Cycle conditions for “Luce rationality”

We extend and refine conditions for “Luce rationality” (i.e., the existence of a Luce – or logit – model) in the context of stochastic choice. When choice probabilities satisfy positivity, the cyclical independence (CI) condition of Ahumada and Ülkü (2018) and Echenique and Saito (2019) is necessary and sufficient for Luce rationality, even if choice is only observed for a restricted set of menus. We adapt results from the cycles approach (Rodrigues-Neto, 2009) to the common prior problem Harsanyi (1967–1968) to refine the CI condition, by reducing the number of cycle equations that need to be checked. A general algorithm is provided to identify a minimal sufficient set of equations. Three cases are discussed in detail: (i) when choice is only observed from binary menus, (ii) when all menus contain a common default; and (iii) when all menus contain an element from a common binary default set. Investigation of case (i) leads to a refinement of the famous product rule.