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 $$U_{j}+m_{j}$$ U j + m j to each alternative in a finite set $$j\in \left\{ 1,\ldots ,J\right\} $$ j ∈ 1 , … , J , where $$\mathbf {U}=\left\{ U_{1},\ldots ,U_{J}\right\} $$ U = U 1 , … , U J is unobserved by the researcher and random with an unknown joint distribution, while the perturbation $$\mathbf {m}=\left( m_{1},\ldots ,m_{J}\right) $$ m = m 1 , … , m J is observed. The decision-maker chooses the alternative that yields the maximum random utility, which leads to a choice probability system $$\mathbf { m\rightarrow }\left( \Pr \left( 1|\mathbf {m}\right) ,\ldots ,\Pr \left( J| \mathbf {m}\right) \right) $$ m → Pr 1 | m , … , Pr J | m . Previous research has shown that the choice probability system is identified from the observation of the relationship $$ \mathbf {m}\rightarrow \Pr \left( 1|\mathbf {m}\right) $$ m → Pr 1 | m . We show that the complete choice probability system is identified from observation of a relationship $$\mathbf {m}\rightarrow \sum _{j=1}^{s}\Pr \left( j|\mathbf {m} \right) $$ m → ∑ j = 1 s Pr j | m , for any $$s