Single-offer reward-effort ambiguity ($U=\alpha R-\kappa E$)

Consider a logistic choice rule $P(\mathrm{accept})=1/(1+\exp (-U))$ with $U=\alpha R-\kappa E$ and fixed noise scale. At one positive offer $(R_0,E_0)$, changes satisfying $\delta \alpha R_0=\delta \kappa E_0$ leave utility and acceptance probability unchanged. One offer therefore cannot identify reward gain and effort cost separately.

For example, $(\alpha ,\kappa )=(0.6,0.2)$ and $(1,1)$ both give $U=1$ at $(R,E)=(2,1)$. Their predictions differ at other offers. Independently varying reward and effort can identify the two coefficients within this model when the design has rank two and choice data are informative. If noise scale is also unknown, its tradeoff with the coefficients requires additional constraints.