the deterministic constant policy.