# Multinomial Logit with different choice set for each individual

I am currently trying to model a discrete choice problem using multinomial logit. The issue is that each individual faces a "different" choice set.

In Stage 1 an individual is told that he can select from a set of alternatives (the number of alternatives and the alternatives themselves may vary between individuals).

In Stage 2 the individual can select any number of the alternatives offered i.e. not mutually exclusive. So individual $i$ is allowed to select from a set of alternatives $\mathbb{A}_i$ that has $J_i$ elements and so can choose from $2^{J_i}$ alternatives.

I am modeling stage 2.

I was thinking of specifying latent utility as follows: $$U_{ij} = \alpha + \beta X_i + \gamma Z_j + \epsilon_{ij}$$ where $\epsilon_{ij}$ is T1EV, iid and $j$ denotes one of the $2^{J_i}$ alternatives.

Then I would specify the choice probabilities by each individual and perform an MLE to recover the coefficients. I am thinking that I should be able to estimate coefficients since they are not varying by individuals and choice.

Will such a procedure work? I cannot include choice specific constants because individuals have different choice sets.

Is there any paper or book that discusses this issue? I have not been able to find any.