# Independence of irrelevant alternatives

Independence of irrelevant alternatives is an axiom often adopted by social scientists as a basic condition of rationality. It appears in theories of voting systems, bargaining theory, and logic. It is controversial for two reasons: first, some mathematicians find it too strict of an axiom; second, experiments by Amos Tversky, Daniel Kahneman, and others have showed that humans make this 'error' all the time.

The axiom states: If A is preferred to B out of the choice set {A,B}, then introducing a third, irrelevant, alternative X (thus expanding the choice set to {A,B,X} ) should not make B preferred to A. In other words, whether A or B is better should not be changed by the availability of X.

In voting systems, independence of irrelevant alternatives is interpreted as, if one candidate (X) wins the election, and a new alternative (Y) is added, only X or Y will win the election.

A less strict property is sometimes 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 win the election if Y is not in the Smith set.

All Condorcet methods fail the former criterion, but some (e.g. Schulze method) satisfy the latter.

Borda count, Coombs' method or Instant-runoff voting do not meet either criterion.

An anecdote which illustrates a violation of this property has been attributed to Sidney Morgenbesser:

After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie."

This anecdote whimsically compares the preferences of a large population to a single person. However, intransitive preferences within a collective seem less unreasonable than intransitive preferences within an individual.

Voting systems which are not independent of irrelevant alternatives suffer from strategic nomination considerations.

Some argue that the independence of irrelevant alternatives criterion, however, is a flawed criterion, on the grounds that IIAC failure can have a positive effect. For example, if a population slightly preferred candidate"B" to candidate "A", but candidate "A"'s supporters were far more loyal, then an introduction of a third candidate could split B's support far more than A's, leading to a win by A. In cases where one candidate's supporters feel they are compromising far more than the other candidate's supporters do, failing IIAC may not be a flaw. In other words, IIAC does not take strength of preference into account. Those who consider IIAC to be flawed find Arrow's famous impossibility theorem to be irrelevant.

Condorcet methods do not fail the IIAC when they have a single Condorcet Winner both before and after the introduction of the new candidate. In other words, the IIAC can never replace one Condorcet Winner with another.

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