A note on identification in discrete choice models with partial observability

This note establishes a new identification result for additive random utility discrete choice models (ARUM). A decision-maker associates a random utility U_{j}+m_{j} to each alternative in a finite set j∈{1,.,J}, where U={U₁,.,U_{J}} is unobserved by the researcher and random with an unknown joint distribution, while the perturbation m=(m₁,.,m_{J}) is observed. The decision-maker chooses the alternative that yields the maximum random utility, which leads to a choice probability system m→(Pr(1|m),.,Pr(J|m)). Previous research has shown that the choice probability system is identified from the observation of the relationship m→Pr(1|m). We show that the complete choice probability system is identified from observation of a relationship m→∑_{j=1}^{s}Pr(j|m), for any s<J. That is, it is sufficient to observe the aggregate probability of a group of alternatives as it depends on m. This is relevant for applications where choices are observed aggregated into groups while prices and attributes vary at the level of individual alternatives.