Discrete choice models are widely used in transportation planning, behavior analysis, and econometrics. I try to gather all my notes about discrete choice modeling for future uses in this series of posts. Like many other notes published in this blog, the main objective is to provide a quick go-to note. If you want to suggest a change or an improvement to this series of posts, you can always contact me on my email.

**Table of Contents**

**Introduction**

### What we mean by model?

Models are the representations of our environment, objects, and any other component which can affect or be affected by the system we are trying to understand. Usually, the model includes all the crucial elements and their relationships; therefore it can be used to understand the consequences of different scenarios.

### What we mean by choice modeling?

The way that a decision is being made can also be modeled. Through these models, we can understand how an entity chooses an action or a sequence of actions. Generally, these models are built using either **revealed **preferences or **stated **by the individuals we are trying to model. We will get back to these preferences in the future.

### What we mean by discrete choice modeling?

Generally speaking, there are two types of choice spaces (sets). You might deal with a **continuous **choice set like when you choose the volume of different ingredients to bake a cake. For example, imagine your ingredients for a particular cake are milk, flour, and butter. You might bake different batches with different combinations of these ingredients. Other people might like different batches, so their choice might be a different combination of these ingredients.

The choice spaces (sets) can also be **discrete**. Like choosing a path in an intersection. You can not have a combination of every path available and you need to pick one. Another example is when you go to a confectionery to choose a cake. You might see different options but yet you have to choose only one cake to bring home. Usually, these models are more mathematically complex compared to the continuous models.

### What are the discrete choice model’s components?

#### Decision maker (*n*)

A person that is making the decisions is considered to e the decision maker we are dealing with. We can also generalize a little bit and consider a group of people to be a decision-maker. In this case, we usually ignore the internal relationships and interactions between these people and consider them as a single entity.

It is obvious that there are different choices available for different decision-makers and they could have different tastes and preferences. Ultimately, we assume that decision-makers are making their choices based on their socioeconomic characteristics. In the case of transportation planning and travel behavior studies, characteristics like income, level of education, sex, age, and household size are being broadly used for various planning purposes.

#### Alternatives/Choices (*C* & *C*_{n})

As previously described, there are multiple-choice sets in front of the decision-makers. In the case of discrete choices, there is always a non-empty, finite, and yet countable set of alternatives. Although it may be possible to identify all possible alternatives available in a model (**universal choices** or * C*), usually the decision-maker may face a subset of them at the moment (

**individual choices**or

**).**

*C*_{n}#### Alternative attributes

We usually compare different alternatives by their characteristics. These characteristics are referred to as attributes. For example, we can consider price, travel time, and level of comfortability as attributes for a transportation modes discrete choice model. Through these attributes, the decision-maker compare different alternatives and choose the one which suits him/her.

We can see these attributes by their natures too. They can be categorized as discrete or continuous. even they might be Boolean. But some choices might have attributes that might be irrelevant to others. For example, when it comes to public transportation modes like transit buses, frequency plays an important role while it would be irrelevant to define such attributes for private cars or private bicycles.

### How the decision maker make his/her decisions?

The key to develop a good model is simplicity. A complicated model that tries to mimic individuals with irrational decision-making process probably is not useful for planning. In the case of choice modeling, we might need to consider our decision-makers rational and trying their best to act in their own interest. In other word, we could consider our decision-makers, **rational **and **self-interested** economic actors.

### How to implement rationality?

Consider two options called i and j from which we expect our decision makers choose:

Keeping these notations in mind, we can introduce some logics that will become handy in developing choice models.

#### Completeness

It simply means that choices are comparable. In other words, alternative *i* would either more preferable (** i>j**) or less preferable (

**) or as preferable as the alternative**

*i*<*j**j*(

**).**

*i*~*j*#### Transitivity

The logic is a global thing. It means that through logic it might be possible to infer the preferability of the new choices. in mathematical form, transitivity could be interpreted as:

if ** i >~ j** and

**then**

*j*>~*k*

*i*>~*k*#### Continuity

The attributes are considered continuous which means that if *k* and *i* are close values and * i >~ j* then we can conclude that

*.*

**k >~ j**### Utility theory

To explore the ideas behind the choice modeling, it might be helpful to discuss a little bit about another theory called the **Utility Theory**. The utility of a choice can be considered as a quality of worthiness or value of the choice to the decision-makers. As this post was meant to be focused on the discrete choice modeling basics, I will not go to the details.

#### Utility function

In many cases, it is possible to find a function that can predict the decision-makers’ probable choice using the choices’ attributes. The function we are talking about is the utility function.