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.

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Independence of irrelevant alternatives

Electoral system

An electoral system or voting system is a set of rules that determine how elections and referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, non-profit organisations and informal organisations. These rules govern all aspects of the voting process: when elections occur, who is allowed to vote, who can stand as a candidate, how ballots are marked and cast, how the ballots are counted, how votes translate into the election outcome, limits on campaign spending, and other factors that can affect the result.

Condorcet winner criterion

An electoral system satisfies the Condorcet winner criterion (pronkɒndɔrˈseɪ) if it always chooses the Condorcet winner when one exists. The candidate who wins a majority of the vote in every head-to-head election against each of the other candidates - that is, a candidate preferred by more voters than any others - is the Condorcet winner, although Condorcet winners do not exist in all cases. It is sometimes simply referred to as the "Condorcet criterion", though it is very different from the "Condorcet loser criterion".

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