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, we show that the cyclical independence (CI) condition of Ahumada and Ulk¨u (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 then 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 (depending on the collection of menus for which choice is observed). 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.