Wednesday, January 16, 2008

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
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.

Local independence
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.
Some text of this article is derived with permission from

IIA in social choice
The independence of irrelevant alternatives (IIA) in econometrics is a property of the multinomial logit and 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), Conditional probit (also called multinomial probit) and mixed logit are alternative models for nominal outcomes that do not violate IIA, but often have assumptions of their own that may be difficult to meet or are computationally infeasible. The conditional 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, 2007), which suggests similarly counterintuitive individual choice behavior.

Independence of irrelevant alternatives IIA in econometrics


Independence of irrelevant alternatives Approval voting

Borda count

Instant-runoff voting

Kemeny-Young method

Minimax Condorcet

Plurality voting system

Range voting

Two-round system