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## Multinomial logit

The most common implementation of destination choice is the multinomial logit (MNL) model. The following discussion assumes familiarity with the general formulation of MNL (provide a link). The destination choice problem is presented with reference to an individual decision-maker, however the model is equally applicable to aggregate, zone-based formulations.

### General specification

This section describes the specification of the destination choice utility function in general terms. More in-depth discussion of the factors that influence destination choice and how to represent them in the MNL model can be found here (link).

Given a trip origin i, and decision-maker m, the utility of each destination j can be written as follows:

In this formulation, the utility of a destination depends on the impedance or spatial separation between the trip origin and the destination, and the attractions at the destination. This is the simplest representation of destination choice utility.

Impedance can be measured by distance, auto travel time, or a generalized cost, among other possible measures of spatial separation. A convenient measure of impedance is the inclusive value, or logsum, of the mode choice model. The mode choice logsums are used when it is desirable to have sensitivity to multi-modal level of service in the destination choice model. The coefficient of the impedance variable(s) can be generic (i.e., the same for all decision-makers), or it can vary for certain types of travelers. For example, it is often found that mothers of pre-school children tend to choose work locations that are closer to home than other workers, all else equal. This is represented in the utility function by a more negative coefficient on distance impedance for women with pre-school children than the coefficient used for other workers.

The attraction variable is commonly referred to as the size term. It measures the activity opportunities at each destination. In the case of a work location model, the size term is typically employment. In the case of a school location model, the size term can be school enrollment. For many other trip purposes, the size term is typically a linear combination of different types of employment, for example:

The size term always enters the utility function in log form. The log formulation is necessary so that the choice probability of a destination is directly proportional to the number of opportunities at the destination. In other words, if the number of jobs at a destination doubles, all else, equal, then the choice probability of this destination also doubles.

The availability of opportunities for some types of trips is sometimes not well captured with employment variables in the size term. Consider for example trips to the beach, to open spaces like parks, or to secondary vacation homes. For these types of attractors, instead of a traditional size variable, it may be preferable to use an indicator or qualitative variable.

The representation of the impedance measure need not be linear in the parameters; in fact, it is common for the marginal disutility with respect to distance to decrease with distance, as shown in the figure below. However, special rules apply when the impedance function is the mode choice logsum. To ensure proper elasticities between mode choice and destination choice, the mode choice logsum coefficient must take values between zero and one. (refer to section on joint dest choice/mode choice).

(show some examples -- work location disutility for multiple types of workers)

Since trip-maker characteristics are the same for all destinations, the way to represent the effect of these variables on destination choice is to interact them with one of the impedance variables, as shown below, or by partially or fully segmenting the model. An example of a partially segmented model is when the worker industry is known, and the size variable becomes a function of worker industry. An example of a fully segmented model is for example when different utility functions are specified for various household car sufficiency segments in a trip-based model.

### Alternative-specific constants

Unlike for many other choice models, it is not common to incorporate alternative-specific constants for every destination zone a destination choice model. Doing so results in some convenient mathematical properties (notably, for MNL models, the *other* parameters of the model will have unbiased estimators even in the presence of non-uniform sampling). However, including a complete set of alternative-specific constants can result in other complications: if the number of zonal alternatives approaches or exceeds the number of sampled destination observations, the model parameters will be completely over-determined and model estimation will simply fail. Even if the model is not completely over-determined, a very large number of observations may be necessary to provide sufficient statistical confidence in the estimated results. If the estimation data set is large relative to the number of zones, these problems may be overcome and it may be reasonable to use a complete set of alternative specific constants, although this is not common when using household surveys in the United States.

Instead of employing a complete set of constants for every alternative, it may sometimes be advisable to include just a partial set of alternatives.

## Gravity models as special case of MNL

(link to Eric's page on theory)