Foundation: Add random component to systematic utility. We only know systematic component. Assume that max of the sums always wins, which because of random component means that the lower systematic utility sometimes ``wins'' anyway.
Specific model depends on the distribution function of the random compoment.
Binary choice:
Gaussian randomness Binary Probit. No closed form
solution.
Gumbel randomness Binary Logit.
Closed form solution
.
Multinomial choice:
Gaussian randomness Multinomial Probit. Not
treated; no closed form solution. Feasible with computers, and has
many theoretical advantages.
Gumbel randomness Multinomial Logit (MNL).
Result again
.
Max likelihood estimation of : Adjust the
so
that the probability for the model to generate the survey is
maximized.