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 in different connotation in several contexts. Although it always attempts to provide an account of rational individual behavior or aggregation of individual preferences, the exact formulation differs widely in both language and exact content. Perhaps the easiest way to understand the axiom is how it pertains to casting a ballot. There the axiom says that if Charlie (the irrelevant alternative) enters a race between Alice and Bob, with Alice (leader) liked better than Bob (runner-up), then the individual voter who likes Charlie less than Alice will not switch his vote from Alice to Bob. Because of this, a violation of IIA is commonly referred to as the "spoiler effect": support for Charlie "spoils" the election for Alice, while it "logically" should not have. After all, Alice was liked better than Bob, and Charlie was liked less than Alice. In collective decision making contexts, the axiom takes a more refined form, and is mathematically intimately tied with Condorcet methods, the Gibbard–Satterthwaite theorem, and the Arrow Impossibility theorem. They all have to do with cyclical majorities between ranked sets, and the related proofs take the same basic form. Behavioral economics has shown the axiom to be commonly violated by humans. In individual choice theory, IIA sometimes refers to Chernoff's condition or Sen's 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.