of the error terms wouldn’t have appropriate statistical characteristics. The MNL approach is to make a maximum likelihood estimate of this functional form. The likelihood function is: we solve for the estimated parameters that max L*. This happens when: The log-likelihood is easier to work with, as the products turn to sums: Consider an example adopted from John Bitzan’s Transportation Economics Notes. Let X be a binary variable that is equal to 1 with probability γ, and equal to 0 with probability (1 − gamma). Then f(0) = (1 − γ) and f(1) = γ. Suppose that we have 5 observations of X, giving the sample {1,1,1,0,1}. To find the maximum likelihood estimator of γ examine various values of γ, and for these values determine the probability of drawing the sample {1,1,1,0,1} If γ takes the value 0, the probability of drawing our sample is 0. If γ is 0.1, then the probability of getting our sample is: f(1,1,1,0,1) = f(1)f(1)f(1)f(0)f(1) = 0.1×0.1×0.1×0.9×0.1 = 0.00009 We can compute the probability of obtaining our sample over a range of γ – this is our likelihood function. The likelihood function for n independent observations in a logit model is where: Yi = 1 or 0 (choosing e.g. auto or not-auto) and Pi = the probability of observing Yi = 1 The log likelihood is thus: In the binomial (two alternative) logit model, , so The log-likelihood function is maximized setting the partial derivatives to zero: The above gives the essence of modern MNL choice modeling. Additional topics Topics not touched on include the “red bus, blue bus” problem; the use of nested models (e.g., estimate choice between auto and transit, and then estimate choice between rail and bus transit); how consumers’ surplus measurements may be obtained; and model estimation, goodness of fit, etc. For these topics see a textbook such as Ortuzar and Willumsen (2001). Returning to roots The discussion above is based on the economist’s utility formulation. At the time MNL modeling was developed there was some attention to psychologist's choice work (e.g., Luce’s choice axioms discussed in his Individual Choice Behavior, 1959). It has an analytic side in computational process modeling. Emphasis is on how people think when they make choices or solve problems (see Newell and Simon 1972). Put another way, in contrast to utility theory, it stresses not the choice but the way the choice was made. It