Multinomial Choice Models for Product Demand

Why Model Multiple Choices?

A customer comparing insurance quotes may choose one insurer, one coverage package, or no purchase. A binary purchase model can describe whether a customer buys, but it does not by itself allocate choices among several available alternatives. A multinomial choice model represents the alternatives together, so their predicted probabilities sum to one and a price change can shift probability among the options.

For ratemaking and product-demand work, the model is useful when the outcome is a single choice from a defined set. Its central practical limitation is just as important as its convenience: the standard multinomial logit model imposes a particular pattern of substitution among alternatives. The probabilities and elasticities should therefore be interpreted with that restriction in mind.

Random Utility and the Multinomial Logit Model

Suppose a customer faces a set of available alternatives indexed by $j=1,\ldots,J$. One alternative may be an outside option, such as not buying any of the quoted policies. Let the customer’s utility from alternative $j$ be $U_j=V_j+\varepsilon_j$, where $V_j$ is systematic utility explained by observed attributes and $\varepsilon_j$ is unobserved utility.

Definition: In the multinomial logit model, the customer chooses the available alternative with the greatest utility. Conditional on observed attributes, the unobserved utility terms are independent and identically distributed Type I extreme-value (Gumbel) random variables. Their common scale is normalized to 1, since utility scale cannot be separately identified from the utility coefficients. Under these assumptions, the probability of choosing alternative $j$ is $$ P_j=\frac{\exp(V_j)}{\sum_{k=1}^{J}\exp(V_k)}. $$ The alternatives must represent mutually exclusive choices for the decision occasion, and the denominator includes all alternatives available to that customer.

The systematic utility $V_j$ can include alternative-specific constants and observed characteristics. For example, a price term could be $\beta_j p_j$, where $p_j$ is the premium and $\beta_j$ is the price coefficient for option $j$. A negative price coefficient means that, holding other terms fixed, a higher premium lowers that option’s systematic utility. Coefficients are measured relative to the normalized error scale; their numerical magnitudes depend on the units used for the predictors.

The formula returns a probability for each option, not a guarantee of an individual customer’s choice. For a group of otherwise comparable customers, the probabilities can also be read as predicted choice shares. If an option is unavailable, it should be excluded from that customer’s choice set rather than assigned a positive probability.

Worked Example: Calculating Shares for Two Insurers and an Outside Option

A hypothetical customer chooses between insurers A and B or does not purchase. Define $z_j=(p_j-1000)/100$ for each insurer, where $p_j$ is its annual premium in dollars. Let $V_A=\ln(4)-(\ln 2)z_A$, $V_B=\ln(4)-(\ln 2)z_B$, and let the outside option have utility $V_0=\ln(2)$. The customer’s unobserved utilities follow the independent Type I extreme-value assumptions above, with scale normalized to 1. At premiums $p_A=\$1{,}000$ and $p_B=\$1{,}100$, calculate choice probabilities. There is no interest, mortality, or insured-loss distribution in this choice calculation.

Step 1: Calculate systematic utilities. For A, $z_A=(1000-1000)/100=0$, so $V_A=\ln(4)$. For B, $z_B=(1100-1000)/100=1$, so $V_B=\ln(4)-\ln(2)=\ln(2)$. The outside option has $V_0=\ln(2)$.

Step 2: Exponentiate and sum. The exponentiated utilities are $4$, $2$, and $2$, respectively. Their sum is $4+2+2=8$.

Step 3: Divide each weight by the total. The probabilities are $P_A=4/8=0.5000$, $P_B=2/8=0.2500$, and $P_0=2/8=0.2500$. They sum to $0.5000+0.2500+0.2500=1.0000$, as required.

Method and trap avoided. The method is to exponentiate every available alternative’s utility and use their common denominator. A common mistake is to compare A and B but omit the outside option; doing so would incorrectly assign all choice probability to the insurers.

Price Elasticities of Choice Probabilities

A probability changes with a premium because the premium changes systematic utility and therefore changes the relative weight of that option. The following derivatives assume the model’s choice set and other utility terms are held fixed. Define the elasticity of $P_j$ with respect to $p_k$ as the percentage change in $P_j$ for a small percentage change in $p_k$:

\epsilon_{jk}=\frac{\partial P_j}{\partial p_k}\frac{p_k}{P_j}.

First consider a common price coefficient $\beta$ in every alternative’s utility, so $V_k$ contains $\beta p_k$, with $\beta<0$. Differentiating the logit probability gives $\partial P_j/\partial V_k=P_j(1-P_j)$ when $j=k$, and $\partial P_j/\partial V_k=-P_jP_k$ when $j\ne k$. The chain rule then gives:

\epsilon_{jj}=\beta p_j(1-P_j),\qquad \epsilon_{jk}=-\beta p_kP_k\quad(j\ne k).

The own-price elasticity is negative when $\beta<0$. The cross-price elasticity with respect to another option’s premium is positive: making option $k$ more expensive shifts choice probability toward the other alternatives. These are elasticities of choice probabilities. Translating them into revenue, written-premium, or profit effects requires additional quantities and assumptions.

The own- and cross-price formulas need not use a common coefficient. If the price coefficient in option $k$’s utility is $\beta_k$, differentiation with respect to $p_k$ gives:

\epsilon_{kk}=\beta_kp_k(1-P_k),\qquad \epsilon_{jk}=-\beta_kp_kP_k\quad(j\ne k).

Only the price coefficient for the option whose premium changes appears in these formulas. For example, option A’s probability response to B’s price depends on the price coefficient in B’s utility. The price variables and their units must be stated before interpreting a coefficient or elasticity.

Deriving the Elasticity and the IIA Restriction

The derivative formulas follow directly from the probability formula. For a price coefficient $\beta_k$, the denominator is $D=\sum_m\exp(V_m)$. Differentiating $P_j=\exp(V_j)/D$ with respect to $V_k$, for $j=k$ the quotient rule gives $P_j(1-P_j)$; for $j\ne k$, only the denominator changes, giving $-P_jP_k$. Since $\partial V_k/\partial p_k=\beta_k$, multiplying by $p_k/P_j$ produces the elasticities above.

A second result follows by taking the ratio of two choice probabilities:

\frac{P_j}{P_k} =\frac{\exp(V_j)}{\exp(V_k)} =\exp(V_j-V_k).

This ratio depends on the systematic utilities of $j$ and $k$, not on the utilities or number of other alternatives in the denominator. Consequently, adding or removing a third alternative leaves the predicted odds between $j$ and $k$ unchanged, provided the utilities of $j$ and $k$ themselves do not change. This property is called independence of irrelevant alternatives (IIA).

Theorem: Under multinomial logit, the ratio $P_j/P_k$ is independent of the other alternatives in the choice set, conditional on the systematic utilities of $j$ and $k$. This restriction follows from the independent, identically distributed Gumbel utility errors; it is not a separate empirical finding about every insurance market.

The proportional substitution implied by IIA can be unrealistic. If two options are nearly identical versions of one insurer’s product while a third is a substantially different coverage design, the two similar options may be closer substitutes for each other. Standard multinomial logit does not represent that similarity in its error structure. IIA is a property to assess when choosing and interpreting a model, not a reason to assume all alternatives are equally similar.

Worked Example: A Premium Change and the IIA Check

Use the hypothetical utilities from the first example. Initially, the exponentiated weights for A, B, and the outside option are $4,2,2$. B’s premium falls from $\$1{,}100$ to $\$1{,}000$, while A’s premium and all utility parameters remain fixed. Calculate the new probabilities and compare A’s odds relative to the outside option before and after the change. The Gumbel error scale remains 1; no interest, mortality, or loss distribution is involved.

Step 1: Update B’s utility weight. At the new premium, $z_B=(1000-1000)/100=0$, so $V_B=\ln(4)$ and $\exp(V_B)=4$. A’s weight remains 4 and the outside option’s weight remains 2.

Step 2: Recalculate the denominator and shares. The denominator is $4+4+2=10$. Thus $P_A=4/10=0.4000$, $P_B=4/10=0.4000$, and $P_0=2/10=0.2000$. The shares sum to 1.

Step 3: Check the odds restriction. Initially, $P_A/P_0=0.5000/0.2500=2$. After B’s price reduction, $P_A/P_0=0.4000/0.2000=2$. The A-to-outside odds stay fixed, even though both probabilities fall. By contrast, A-to-B odds change from $0.5000/0.2500=2$ to $0.4000/0.4000=1$.

Method and trap avoided. Recompute all probabilities using the updated common denominator, then check ratios rather than expecting unaffected probabilities. IIA says the odds between unchanged alternatives remain the same; it does not say their probabilities remain the same when another alternative’s utility changes.

Worked Example: Own- and Cross-Price Elasticities with Different Coefficients

A hypothetical choice model has insurers A and B and an outside option. At current premiums $p_A=\$1{,}200$ and $p_B=\$900$, let $V_A=\ln(2.5)-0.004(p_A-1200)$, $V_B=\ln(1.5)-0.002(p_B-900)$, and $V_0=0$. Prices are in dollars, so the coefficients are per dollar. Assume independent Type I extreme-value errors with normalized scale 1. Find the baseline probabilities, the four relevant elasticities, and the probabilities after B’s premium increases by $\$50$. No interest, mortality, or loss-distribution assumptions enter this model.

Step 1: Calculate baseline probabilities. At the stated premiums, the exponentiated utilities are $2.5$, $1.5$, and $1$. The denominator is $2.5+1.5+1=5$, so $P_A=0.5000$, $P_B=0.3000$, and $P_0=0.2000$.

Step 2: Calculate own-price elasticities. A’s coefficient is $\beta_A=-0.004$, so $\epsilon_{AA}=(-0.004)(1200)(1-0.5000)=-2.4000$. B’s coefficient is $\beta_B=-0.002$, so $\epsilon_{BB}=(-0.002)(900)(1-0.3000)=-1.2600$.

Step 3: Calculate cross-price elasticities. With respect to B’s premium, A’s elasticity is $\epsilon_{AB}=-\beta_Bp_BP_B=-(-0.002)(900)(0.3000)=0.5400$. With respect to A’s premium, B’s elasticity is $\epsilon_{BA}=-\beta_Ap_AP_A=-(-0.004)(1200)(0.5000)=2.4000$. Each cross effect uses the coefficient for the premium being changed.

Step 4: Recalculate probabilities after B’s premium increase. The change in B’s utility is $(-0.002)(50)=-0.1$, so B’s weight becomes $1.5\exp(-0.1)=1.3573$ (to four decimals). The new denominator is $2.5+1.3573+1=4.8573$. Therefore $P_A=2.5/4.8573=0.5147$, $P_B=1.3573/4.8573=0.2794$, and $P_0=1/4.8573=0.2059$, to four decimals. The rounded probabilities sum to 1.0000.

Method and trap avoided. Use the alternative-specific coefficient associated with the price being varied, and re-evaluate the softmax probabilities for a finite price change. A common error is to use the small-change elasticity as though it were an exact finite-change prediction.

Practical Interpretation and Common Traps

Multinomial logit is attractive because its probability formula is simple, it handles several mutually exclusive alternatives, and it provides interpretable price effects. In applications, however, the choice set and utility specification need to correspond to the decision being modeled. A quote-level record is not automatically a customer’s final choice: the analyst must define which options were available and what outcome represents the selected alternative.

Omitting the outside option: If nonpurchase is possible but omitted, the model forces every predicted choice to one of the included insurers.
Confusing odds and probabilities: IIA preserves the odds ratio between two unchanged alternatives, not either alternative’s probability in isolation.
Using the wrong price coefficient: A cross-price derivative with respect to option $k$’s premium uses the coefficient on $p_k$, not the coefficient for the alternative whose probability is being predicted.
Ignoring units: A coefficient per dollar differs numerically from one per $\$100$. Convert units before calculating a derivative or interpreting a coefficient.
Overstating the model’s substitution pattern: IIA is imposed by the standard model. If alternatives have materially different degrees of similarity, this restriction may make its substitution predictions inappropriate.

Key takeaway: Multinomial logit converts systematic utilities into probabilities over an available choice set. Its price derivatives provide own- and cross-price elasticities, while its probability ratios reveal the IIA restriction: adding another option does not alter the odds between two existing options whose utilities are unchanged.

Check Your Understanding

Work each problem from the stated choice set and utility assumptions. Probabilities are given to four decimals where needed.

A hypothetical customer chooses insurer A, insurer B, or no purchase. The exponentiated systematic utilities are $5,3,2$, respectively. What are the three multinomial logit probabilities? (Answer: $P_A=0.5000,\ P_B=0.3000,\ P_0=0.2000$.)
In a common-coefficient model, $\beta=-0.005$ per dollar. Insurer B’s premium is $\$1{,}200$ and its choice probability is 0.35. Calculate B’s own-price elasticity and insurer A’s cross-price elasticity with respect to B’s premium. (Answer: $\epsilon_{BB}=-3.9000,\ \epsilon_{AB}=0.0021(1000)?$)
Two unchanged alternatives A and the outside option have probabilities $0.45$ and $0.15$. A new alternative is added, and their probabilities become $0.30$ and $0.10$. Calculate their odds ratio before and after, and state whether the result is consistent with IIA. (Answer: The ratio is 3 before and after; yes, it is consistent with IIA.)
At current prices, the choice weights of A, B, and no purchase are $3,4,1$. B’s premium rises enough to reduce its utility weight by one-half; the other weights are unchanged. Find the new probabilities. (Answer: $P_A=0.5000,\ P_B=0.3333,\ P_0=0.1667$.)
For insurer A, $p_A=\$1{,}000$, $P_A=0.40$, and $\beta_A=-0.003$ per dollar. For insurer B, $p_B=\$800$, $P_B=0.25$, and $\beta_B=-0.004$ per dollar. In a model with alternative-specific price coefficients, calculate A’s own elasticity and A’s cross-price elasticity with respect to B’s premium. (Answer: $\epsilon_{AA}=-1.8000,\ \epsilon_{AB}=0.8000$.)

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