Independence of irrelevant alternatives

The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, 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 an account of rational individual behavior or aggregation of individual preferences, the exact formulations differ from context to context.

In individual choice theory, IIA sometimes refers to Chernoff's condition or Sen's property α (alpha): if an alternative x is chosen from a set T, and x is also an element of a subset S of T, then x must be chosen from S. That is, eliminating some of the unchosen alternatives shouldn't affect the selection of x as the best option.

In social choice theory, Arrow's IIA is one of the conditions in Arrow's impossibility theorem, which states that it is impossible to aggregate individual rank-order preferences ("votes") satisfying IIA in addition to certain other reasonable conditions. Arrow defines IIA thus:

Another expression of the principle:

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.

In social choice theory, IIA is also defined as:

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 irrelevant to the choice between A and B.

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