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. A decision‐maker associates a random utility $U_j+m_j$ to each alternative in a finite set $j\in \left\{1,\dots ,J\right\}$ , where $\mathbf{U}=\left\{U_1,\dots ,U_J\right\}$ is unobserved by the researcher and random with an unknown joint distribution, while the perturbation $\mathbf{m}=\left(m_1,\dots ,m_J\right)$ is observed. The decision‐maker chooses the alternative that yields the maximum random utility, which leads to a choice probability system $\mathbf{m}\to \left(Pr\left(1|\mathbf{m}\right),\dots ,Pr\left(J|\mathbf{m}\right)\right)$ . Previous research has shown that the choice probability system is identified from the observation of the relationship $\mathbf{m}\to Pr\left(1|\mathbf{m}\right)$ . We show that the complete choice probability system is identified from observation of a relationship $\mathbf{m}\to {\sum }_{j=1}^{s}Pr\left(j|\mathbf{m}\right)$ , for any $s<J$ . That is, it is sufficient to observe the aggregate probability of a group of alternatives as it depends on $\mathbf{m}$ . This is relevant for applications where choices are observed aggregated into groups while prices and attributes vary at the level of individual alternatives.