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 (jin left{ 1,ldots ,Jright} ), where (mathbf {U}=left{ U_{1},ldots ,U_{J}right} ) is unobserved by the researcher and random with an unknown joint distribution, while the perturbation (mathbf {m}=left( m_{1},ldots ,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 { mrightarrow }left( Pr left( 1|mathbf {m}right) ,ldots ,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}rightarrow Pr left( 1|mathbf {m}right) ). We show that the complete choice probability system is identified from observation of a relationship (mathbf {m}rightarrow sum _{j=1}^{s}Pr left( j|mathbf {m} right) ), for any (s. 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.