Proposition: Binary reduction and attainment

For every $W_e\in {\mathsf{W}}_3$,

The maximum exists and belongs to $[0,1]$.

Proof. For finite probability laws,

where $Z$ ranges over subsets of the outcome alphabet. For any such subset, the coarse-grained receiver element $B={\sum }_{z\in Z}{B}_z$ belongs to ${\mathsf{B}}_b$. Thus every witness in (eq:strength-definition) is bounded by the right side of (eq:strength-maximum).

Conversely, every $B\in {\mathsf{B}}_b$ can be completed to a binary instrument. Put

Then $B^{\mathrm{c}}\ge 0$, and

The binary law for $\{B,B^{\mathrm{c}}\}$ has total-variation distance equal to the absolute difference of the $B$-outcome probabilities. This gives the reverse inequality.

The sets in (eq:local-sets) are closed and bounded: their elements are positive with trace at most two. Their finite product is compact, and the objective is continuous. It therefore attains its supremum. Finally, each compared binary outcome probability lies in $[0,1]$, so the resulting distance is at most one.