Summary

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 $\leadsto$ Binary Probit. No closed form solution.

Gumbel randomness $\leadsto$ Binary Logit. Closed form solution $P_A \propto e^{V_A}$.

Multinomial choice:

Gaussian randomness $\leadsto$ Multinomial Probit. Not treated; no closed form solution. Feasible with computers, and has many theoretical advantages.

Gumbel randomness $\leadsto$ Multinomial Logit (MNL). Result again $P_A \propto e^{V_A}$.

Max likelihood estimation of $\beta$: Adjust the $\beta$ so that the probability for the model to generate the survey is maximized.