The multinomial logit (MNL) model, derived by McFadden (1974, Nobel Prize 2000), is the workhorse model of discrete choice in economics, marketing, and transportation. It expresses the probability of choosing each alternative as a softmax function of systematic utility, and is mathematically equivalent to Luce's Choice Axiom applied to utilities measured on a ratio scale.
The Model
εᵢ ~ iid Gumbel (Type I extreme value)
P(choose i) = exp(Vᵢ) / Σⱼ exp(Vⱼ)
Vᵢ = β₁x_{i1} + β₂x_{i2} + ... (linear in parameters)
IIA Property and Limitations
The MNL inherits the IIA property from its Gumbel error assumption: the odds ratio P(i)/P(j) = exp(Vᵢ − Vⱼ) is independent of other alternatives in the choice set. This makes estimation simple but produces the "red bus/blue bus" problem — adding a second bus route halves the market share of the existing bus rather than drawing proportionally from all modes. Nested logit, mixed logit, and probit models relax IIA by allowing correlated errors.
In psychology, the softmax function appears in reinforcement learning models (as the action selection rule), neural network models (as the output layer for classification), and rational models of categorization. The temperature parameter τ in softmax P(i) = exp(Vᵢ/τ)/Σexp(Vⱼ/τ) controls the tradeoff between exploitation (low τ, choosing the best option) and exploration (high τ, choosing randomly).