IIA, Substitution, and the Logit Demand Pitfall

Why Substitution Patterns Matter

A demand model does more than predict which products sell today. It also predicts where consumers go when a product becomes more expensive or disappears. Those predictions matter for product design, market definition, and counterfactual analysis. The multinomial logit model is tractable, but it imposes a particularly restrictive rule: substitution away from one product is allocated to the remaining alternatives in proportion to their existing shares.

This restriction is the independence of irrelevant alternatives, or IIA. It does not mean that consumers regard every product as equally attractive. Rather, it means that the odds of choosing any two particular products do not depend on which other alternatives are available. That property can make predictions implausible when some products are much closer substitutes for one another than for the rest of the market.

Logit Choice and the IIA Property

Consider a consumer choosing one alternative from a set $J$, which includes an outside option such as not purchasing. Let $U_{ij}$ be consumer $i$'s utility from alternative $j$. In the standard multinomial logit model, utility is the sum of a deterministic component $\delta_j$ and a random component $\varepsilon_{ij}$:

U_{ij}=\delta_j+\varepsilon_{ij}.

The model assumes that, for each consumer, the $\varepsilon_{ij}$ terms are independent and identically distributed Type I extreme value errors across alternatives, with a common scale normalized to one. The consumer chooses the available alternative with the highest utility. Under these assumptions, the probability of choosing $j$, also called its choice share in this setting, is

s_j=\frac{\exp(\delta_j)}{\sum_{k\in J}\exp(\delta_k)}.

The deterministic utility $\delta_j$ summarizes the product's systematic attractiveness, including observed characteristics and price effects. The formula applies to the choice set that includes the outside option. It is the same logit structure used for market shares and utility inversion in earlier material; the focus here is what it implies about substitution.

Definition: The independence of irrelevant alternatives (IIA) property is the claim that, for any two available alternatives $j$ and $k$, their choice-probability ratio does not depend on the attributes or availability of other alternatives: $$ \frac{s_j}{s_k}=\exp(\delta_j-\delta_k). $$

The right-hand side depends only on the deterministic utilities of $j$ and $k$. Adding, removing, or changing another option does not change their relative odds, as long as $j$ and $k$ themselves are unchanged and available. IIA is therefore a restriction on the pattern of substitution, not a claim that the alternatives have identical utilities or identical shares.

Why IIA Implies Proportional Substitution

Write $w_j=\exp(\delta_j)$ for the positive weight of alternative $j$. Its share is its weight divided by the sum of all available weights. Suppose alternative $r$ is removed, with all other alternatives' utilities unchanged. Before removal, the total weight is $W=\sum_{k\in J}w_k$, and after removal it is $W-w_r$. For any remaining alternative $j$, its old and new shares are

s_j=\frac{w_j}{W}, \qquad s_j^{(-r)}=\frac{w_j}{W-w_r} =\frac{s_j}{1-s_r}.

Thus every remaining alternative's share is multiplied by the same factor, $1/(1-s_r)$. The amount each gains depends on its old share: alternatives with larger initial shares gain more, in the same proportions as their initial shares. The outside option is one of these alternatives, so it also receives part of the reallocated share.

Worked Example: Removing a Product From a Market

Suppose a hypothetical market has three products, A, B, and C, and an outside option. Their initial shares are 0.300, 0.200, 0.100, and 0.400, respectively. Product A is withdrawn. Under logit, calculate the remaining shares and the gains for each alternative.

The remaining alternatives have total initial share $1-0.300=0.700$. Each remaining share is divided by 0.700:

s_B^{(-A)}=\frac{0.200}{0.700}=\frac{2}{7}\approx0.2857, \quad s_C^{(-A)}=\frac{0.100}{0.700}=\frac{1}{7}\approx0.1429, \quad s_0^{(-A)}=\frac{0.400}{0.700}=\frac{4}{7}\approx0.5714.

Here $s_0$ denotes the outside-option share. B gains $2/7-1/5=3/35\approx0.0857$, C gains $1/7-1/10=3/70\approx0.0429$, and the outside option gains $4/7-2/5=6/35\approx0.1714$. These gains sum to $0.300$, exactly A's initial share. They are divided in the same 2:1:4 ratio as the initial shares of B, C, and the outside option.

The calculation is a model prediction, not evidence that actual consumers would switch in these proportions. Whether the prediction is credible depends on how plausible IIA is for this market.

The same proportional rule applies when a product becomes less attractive rather than disappearing, although the exact share changes are then calculated using the new utilities and the logit formula. A price increase that reduces one product's deterministic utility lowers its weight. Because the other products' weights are unchanged, all their shares rise by the same proportion relative to their own starting shares.

Worked Example: A Price Change That Halves One Product's Weight

Suppose four alternatives—A, B, C, and the outside option—have initial logit weights 3, 2, 1, and 4. The total weight is 10, so the initial shares are 0.300, 0.200, 0.100, and 0.400. A hypothetical price increase reduces A's weight from 3 to 1.5, while the other weights stay fixed. Find the new shares and the change for each alternative.

The new total weight is $1.5+2+1+4=8.5$. The shares are therefore

s_A'=\frac{1.5}{8.5}=\frac{3}{17}\approx0.1765,\quad s_B'=\frac{2}{8.5}=\frac{4}{17}\approx0.2353, $$ $$ s_C'=\frac{1}{8.5}=\frac{2}{17}\approx0.1176,\quad s_0'=\frac{4}{8.5}=\frac{8}{17}\approx0.4706.

A's share falls by $3/10-3/17=21/170\approx0.1235$. B gains $4/17-1/5=3/85\approx0.0353$; C gains $2/17-1/10=3/170\approx0.0176$; and the outside option gains $8/17-2/5=6/85\approx0.0706$. The gains sum to $21/170$, A's share loss. B, C, and the outside option gain in proportion to their initial shares, $0.200:0.100:0.400$, or $2:1:4$.

This example illustrates a finite change in predicted shares. It does not establish that the price change caused a particular observed market outcome; that would require an appropriate demand analysis and assumptions about the price-setting process.

When the Substitution Rule Looks Implausible

Consider a hypothetical market with two nearly identical versions of a product, A and B, a distinct rival C, and the outside option. Suppose all four have equal initial shares of 0.250. If B disappears, logit predicts that A, C, and the outside option each rise to $1/3$, or about 0.3333. Each gains $1/3-1/4=1/12\approx0.0833$, so each receives one-third of B's lost share.

That allocation follows from IIA: the remaining alternatives had equal initial shares. Yet if A and B are almost identical while C is substantially different, it may be more plausible that many of B's former customers switch to A than to C or to not purchasing. Logit cannot express that preference pattern through its proportional substitution rule alone. The example does not prove that the equal allocation is always wrong; it identifies a strong modeling assumption that should be checked against the products and the intended counterfactual.

The restriction can matter even when a model fits current market shares. Shares describe one observed allocation; counterfactual predictions depend on how the model allocates consumers when the choice set or product utilities change. A model can reproduce baseline shares and still imply questionable substitution.

Worked Example: Checking Whether Proportional Reallocation Is Credible

Suppose a hypothetical market has two close substitutes, X and Y, plus a distinct product Z and an outside option. Initial shares are 0.240 for X, 0.160 for Y, 0.300 for Z, and 0.300 for the outside option. If X disappears, calculate the logit prediction and assess what feature of it deserves scrutiny.

The remaining share is $1-0.240=0.760$. The predicted shares are

s_Y^{(-X)}=\frac{0.160}{0.760}\approx0.2105,\quad s_Z^{(-X)}=\frac{0.300}{0.760}\approx0.3947,\quad s_0^{(-X)}=\frac{0.300}{0.760}\approx0.3947.

Y gains about $0.2105-0.160=0.0505$. Z gains about $0.3947-0.300=0.0947$, and the outside option also gains about $0.0947$. Together these gains are approximately 0.240, X's initial share. The prediction allocates less of X's lost share to its close substitute Y than to either Z or the outside option because Y's initial share was smaller.

That feature is the diagnostic. If market knowledge or other evidence suggests that X's customers are especially likely to choose Y, the logit prediction may be too diffuse. The baseline shares alone do not tell us whether that substitution pattern is appropriate.

What the Limitation Motivates

The issue is not that logit is unusable. Its closed-form choice probabilities and share relationships make it a valuable benchmark, and its assumptions may be reasonable in some applications. The issue is that the standard model represents consumers with the same deterministic utility differences and imposes IIA through independent utility shocks. It cannot separately encode that some consumers regard X and Y as very similar while others have different preferences.

A richer demand model can allow preferences to vary across consumers, so that different consumers can rank products differently. When these consumer-level choice probabilities are aggregated, substitution patterns need not follow the simple proportional rule of standard logit. Modeling this heterogeneity requires additional assumptions and estimation work; the central point here is why such extensions can be useful, not how to estimate them.

Common Mistake and Key Takeaway

Wrong reasoning: “IIA means every remaining product gains the same amount when one product disappears.” Correction: IIA predicts gains in proportion to the remaining alternatives' initial shares. Equal gains occur only when those initial shares are equal.
Wrong reasoning: “The withdrawn product's customers all switch to the closest substitute.” Correction: standard logit does not identify a uniquely closest substitute; it allocates lost share across all remaining alternatives, including the outside option, according to their initial shares.
Wrong reasoning: “A good fit to observed shares guarantees credible substitution predictions.” Correction: matching a baseline allocation does not by itself validate the model's counterfactual reallocation rule.

Key takeaway: Multinomial logit fixes pairwise choice odds independently of other alternatives. Consequently, when one product's attractiveness falls or it is removed, the remaining alternatives gain share in proportion to their previous shares. This is a transparent benchmark—and a potentially serious limitation when consumers view some products as especially close substitutes.

Check Your Understanding

Answer each question using the logit substitution rule.

State IIA in terms of the ratio of two choice probabilities. Which utilities determine that ratio?
A product with share 0.20 is removed. What factor multiplies each remaining alternative's share, including the outside option?
Why does IIA not imply that all alternatives gain the same number of share points?
Two equal-share products are close substitutes, while a third product is very different. What proportional reallocation does logit predict if one of the close substitutes disappears?
Why can a model that reproduces observed market shares still produce implausible counterfactual substitution?