Substitution Patterns In Multinomial Logit

Why Utility Changes Matter for Substitution

When an alternative becomes more attractive, its choice probability rises, but the probabilities of the other alternatives must fall: exactly one alternative is chosen. The multinomial logit model gives a precise account of how this probability is reallocated. Its distinctive implication is that a utility change in one alternative draws probability from the others in proportion to their existing choice probabilities.

This result is about changes in the alternatives’ utility indices, holding the choice set and all other utilities fixed. It is not yet a formula for how a change in a covariate affects choice probabilities; that requires connecting the covariate to utilities. The distinction matters because the response to a covariate depends both on the utility specification and on the probability response derived here.

Model and Utility-Response Derivatives

Definition: For a decision maker facing a fixed choice set $j=1,\ldots,J$, let $V_j$ be alternative $j$'s systematic utility. The multinomial logit model assumes that the random utility errors are independent and identically distributed type-I extreme value variables, with their scale normalized. Its choice probabilities are $$ P_j=\frac{\exp(V_j)}{\sum_{k=1}^{J}\exp(V_k)}. $$ A utility response changes one or more $V_j$'s while holding the choice set and the other specified quantities fixed.

The scale normalization is important: utility is measured in the model's normalized utility units. The response derivatives below are per unit change in those units. A common additive shift in all utilities cannot affect probabilities, because it multiplies every exponential by the same factor.

To find how $P_i$ responds to $V_j$, differentiate the probability formula. The result is

\frac{\partial P_i}{\partial V_j} =P_i\left(\mathbf{1}\{i=j\}-P_j\right) = \begin{cases} P_i(1-P_i), & i=j,\\ -P_iP_j, & i\ne j. \end{cases}

Here $\mathbf{1}\{i=j\}$ equals one when $i=j$, and zero otherwise. Thus an alternative's own utility has a positive effect on its probability, while a different alternative's utility has a negative effect on it. The derivative is local: it describes the rate of change at the current utility values, not necessarily the exact effect of a large finite change.

Worked Example: A Small Increase in One Alternative’s Utility

Suppose three hypothetical alternatives A, B, and C have current choice probabilities $0.50$, $0.30$, and $0.20$. Consider a small increase in B's utility, with A's and C's utilities unchanged.

For B, the own-utility derivative is $$ \frac{\partial P_B}{\partial V_B}=P_B(1-P_B)=0.30(0.70)=0.21. $$ For A and C, the cross-utility derivatives are $$ \frac{\partial P_A}{\partial V_B}=-P_AP_B=-(0.50)(0.30)=-0.15, \qquad \frac{\partial P_C}{\partial V_B}=-P_CP_B=-(0.20)(0.30)=-0.06. $$

For an increase of $0.01$ normalized utility units, the first-order approximations are therefore a $0.0021$ rise in B's probability, a $0.0015$ fall in A's, and a $0.0006$ fall in C's. These are changes of 0.21, 0.15, and 0.06 percentage points, respectively. They sum to zero, as probability changes must.

Why Probability Losses Are Proportional

For a utility increase in alternative $j$, every other alternative $i$ loses probability at a rate of $P_iP_j$ per unit increase. The common factor $P_j$ means that, among the alternatives losing probability, their losses are proportional to their own current probabilities. For two alternatives $i$ and $k$, both different from $j$,

\frac{-\partial P_i/\partial V_j}{-\partial P_k/\partial V_j} =\frac{P_iP_j}{P_kP_j} =\frac{P_i}{P_k}.

This local proportionality has an exact finite-change counterpart. Let the utility of $j$ rise by $\delta$, with every other utility fixed, and set $q=\exp(\delta)$. The numerator for $j$'s probability is multiplied by $q$, while the original denominator is increased by $(q-1)\exp(V_j)$. Dividing by the original denominator gives

P_j'=\frac{qP_j}{1+(q-1)P_j}, \qquad P_i'=\frac{P_i}{1+(q-1)P_j}\quad(i\ne j).

Consequently, each other alternative's probability changes by the same multiplicative factor. If $\delta>0$, they all lose probability, and the ratio of any two losing probabilities is unchanged:

\frac{P_i-P_i'}{P_k-P_k'}=\frac{P_i}{P_k} \qquad (i,k\ne j).

Equivalently, the probability taken from the other alternatives is distributed among them according to their baseline probability shares. A utility decrease in $j$ reverses the direction: the alternatives other than $j$ gain probability in those same proportions.

Worked Example: Exact Substitution After a Finite Utility Increase

Use the hypothetical probabilities $P_A=0.50$, $P_B=0.30$, and $P_C=0.20$. Increase B's utility by $0.10$ normalized units and leave A's and C's utilities unchanged. The multiplier is $$ q=\exp(0.10)=1.10517, \qquad D=1+(q-1)P_B=1+(0.10517)(0.30)=1.03155. $$

The new probabilities are $$ P_B'=\frac{(1.10517)(0.30)}{1.03155}=0.3214, \qquad P_A'=\frac{0.50}{1.03155}=0.4847, \qquad P_C'=\frac{0.20}{1.03155}=0.1939. $$ They sum to approximately 1.0000. B gains $0.0214$, or 2.14 percentage points. A loses $0.0153$, or 1.53 percentage points, and C loses $0.0061$, or 0.61 percentage points.

The losses are in the ratio $0.0153:0.0061$, approximately $2.5:1$, which matches A's and C's baseline probability ratio $0.50:0.20=2.5:1$. The exact finite change therefore illustrates the same proportional allocation as the derivative, without relying on a small-change approximation.

Several Utilities Can Change Together

The derivative formula also gives the response when several utilities change at once. For small changes $dV_j$, the first-order probability change for alternative $i$ is

dP_i =\sum_{j=1}^{J}\frac{\partial P_i}{\partial V_j}\,dV_j =P_i\left(dV_i-\sum_{j=1}^{J}P_j\,dV_j\right).

The quantity $\sum_j P_jdV_j$ is the probability-weighted average of the utility changes. An alternative's probability rises to first order when its own utility change exceeds this average, and falls when it is below the average. This expression also makes the invariance to a common shift explicit: if every utility changes by the same amount $c$, then $dV_i-\sum_j P_jdV_j=c-c=0$ for every $i$.

Worked Example: Utility Changes in More Than One Alternative

Suppose hypothetical alternatives A, B, and C have probabilities $0.20$, $0.50$, and $0.30$. Consider small utility changes $dV_A=0.40$, $dV_B=0$, and $dV_C=0$, in normalized utility units. The probability-weighted average change is $$ \sum_jP_jdV_j=(0.20)(0.40)+(0.50)(0)+(0.30)(0)=0.08. $$

The first-order changes are therefore $$ dP_A=0.20(0.40-0.08)=0.064,\qquad dP_B=0.50(0-0.08)=-0.040,\qquad dP_C=0.30(0-0.08)=-0.024. $$ The changes sum to $0.064-0.040-0.024=0$. To first order, A gains 6.4 percentage points, while B and C lose 4.0 and 2.4 percentage points. The losses are in the ratio $4.0:2.4=0.50:0.30$, the baseline probability ratio of B to C.

These are derivative-based approximations for small changes, not exact finite-change predictions. In particular, the numerical changes are large enough that a user applying them to actual probabilities should check whether the small-change approximation is adequate.

Derivation and Probability Conservation

The derivative follows directly from the quotient rule. Write $S=\sum_k\exp(V_k)$, so that $P_i=\exp(V_i)/S$. Differentiating the numerator with respect to $V_j$ gives $\mathbf{1}\{i=j\}\exp(V_i)$; differentiating the denominator gives $\exp(V_j)$. Hence

\frac{\partial P_i}{\partial V_j} = \frac{\mathbf{1}\{i=j\}\exp(V_i)S-\exp(V_i)\exp(V_j)}{S^2} = P_i\left(\mathbf{1}\{i=j\}-P_j\right).

Adding the response across all outcomes to a change in $V_j$ yields

\sum_i\frac{\partial P_i}{\partial V_j} =P_j-\sum_iP_iP_j =P_j-P_j\sum_iP_i =0,

because the choice probabilities sum to one. Thus the increase in the probability of $j$ is exactly balanced by decreases in the probabilities of the other alternatives, in the derivative sense.

Interpretation and a Common Pitfall

The proportional substitution rule is both useful and restrictive. It says that, if one alternative becomes more attractive, the model assigns losses among all the other alternatives in proportion to their probabilities before the change. An alternative with twice the baseline probability of another loses twice as much probability in response to the same utility increase. This is a property of the multinomial logit probability formula, not a general law of choice behavior or a claim that observed individuals literally switch in those proportions.

The result depends on the model assumptions and the specified choice set. It also concerns utility changes themselves. If a covariate changes, its effect on probabilities depends on how that covariate enters each alternative's utility, as well as on the current probabilities. A coefficient cannot be read directly as a probability change. The response derivatives are one ingredient for that later calculation, not a substitute for it.

Common mistake: “Alternative B's utility rises, so the probability it gains must come equally from every other alternative.”
Correction: Its probability gain is balanced by losses, but those losses are not generally equal. Under multinomial logit, each losing alternative's loss is proportional to its baseline probability.
Common mistake: “The cross-utility derivative is the effect of changing any predictor.”
Correction: It is the effect of changing $V_j$. A predictor's probability effect also depends on its utility coefficients and on which alternatives' utilities it changes.

Key takeaway: In multinomial logit, $\partial P_i/\partial V_i=P_i(1-P_i)$, while $\partial P_i/\partial V_j=-P_iP_j$ for $i\ne j$. A change in one alternative's utility reallocates probability across the others in proportion to their current probabilities.

Check Your Understanding

Use the multinomial logit utility-response formulas to answer these questions.

If an alternative has probability $0.40$, what is its own-utility derivative? Interpret the result per unit increase in its normalized utility, holding the other utilities fixed.
Three alternatives have probabilities $0.20$, $0.50$, and $0.30$. If the first alternative's utility rises, what is the ratio of the local probability losses for the second and third alternatives?
Why must the utility-response derivatives across all alternatives sum to zero for a change in one utility?
For a finite utility increase $\delta$ in alternative $j$, state the exact updated probability formula for an alternative $i\ne j$.
Explain why a utility-response derivative is not automatically the probability effect of changing a covariate.