In rational choice theory and economics
In rational choice theory and economics, 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 the axioms generalizes IIA to random events:
- If , then for any and ,
where p is a probability, pL+(1-p)N means a gamble with probability p of yielding L and probability (1-p) of yielding N, and means that M is preferred over L. This axiom says that if an outcome (or lottery ticket) L is worse than M, then adding with probability p of receiving L rather than N is considered to be not as good as having a chance with probability p of receiving M rather than N.
In economics, the axiom is further connected to the theory of revealed preferences. Economists often invoke IIA when building descriptive models of behavior to ensure agents have well-defined preferences that can be used for making testable predictions. If agents' behavior can change depending on irrelevant circumstances, economic models could be made unfalsifiable by claiming some irrelevant circumstance must have changed when repeating the experiment. Often, the axiom is justified by arguing that irrational agents will be money pumped until they are bankrupt, at which point their preferences become unobservable or irrelevant to the rest of the economy.
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 by reference to the Dutch Book arguments.
Behavioral economics introduces models that weaken or remove the assumption of IIA, providing greater accuracy at the cost of greater complexity. Behavioral economics has shown the axiom is commonly violated in human decisions; for example, inserting a $5 medium soda between a $3 small and $6 large can make customers perceive the large as a better deal (because it's "only $1 more than the medium").
IIA is a direct consequence of the multinomial logit and conditional logit[clarification needed] models in econometrics[citation needed], meaning such models cannot precisely describe situations where consumers violate IIA.
Voting and 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."
Arrow's impossibility theorem shows that no reasonable (non-dictatorial, Pareto-efficient) ranked-choice voting voting system with more than two outcomes can satisfy IIA, even if voters are perfectly honest.
However, Arrow's theorem does not apply to cardinal voting methods. Thus some cardinal methods can pass IIA: approval voting, range voting, and median voting rules like majority judgment 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. Thus, if the voters change their rating scales depending on the candidates who are running, IIA does not necessarily imply that the outcome is independent of non-winning candidates.
Other methods that pass IIA include random pair, random candidate, and random dictatorship.