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My question is: does there exist an extension of the multinomial logit that considers a sequence of choices? I.e., first the person chooses one of the different alternatives presented, subsequently the person has to taken another decision between different alternatives.

I'm interested in the marginal and conditional probability of the choices. The choices in the first step are not mutually exclusive. Can someone help me? Thanks in advance

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When you say "the person has to taken another decision between different alternatives," I get the impression that the set of possible choices depends on the last choice made. Because of this the model I would suggest is the Markov Model.

In its simplest form, the Markov Model involves a random walk on a graph. At each node on the graph there is a multinomial distribution which captures the probability of transitioning to other nodes (states) in the graph. Because each transition is governed by a lone multinomial distribution, this makes the process "memoryless," in the sense that your probability of going to a given state depends only on your current state.

If you would like to make things a bit more exciting there is no reason that at each node you could not have a multinomial distribution conditioned on some features. For instance maybe one demographic is more likely to transition $A \rightarrow B$ than others.

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  • $\begingroup$ Yes, in the sense that the alternatives in the second choice are the same independently by what the person has choosen in the first step..but the probability of the choices are affected by the first choice. I'm sorry but I Was wrong: also the first set of choices respects the IIA assumption. Given this, do you suggest Markov model (and if so, can you give me some good references) or is there an extension of multinomial logit? An additional difficulty is that the set of choices in the first step are not mutually exclusive..Thanks in advance! $\endgroup$
    – Andrea
    Commented Sep 28, 2015 at 19:29
  • $\begingroup$ @Andrea Sorry, I had a lot of trouble reading the first few lines of the comment. Could you reword them? $\endgroup$
    – jlimahaverford
    Commented Sep 28, 2015 at 19:36
  • $\begingroup$ I'm sorry for my English. I mean that the set of choices in the second step does not depend on the last choice made. However, the probability of choose one alternative instead of another one depends on the last choice made. $\endgroup$
    – Andrea
    Commented Sep 28, 2015 at 20:04
  • $\begingroup$ You're describing a stochastic process with memory. It's markov only if memory is finite length $\endgroup$
    – Aksakal
    Commented Sep 28, 2015 at 20:13
  • $\begingroup$ Given that the set of choices is not changing, and moreover that the choices are not mutually exclusive, I would suggest doing ordinary mutlinomial logit, but including recent choices as additional features. $\endgroup$
    – jlimahaverford
    Commented Sep 28, 2015 at 20:26
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Are you looking for something like Multinomial Conditional Logit? You may find interesting discussions in Stata or in R

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It appears that you're looking for markov decision processes.

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