Why a Pairwise Odds Restriction Matters
A person choosing among several options does not usually assign each option a probability in isolation: the probabilities must add to one across the available choices. Adding an option can therefore change the probability of choosing an existing one. Multinomial logit imposes a sharper restriction: although those probabilities can change, the odds between any two existing alternatives stay fixed when other options are added or removed, provided the pair’s systematic utilities remain unchanged. This restriction is called the independence of irrelevant alternatives property, or IIA.
“Independence” here does not mean that people choose independently, nor does “irrelevant” mean that a new option has no effect on anyone’s choice probability. It refers specifically to the pairwise probability ratio: in the model, that ratio does not depend on which other alternatives are available. The restriction is a feature of multinomial logit, not a general law of individual choice.
Multinomial Logit and the IIA Definition
For decision maker \(i\), let \(C_i\) be the set of alternatives available to choose from. Write \(V_{ij}\) for the systematic utility of alternative \(j\), and let \(P_{ij}\) be the probability that decision maker \(i\) chooses \(j\). In the standard multinomial logit model, each available alternative has a systematic utility and an alternative-specific random component; the random components are independent and identically distributed type-I extreme-value variables. Under this assumption, the choice probability is
The denominator sums the exponentiated systematic utilities over the entire available choice set. The probabilities are positive and sum to one as long as the set is finite and each \(V_{ij}\) is finite. In the usual random-utility interpretation, \(V_{ij}\) may depend on observed characteristics of the person and alternative, as well as estimated model parameters.
For a multinomial outcome, \(P_{ij}/P_{ik}\) is the relative probability, or odds, of choosing \(j\) rather than \(k\). It is not the same object as either probability alone. IIA says the odds are invariant; it does not say that \(P_{ij}\) or \(P_{ik}\) must be invariant.
Deriving the Pairwise Odds
The result follows directly from the multinomial logit probability formula. Take two alternatives \(j\) and \(k\) that are both available. Divide their probabilities. Each has the same denominator, the sum over \(C_i\), so that common term cancels.
No utility for a third alternative appears in the final expression. If an alternative \(r\) is added, its \(\exp(V_{ir})\) enters both probability denominators, and cancels in the ratio just as the other denominator terms do. If \(r\) is removed, its term disappears from both denominators and still cancels. Thus the ratio is unchanged so long as \(V_{ij}\) and \(V_{ik}\) themselves are unchanged.
This last condition is important. If the model specifies that a pair’s systematic utilities change when the menu changes, then the ratio can change through \(V_{ij}-V_{ik}\). That is not a contradiction of the algebra: it is a change to the quantities being held fixed in the IIA comparison. The standard IIA statement compares choice sets while holding the pair’s systematic utilities and model parameters fixed.
Worked Example: Adding a Meal Option
Suppose a hypothetical meal-choice model assigns exponentiated systematic utilities of 6 to soup and 3 to salad. Initially these are the only two available choices. A sandwich is then added, with exponentiated systematic utility 3. Calculate the probabilities before and after the addition, and compare the soup-to-salad odds.
Before the sandwich is added. The denominator is \(6+3=9\). Thus the soup and salad probabilities are \(6/9=2/3\) and \(3/9=1/3\). Their odds are
After the sandwich is added. The new denominator is \(6+3+3=12\). The probabilities are \(6/12=1/2\) for soup, \(3/12=1/4\) for salad, and \(3/12=1/4\) for the sandwich. The soup-to-salad odds are
The probabilities of soup and salad both fell, but their ratio stayed at 2. That is IIA. The calculation assumes the addition of the sandwich does not change the systematic utilities assigned to soup or salad.
Worked Example: Adding a Similar Transport Option
Consider a hypothetical choice among the Blue Bus, Red Bus, and walking. Suppose the exponentiated systematic utilities are 2, 2, and 4, respectively. A new Green Bus option is introduced with exponentiated systematic utility 2. Compare the combined probability of choosing a bus before and after the new option, and calculate the Blue-to-Red odds.
Before the Green Bus is available, the denominator is \(2+2+4=8\). The Blue Bus and Red Bus probabilities are each \(2/8\), and walking has probability \(4/8\). The probability of choosing either bus is
The Blue-to-Red odds are \((2/8)/(2/8)=2/2=1\).
After the Green Bus is added, the denominator is \(2+2+2+4=10\). Each bus has probability \(2/10\), and walking still has probability \(4/10\). The combined probability of choosing a bus is
The Blue-to-Red odds are \((2/10)/(2/10)=2/2=1\), unchanged from before. In this example the probability of choosing some bus rises from \(0.5000\) to \(0.6000\), while the pairwise odds between the Blue and Red buses remain fixed. The calculation illustrates the distinction between a probability for one option or group and the odds for a particular pair.
Worked Example: Removing an Alternative
A hypothetical choice set contains three alternatives, A, B, and C, with exponentiated systematic utilities 5, 2, and 3. Alternative C becomes unavailable, while the utilities of A and B remain fixed. Find the A-to-B odds before and after C is removed, and the probabilities after removal.
Before removal, the denominator is \(5+2+3=10\), so \(P_A=5/10\) and \(P_B=2/10\). Their odds are
After removal, the denominator is \(5+2=7\). The probabilities become \(P_A=5/7\) and \(P_B=2/7\), giving
The individual probabilities changed because the available choices now consist only of A and B. But their ratio did not change. This result depends on treating the removal as a change in availability alone, without changing either remaining alternative’s systematic utility.
What IIA Does and Does Not Say
The cancellation in the derivation is exact, but its interpretation should be kept within scope. IIA is a restriction on the probabilities generated by the multinomial logit model. It says that the relative odds for a pair are determined by their systematic utility difference, not by the other available alternatives. If that difference is zero, the pairwise odds equal one; if it is positive, the odds favor \(j\) over \(k\); and if it is negative, they favor \(k\).
IIA does not mean that an added alternative has no effect on the choice probabilities of existing options. Since the denominator changes, those probabilities generally do change. Nor does IIA mean that options that seem very similar to a person are treated as identical. The model’s pairwise odds are governed by the specified systematic utilities and the common multinomial logit structure.
The property can be a strong modelling restriction. For instance, if a new option is a close variant of one existing alternative, an analyst might expect it to draw disproportionately from that similar option rather than from every alternative in proportion to the multinomial logit probabilities. IIA rules out such menu-dependent changes in the pairwise odds when the systematic utilities are held fixed. Whether that restriction is suitable depends on the application and how utilities are specified.
Common Mistakes and a Practical Check
- Confusing odds with probabilities. IIA does not claim that the probability of choosing A is unaffected by adding B. Check the ratio \(P_A/P_B\), not \(P_A\) by itself.
- Using rounded probabilities to claim exact equality. Calculate the ratio from exact fractions when possible. For example, \((6/9)/(3/9)=6/3=2\) exactly. Rounded decimal displays can produce a small apparent discrepancy.
- Ignoring a changed utility. The IIA comparison assumes the systematic utilities of the two alternatives being compared do not change with the choice set. If they do, the odds can change because their utility difference changed.
- Treating IIA as an empirical truth. It is a property implied by multinomial logit’s probability structure and its assumptions, not a guarantee that real choices satisfy the restriction.
- Calling all other alternatives literally irrelevant. Other alternatives affect the common denominator and can change individual choice probabilities. They are “irrelevant” only to the pairwise probability ratio under the stated conditions.
A useful way to check a claim about IIA is to write the two probabilities with their full choice-set denominators, divide, and verify that the common denominator cancels. Then confirm that the systematic utilities for the pair have been held fixed. This makes clear both why the restriction holds and what conditions its interpretation requires.
Check Your Understanding
Answer each question using the multinomial logit formula and state any condition needed for the comparison.
- In one sentence, define the independence of irrelevant alternatives property for two available alternatives.
- Suppose the exponentiated systematic utilities for A and B are 8 and 4. What are the A-to-B odds, regardless of the utilities of other available alternatives?
- A third option is added, and the probabilities of both A and B decrease. Does this alone violate IIA? Explain.
- In the meal example, why are the soup-to-salad odds equal to 2 both before and after the sandwich is added?
- Name one condition that must hold for an added or removed alternative not to change the IIA odds between an existing pair.