Independence of irrelevant alternatives

Independence of irrelevant alternatives (IIA), also known as binary independence, the independence axiom, is an axiom of decision theory and economics describing a necessary condition for rational behavior. The axiom says that a choice between $$A$$ and $$B$$ should not depend on the quality of a third, unrelated outcome $C$.

The axiom is deeply connected to several of the most important results in social choice, welfare economics, ethics, and decision theory. Among these results are Arrow's impossibility theorem, the VNM utility theorem, Harsanyi's utilitarian theorem, and the Dutch book theorems.

Violations of IIA in individual behavior (caused by irrationality) are called menu effects or menu dependence. Violations of IIA in social choice 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.

Decision theory
In rational choice theory, IIA is one of the von Neumann-Morgenstern axioms, four axioms that together characterize rational choice under uncertainty (and establish that it can be represented as maximizing expected utility). One of these axioms generalizes IIA to random events.

Say we have two possible outcomes, $$\,\text{Bad} \prec \text{Good}$$ (i.e. Good is preferred to Bad), and also an irrelevant outcome $$N$$. Let $$p$$ be any probability. IIA states that a random probability $$p$$ of receiving $$N$$ rather than $$\text{Bad}$$ or $$\text{Good}$$ does not affect our decision.

Here we write $$\, (1-p) A + p B$$ to mean a gamble (or lottery) with a probability $$1-p$$ of resulting in $$A$$ and probability $$p$$ of resulting in $$B$$. Then, for any $$N$$ and $$p\in[0,1]$$ (with the "irrelevant" part of the lottery underlined):

$$\, (1-p) \, \text{Bad} + \underline{p N} \prec (1-p) \, \text{Good} + \underline{p N}$$.

In other words, the probabilities involving $$N$$ cancel out, because the probability of $$N$$ is unchanged, regardless of which lottery we pick.

In prescriptive (or normative) models, independence of irrelevant alternatives is used together with the other VNM axioms to develop a theory of how rational agents should behave, often relying on the Dutch Book arguments.

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.

IIA is a direct consequence of the multinomial logit model in empirical econometrics.

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, known as a menu effect. 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 the positive (not normative) assumption of IIA. This provides greater accuracy, at the cost of making the model more complex and more difficult to falsify.

Social choice
In social choice theory and election science, 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-choice voting voting system can satisfy IIA, even when voters are perfectly honest. However, Arrow's theorem does not apply to rated voting methods, which can (and typically 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.

Examples of failure
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-choice voting worked out afterwards. Two years afterwards the ISU switched to a pairwise ranked-choice system.

At the 2002 Winter Olympics, that system produced another IIA failure. 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 range voting.