Discrete Choice Models

Definition

A discrete choice model describes a decision-maker's selection from a finite set of mutually exclusive alternatives $\{1, \ldots, J\}$. The random utility model (McFadden 1974) specifies utility $U_{ij} = V_{ij} + \varepsilon_{ij}$ for alternative $j$, where $V_{ij}$ is the systematic component (function of observed attributes and individual characteristics) and $\varepsilon_{ij}$ is an unobserved idiosyncratic shock. The decision-maker chooses $j^* = \arg\max_j U_{ij}$.

Key Ideas

• Logit model: if $\varepsilon_{ij} \overset{\text{iid}}{\sim}$ Type-I Extreme Value, $\Pr(j^*=j) = \exp(V_{ij}) / \sum_k \exp(V_{ik})$ (McFadden 1973). Computationally tractable but exhibits the Independence of Irrelevant Alternatives (IIA) property.
• Probit model: if $(\varepsilon_{i1},\ldots,\varepsilon_{iJ}) \sim \mathcal{N}(0,\Sigma)$, the choice probability is a multivariate normal integral; IIA is avoided when $\Sigma$ has off-diagonal elements. See Multinomial Probit.
• Mixed logit / random-coefficient logit: allows parameters $\beta$ to vary randomly across individuals from a mixing distribution $p(\beta|\tau)$; nests logit and is IIA-free. Estimation by simulation (Geweke-Hajivassiliou-Keane (GHK) simulator) or Hierarchical Bayes (HB).
• IIA problem: in standard logit, the ratio of probabilities for any two alternatives depends only on those two alternatives, not on the rest of the choice set. This generates counter-intuitive predictions when alternatives are close substitutes.

How It Works

Estimation typically proceeds by maximum likelihood (probit/logit) or Markov Chain Monte Carlo (MCMC) for hierarchical Bayes specifications. Individual-specific parameters are recovered from the posterior $p(\beta_i | y_i, \tau)$ via Gibbs sampling.

Why It Matters

Discrete choice models are the workhorse of applied microeconomics and marketing, enabling estimation of willingness to pay, elasticities, and counterfactual market shares from observed choices.

Related

• Multinomial Probit
• Binary Probit
• Conjoint Analysis
• Nonhomothetic Preferences
• Simulation-Based Estimation
• Consumer Heterogeneity