Games - Independence of irrelevant alternatives

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Independence of irrelevant alternatives (IIA) 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 human behaviour rarely adheres to this axiom.

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. Local IIA was devised as a criterion for certain Condorcet methods.

None of the seriously proposed ranked methods satisfy IIA, but some Condorcet methods (e.g. ranked pairs, Schulze method) satisfy the local IIA.

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, particularly when there exists preferences that create a Condorcet voting paradox. If cherry pie were in the Smith set, for instance, it would be the case that apple pie is preferred to blueberry pie, blueberry pie to cherry pie, but cherry pie to apple pie.

Voting systems which are not independent of irrelevant alternatives suffer from strategic nomination considerations, but these considerations aren't generally considered serious unless the voting system specifically fails the (easier to satisfy) independence of clones criterion.

Some argue that the independence of irrelevant alternatives criterion (IIAC), 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.

Some text of this article is derived with permission from http://condorcet.org/emr/criteria.shtml

IIA in econometrics

The independence of irrelevant alternatives is an assumption of the multinomial logit model in econometrics; outcomes that could theoretically violate IIA (such as the outcome of multicandidate elections, or per Arrow any choice made by humans) may make multinomial logit an invalid estimator. Mixed logit, conditional logit, and multinomial probit are alternative models for nominal outcomes that do not violate IIA, but often have assumptions of their own that may be difficult to meet or are computationally infeasible.

As an alternative, the conditional logit model may be used if the structure of decision-making can be reduced to a series of binary choices; this approach also avoids the IIA assumption.

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