 Choice-value scale non-identifiability

If $P(a)=\exp (\beta Q(a))/{\sum }_b\exp (\beta Q(b))$, then for every $c>0$, replacing $Q$ by $cQ$ and $\beta$ by $\beta /c$ leaves each exponent unchanged. The same choices therefore cannot identify both an unrestricted value scale and inverse temperature. Fixing a value scale or adding independent measurements is needed to break this equivalence.