Independence of Irrelevant Alternatives

The independence of irrelevant alternatives (IIA), also known as binary independence is an axiom of decision theory and various social sciences. The term is used with different meanings in different contexts. Although they all attempt to provide a rational account of individual behavior or aggregation of individual preferences, the exact formulations differ from context to context.

In individual choice theory, the name "IIA" is sometimes used to refer to Chernoff's condition or Sen's property α (alpha): if an alternative x chosen from a set T is an element of a subset S of T, then x must be chosen from S.

In social choice theory, Arrow's IIA is well known as one of the conditions in Arrow's impossibility theorem: the social preferences between alternatives x and y depend only on the individual preferences between x and y. Kenneth Arrow (1951) shows the impossibility of aggregating individual rank-order preferences ("votes") satisfying IIA and certain other reasonable conditions.

There are other requirements that go by the name of "IIA".

One such requirement is as follows: 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 preferable to A. In other words, preferences for A or B should not be changed by the inclusion of X, i.e., X is irrelevant to the choice between A and B. This formulation appears in bargaining theory, theories of individual choice, and voting theory. Some theorists find it too strict an axiom; experiments by Amos Tversky, Daniel Kahneman, and others have shown that human behavior rarely adheres to this axiom.

A distinct formulation of IIA is found in 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 if only preferences for X change, the voting rule must not lead to B's being selected over A. 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.