**Discrete choice models** are used, primarily in economics, to model the choices made between well-defined alternatives. Typically they model the choices a consumer might make between a finite set of alternate products or services.

The set of all alternatives available is called the **choice set**. If there are just two alternatives, it is called a **binary choice**. A discrete choice model with more than two alternatives is called a **multinomial discrete choice model. **

## The Basis for Discrete Choice Models

The basis of discrete choice models is the assumption that the consumer (or chooser, whoever that might be) will look at the choice set and decide what level of utility each alternative will give, and then choose the alternative that offers them the greatest utility. These utility levels are different for each consumer.

We write the utility level of an alternative j for a consumer i as u_{i j}. For purposes of analysis, we can write:

u_{i j}= v_{i j} + ε_{i j},

i = 1, … N, and j = 1, … , J

Where:

- The subscript i denotes the index for the individual (the consumer in most cases),
- The subscript j denotes the index for the alternatives,
- N is the number of individuals,
- J is the number of alternatives in the model,

v_{i j} is a utility function which relates all relevant observed factors to the utility level of the item. Part of the discrete choice model is that we assume v_{i j} is linear in all relevant parameters. ε_{i j} is an error term, and can be considered to represent all unobserved characteristics of utility.

We can break v_{i j} down further in terms of two vectors:

v_{i j} = β x’_{i j}

Here x’_{i j} is a vector that represents all the observed attributes which relate, in some way and in some level, to the utility of the choice j. The vector β is a vector of equivalent dimension, made up of fixed regression coefficients; it factors in how much each of the attributes of x’_{i j} relate to the overall utility level.

## The Choice Set

In discrete choice models, a choice set must contain a finite number of alternatives. These choices have to be mutually exclusive; there can’t be a way of choosing more than one. They also have to be collectively exhaustive. That means the individual (consumer, or whoever it is making the choice) must choose one. If picking nothing or ‘going elsewhere to shop’ is an option, it must be represented as such in the choice set.

## References

SAS/STAT(R) User Guide: Discrete Choice Models

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