Independence of irrelevant alternatives

For the same principle in social choice, see Spoiler effect.

Independence of irrelevant alternatives (IIA) is a major axiom of decision theory which codifies the intuition that a choice between $A$ and $B$ should not depend on the quality of a third, unrelated outcome ${\textstyle C}$. There are several different variations of this axiom, which are generally equivalent under mild conditions. As a result of its importance, the axiom has been independently rediscovered in various forms across a wide variety of fields, including economics, cognitive science, social choice, fair division, rational choice, artificial intelligence, probability, and game theory. It is closely tied to many of the most important theorems in these fields, including Arrow's impossibility theorem, the Balinski-Young theorem, and the money pump arguments.

In behavioral economics, failures of IIA (caused by irrationality) are called menu effects or menu dependence. Violations of IIA in social choice, elections, and sports are called spoiler effects.

Motivation

This is sometimes explained with a short story by philosopher Sidney Morgenbesser:

Morgenbesser, ordering dessert, is told by a waitress that he can choose between blueberry or apple pie. He orders apple. Soon the waitress comes back and explains cherry pie is also an option. Morgenbesser replies "In that case, I'll have blueberry."

Independence of irrelevant alternatives rules out this kind of arbitrary behavior, by stating that:

If A(pple) is chosen over B(lueberry) in the choice set {A, B}, introducing a third option C(herry) must not result in B being chosen over A.

By field

Economics

In economics, the axiom is connected to the theory of revealed preferences. Economists often invoke IIA when building descriptive (positive) models of behavior to ensure agents have well-defined preferences that can be used for making testable predictions. If agents' behavior or preferences are allowed to change depending on irrelevant circumstances, any model could be made unfalsifiable by claiming some irrelevant circumstance must have changed when repeating the experiment. Often, the axiom is justified by arguing that any irrational agent will be money pumped until going bankrupt, making their preferences unobservable or irrelevant to the rest of the economy.

Behavioral economics

While economists must often make do with assuming IIA for reasons of computation or to make sure they are addressing a well-posed problem, experimental economists have shown that real human decisions often violate IIA. For example, the decoy effect shows that inserting a $5 medium soda between a $3 small and $5.10 large can make customers perceive the large as a better deal (because it's "only 10 cents more than the medium"). Behavioral economics introduces models that weaken or remove many assumptions of consumer rationality, including IIA. This provides greater accuracy, at the cost of making the model more complex and more difficult to falsify.

Social choice

Main article: Spoiler effect

In social choice theory, independence of irrelevant alternatives is often stated as "if one candidate (X) would win an election without a new candidate (Y), and Y is added to the ballot, then either X or Y should win the election." Situations where Y affects the outcome are called spoiler effects. Arrow's impossibility theorem shows that no reasonable (non-random, non-dictatorial) ranked voting system can satisfy IIA. However, Arrow's theorem does not apply to rated voting methods, which can (and often do) pass IIA. Approval voting, score voting, and median voting all satisfy the IIA criterion and Pareto efficiency. Note that if new candidates are added to ballots without changing any of the ratings for existing ballots, the score of existing candidates remains unchanged, leaving the winner the same. Generalizations of Arrow's impossibility theorem show that if the voters change their rating scales depending on the candidates who are running, the outcome of cardinal voting may still be affected by the presence of non-winning candidates.

Other methods that pass IIA include sortition and random dictatorship.

Common voting methods

Deterministic voting methods that behave like majority rule when there are only two candidates can be shown to fail IIA by the use of a Condorcet cycle:

Consider a scenario in which there are three candidates A, B, & C, and the voters' preferences are as follows:

25% of the voters prefer A over B, and B over C. (A>B>C)

40% of the voters prefer B over C, and C over A. (B>C>A)

35% of the voters prefer C over A, and A over B. (C>A>B)

(These are preferences, not votes, and thus are independent of the voting method.)

75% prefer C over A, 65% prefer B over C, and 60% prefer A over B. The presence of this societal intransitivity is the voting paradox. Regardless of the voting method and the actual votes, there are only three cases to consider:

• Case 1: A is elected. IIA is violated because the 75% who prefer C over A would elect C if B were not a candidate.
• Case 2: B is elected. IIA is violated because the 60% who prefer A over B would elect A if C were not a candidate.
• Case 3: C is elected. IIA is violated because the 65% who prefer B over C would elect B if A were not a candidate.

For particular voting methods, the following results hold:

• Instant-runoff voting, the Kemeny-Young method, Minimax Condorcet, Ranked Pairs, top-two runoff, First-past-the-post, and the Schulze method all elect B in the scenario above, and thus fail IIA after C is removed.
• The Borda count and Bucklin voting both elect C in the scenario above, and thus fail IIA after A is removed.
• Copeland's method returns a three-way tie. If A is removed, then B becomes the sole winner, making C lose. Hence it, too, fails IIA.

Stochastic choice

Main article: Luce's choice axiom

Sports

Main article: Elimination from postseason contention § Spoiler effect

On two occasions the failure of IIA has led the International Skating Union (ISU), which regulates figure skating, to change the voting method its judges use during competition. The first was at the 1995 World Figure Skating Championships, when Michelle Kwan's fourth-place performance near the end of the ladies' competition resulted in Surya Bonaly and Nicole Bobek exchanging second and third place, even though they had already skated, due to the way the ranked voting rule worked out afterwards. Two years afterwards the ISU switched to a pairwise-comparison method. However, at the 2002 Winter Olympics, the system produced another IIA failure regardless. Kwan had been ahead of Sarah Hughes, the eventual gold medal winner, until Irina Slutskaya skated, whereupon she and Hughes exchanged places in the rankings. Two years later the ISU adopted score voting to prevent the recurrence of such paradoxes.

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