Definitions

# Independence of irrelevant alternatives

Independence of irrelevant alternatives (IIA) is a term for an axiom of decision theory and various social sciences. Although exact formulations of IIA differ, intentions of the usages are similar in attempting to provide a rational account of individual behavior or aggregation of individual preferences.

IIA is also known as "Chernoff's condition" or Sen's property α (alpha)

Under one common formulation, the axiom states that:

If A is preferred to B out of the choice set {A,B}, then introducing a third alternative X, thus expanding the choice set to {A,B,X}, must not make B preferred to A.
In other words, whether A or B is better should not be changed by the availability of X, which is irrelevant to the choice between A and B. This formulation appears in bargaining theory, theories of individual choice, and voting theory. It is controversial for two reasons: first, some theorists find it too strict an axiom; second, experiments by Amos Tversky, Daniel Kahneman, and others have shown that human behaviour rarely adheres to this axiom.

A distinct formulation of IIA is that of social choice theory:

If A is selected over B out of the choice set {A,B} by a voting rule for given voter preferences of A, B, and an unavailable third alternative X, then B must not be selected over A by the voting rule if only preferences for X change.
In other words, whether A or B is selected should not be affected by a change in the vote for an unavailable X, which is thus irrelevant to the choice between A and B. Kenneth Arrow (1951) shows the impossibility of aggregating individual preferences ("votes") satisfying IIA and certain other apparently reasonable conditions.

## Voting theory

In voting systems, independence of irrelevant alternatives is often interpreted as, if one candidate (X) wins the election, and a new alternative (Y) is added, only X or Y will win the election.

Approval voting and range voting satisfy the independence of irrelevant alternatives criterion, largely because they are cardinal rather than ordinal voting systems and votes for one candidate do not restrict votes for another. Another cardinal system, cumulative voting, does not satisfy the criterion.

An anecdote which illustrates a violation of this property has been attributed to Sidney Morgenbesser:

After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie."

All voting systems have some degree of inherent susceptibility to strategic nomination considerations. Some regard these considerations as less serious unless the voting system specifically fails the (easier to satisfy) independence of clones criterion.

### Local independence

A related criterion proposed by H. P. Young and A. Levenglick is called local independence of irrelevant alternatives. It says that if one candidate (X) wins an election, and a new alternative (Y) is added, X will still win the election if Y is not in the Smith set. In other words, the outcome of the election is independent of alternatives which are not in the Smith set. Note that this neither implies nor is implied by the independence of irrelevant alternatives criterion; in fact, the two are mutually exclusive.

No deterministic ranked methods satisfy IIA, but local IIA is satisfied by some methods which always elect from the Smith set, such as ranked pairs and Schulze method.

### Criticism

Many voting theorists believe that IIA is too strong a condition. The majority criterion is incompatible with IIA. Consider

7 votes for A > B > C
6 votes for B > C > A
5 votes for C > A > B

Without loss of generality, suppose a voting system satisfying the majority criterion chooses A. If B drops out of the race, the remaining votes will be:

7 votes for A > C
11 votes for C > A

and C will win under the majority criterion, even though the change (B dropping out) concerned an "irrelevant" alternative candidate who did not win in the original circumstance. This is an example of the spoiler effect.

### IIA in social choice

From Kenneth Arrow (1951, pp. 15, 23, 27), each "voter" i in the society has an ordering Ri that ranks the (conceivable) objects of social choice—X, Y, and Z in simplest case -- from lowest on up (say candidates for public office or social states from different economic policies). The voting rule in turn maps the set of voter orderings (R1, ...,Rn) onto a social ordering R that determines the social ranking of X, Y, and Z. In the first edition of his book, Arrow made a mistake in interpreting the IIA axiom. This mistake was later corrected but it reappears from time to time in various publications. Let's take a look at Arrow's original mistake. Among the objects of choice, he distinguished those that by hypothesis are specified as feasible and infeasible. Consider two possible sets of voter orderings ($R_1$, ...,$R_n$ ) and ($R_1\text{'}$, ...,$R_n\text{'}$) such that the ranking of X and Y for each voter i is the same for $R_i$ and $R_i\text{'}$. The voting rule generates corresponding social orderings R and R'. Now suppose that X and Y are feasible but Z is infeasible (say, the candidate is not on the ballot or the social state is outside the production possibility curve). Arrow required that the voting rule that R and R' select the same (top-ranked) social choice from the feasible set (X, Y), and that this requirement holds no matter what the ranking is of infeasible Z relative to X and Y in the two sets of orderings. In fact, the IIA axiom does not allow to "remove" an alternative from the available set (or a candidate from the ballot). It says nothing about what would happen in such a case. All alternatives are assumed "feasible." IIA requires that whenever a pair of alternatives is ranked the same way in two preference profiles (that are over the same sets of alternatives), then the voting rule must order these two alternatives identically. Paramesh Ray (1973) showed that Arrow's IIA does not imply an IIA similar to that at the top of this article nor conversely.

## IIA in econometrics

The independence of irrelevant alternatives (IIA) in econometrics is a property of the multinomial logit (also called conditional logit) models in econometrics; outcomes that could theoretically violate this IIA (such as the outcome of multicandidate elections or any choice made by humans) may make multinomial logit and conditional logit invalid estimators.

IIA implies that adding another alternative or changing the characteristics of a third alternative does not affect the relative odds between the two alternatives considered. This implication is not realistic for applications with similar alternatives. Many examples have been constructed to illustrate this problem, including Beethoven/Debussy (Debreu 1960; Tversky 1972), Bicycle/Pony (Luce and Suppes 1965), and Red Bus/Blue Bus (McFadden 1974).

Consider the Red Bus/Blue Bus example. Commuters initially face a decision between two modes of transportation: car and red bus. Suppose that a consumer chooses between these two options with equal probability, 0.5, so that the odds equal 1. Now suppose a third mode, blue bus, is added. Assuming bus commuters do not care about the color of the bus, consumers are expected to choose between bus and car still with equal probability. But IIA implies that this is not the case: the probability of commuters that take each of the three modes equals one third. (Based on Wooldridge 2002, pp. 501-2.)

Many modeling advances have been motivated by a desire to alleviate the concerns raised by IIA. Generalized extreme value (McFadden 1978), multinomial probit (also called conditional probit) and mixed logit are alternative models for nominal outcomes which relax IIA, but these models often have assumptions of their own that may be difficult to meet or are computationally infeasible. The multinomial probit model has as a disadvantage that it makes calculation of maximum likelihood infeasible for more than five alternatives as it involves multiple integrals. IIA can also be relaxed by specifying a hierarchical model, ranking the choice alternatives. The most popular of these is called the nested logit model (McFadden 1984).

Generalized extreme value and multinomial probit models possess another property, the Invariant Proportion of Substitution (Steenburgh 2008), which suggests similarly counterintuitive individual choice behavior.

## Examples

### Borda count

5 voters rank 5 alternatives [A, B, C, D, E].

3 voters rank [A>B>C>D>E]. 1 voter ranks [C>D>E>B>A]. 1 voter ranks [E>C>D>B>A].

Borda count (a=0, b=1): C=13, A=12, B=11, D=8, E=6. C wins.

Now, the voter who ranks [C>D>E>B>A] instead ranks [C>B>E>D>A]; and the voter who ranks [E>C>D>B>A] instead ranks [E>C>B>D>A]. Note that they change their preferences only over the pairs [B, D] and [B, E].

The new Borda count: B=14, C=13, A=12, E=6, D=5. B wins.

Note that the social choice has changed the ranking of [B, A], [B, C] and [D, E]. The changes in the social choice ranking are dependent on irrelevant changes in the preference profile. In particular, B now wins instead of C, even though no voter changed their preference over [B, C].

### Instant-runoff voting

5 voters rank 3 alternatives [A, B, C].

2 voters rank [A>B>C]. 2 voters rank [C>B>A]. 1 voter ranks [B>A>C].

Round 1: A=2, B=1, C=2; B eliminated. Round 2: A=3, C=2; A wins.

Now, the two voters who rank [C>B>A] instead rank [B>C>A]. Note that they only change their preferences over B and C.

Round 1: A=2, B=3, C=0; C eliminated. Round 2: A=2, B=3; B wins.

Note that the social choice ranking of [A, B] is dependent on preferences over the irrelevant alternatives [B, C].

### Plurality voting system

7 voters rank 3 alternatives [A, B, C].

3 voters rank [A>B>C] 2 voters rank [B>A>C] 2 voters rank [C>B>A]

The plurality winner is A, with 3 votes out of 7.

Now, the 2 voters who rank [C>B>A] instead rank [B>C>A]; note that they only change their preferences over [B, C].

The new plurality winner is B, with 4 votes out of 7.

The social choice has swapped B ahead of A, even though the only change in the preference profile was between B and C, which are irrelevant alternatives.

### Two-round system

A probable example of the two-round system's failing this criterion was the 1991 Louisiana gubernatorial election. Polls leading up to the election suggested that, had the runoff been Edwin Edwards v Buddy Roemer, Roemer would have won. However, in the actual election, David Duke managed to finish second and make the runoff instead of Roemer, a runoff which Edwards then won by a large margin. Thus, the presence of Duke in the election changed which of the non-Duke candidates won.

## References

• Kenneth J. Arrow (1963), Social Choice and Individual Values
• Paramesh Ray (1973). "Independence of Irrelevant Alternatives," Econometrica, Vol. 41, No. 5, pp. 987-991. Discusses and deduces the (not always recognized) differences between various formulations of IIA.
• Peter Kennedy (2003), A Guide to Econometrics, 5th ed.
• G.S. Maddala (1983). Limited-dependent and Qualitative Variables in Econometrics