Transition choice probabilities in logit

We consider the widely used multinomial logit model with i.i.d. Gumbel random terms. Transition choice probabilities, i.e. probabilities of choosing alternative i in the first choice and alternative j in the second, are available in analytic form in the two extreme cases where the random terms of each alternative are independent or perfectly correlated across choices. We extend these results and provide the transition probabilities in analytic form in the case where the random terms follow a bi-extremal distribution with correlation coefficient varying in the full positive range between zero and one.