Random Utility, Choice Sets, and Normalization

Conceptual map

IV-01.01Consumers and alternatives
IV-01.02Random utility
IV-01.03Logit probabilities
IV-01.04Normalization
IV-01.05Executable model objects

1. Choice probabilities follow from a complete utility contract

Alternatives, systematic utilities, shock law, location, and scale must be fixed before probabilities or elasticities are interpreted.

Consumer $$i$$ chooses one alternative from a finite nonempty set $\mathcal{J}$ . Utility is $U_{ij}=V_{ij}+\sigma$ $\varepsilon _{ij}$ , where $\sigma>0$ and the $\varepsilon _{ij}$ are iid standard type-I extreme value. Ties have probability zero. The maximum-utility rule gives

Proposition 1 · Multinomial logit probability

\[P_{ij}=\operatorname{exp}(V_{ij}/\sigma )/\sum _{k\in \mathcal{J}}\operatorname{exp}(V_{ik}/\sigma ).\](1)

Proof sketch. Condition on $\varepsilon _{ij}$ and integrate the product of extreme-value distribution functions for every competing shock. The resulting integral equals the softmax expression. ∎

Differentiation yields $\partial P_{ij}/\partial V_{ik}=P_{ij}(1{j=k}-P_{ik})/\sigma$ .

2. Location invariance and scale normalization are distinct

Adding a common $c_{i}$ to every $V_{ij}$ cancels from numerator and denominator in (1). Choice data therefore identify utility differences. One may set an outside-option utility to zero or impose another location normalization.

Changing $\sigma$ alters $V/\sigma$ and generally changes probabilities and derivatives. Multiplying both systematic utility and $\sigma$ by the same positive constant leaves choices unchanged; changing one while holding the other fixed changes the model.

3. A three-alternative calculation

Take systematic utilities (0,1,2). At $\sigma=1$ , shares are approximately (0.09003,0.24473,0.66524). Adding five to every utility produces the same shares. At $\sigma=2$ , shares become approximately (0.18632,0.30720,0.50648). The last vector is less concentrated because utility differences are smaller relative to shock scale.

A numerically stable implementation subtracts $\operatorname{max}_{j}V_{ij}/\sigma$ before exponentiation. This changes neither ratios nor shares and prevents overflow. The same stabilization must be applied within each observation-specific choice set.

4. Choice-set and scale boundaries

Failure case · Coefficient compared across different shock scales

A price coefficient of $$-1$$ with $\sigma=1$ implies twice the utility sensitivity of the same coefficient with $\sigma=2$ . Coefficients require a scale convention before behavioral comparison.

An empty choice set leaves the maximization undefined. An alternative assigned systematic utility $-\infty$ has zero share as a limit and should be removed from finite exponentiation. As $\sigma$ approaches zero, choices concentrate on systematic-utility maximizers and deterministic ties need a tie rule; as $\sigma$ grows, finite utility differences vanish relative to shocks and shares approach uniformity over the available set.

5. Implementation, exercises, and sources

Store the choice set by observation, utility formula, outside option, shock family, location normalization, and scale. Verify shares are nonnegative and sum to one, compare analytic derivatives with finite differences, and keep simulation draws fixed across counterfactuals.

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Exercises

Reproduce all shares in Section 3.
Verify the derivative formula and its zero column sum.
Take $\sigma$ toward zero under a unique maximizer and under a systematic tie.

Partial solutions

1. Divide (1, $e,e^{2}$ ) or (1, $e^{1/2},e$ ) by its sum. 2. Summing derivatives over $$j$$ gives $P_{ik}-P_{ik}\sum P_{ij}=0$ . 3. A unique maximizer receives limiting probability one; tied maximizers split mass under symmetric iid shocks.

Daniel McFadden (1974), “Conditional Logit Analysis of Qualitative Choice Behavior,” in Frontiers in Econometrics.Random-utility logit probabilities.
Kenneth Train, Discrete Choice Methods with Simulation, Chapters 2–3.Utility normalization and simulation.
Moshe Ben-Akiva and Steven Lerman, Discrete Choice Analysis.Choice sets and identification conventions.

6. Audit checkpoint

State consumers, nonempty choice sets, systematic utility, shock distribution, location and scale normalizations, tie behavior, outside option, unavailable alternatives, numerical stabilization, and probability and derivative residuals.

7. Scope boundary

The chapter develops finite multinomial logit as the executable baseline. Random coefficients and nested dependence require additional integration and identification conditions.