Английская Википедия:Independence of irrelevant alternatives
Шаблон:Short description Шаблон:For
Independence of irrelevant alternatives (IIA), also known as binary independence[1] or the independence axiom, is an axiom of decision theory and economics describing a necessary condition for rational behavior. The axiom says that adding "pointless" (rejected) options should not affect the outcome of a decision. 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.
The axiom is deeply connected to several of the most important results in social choice theory, welfare economics, ethics, and rational choice theory; among these are Arrow's Impossibility theorem, the Von Neumann–Morgenstern utility theorem, Harsanyi's utilitarian theorem, and the Dutch book theorems.
By field
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 <math>\,L\prec M</math>, then for any <math>\,N</math> and <math>\,p\in(0,1]</math>,
- <math>\,pL+(1-p)N \prec pM+(1-p)N,</math>
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 <math>\,L\prec M</math> 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 logitШаблон:Clarify models in econometricsШаблон:Citation needed, meaning such models cannot precisely describe situations where consumers violate IIA.
Voting and social choice
Шаблон:Main 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 can satisfy IIA, even if voters are perfectly honest.
However, cardinal voting systems are not affected by Arrow's theorem. Approval voting, range voting, median voting, and random dictatorship 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.
Despite being a cardinal system (and therefore not subject to Arrow's theorem), cumulative voting does not satisfy IIA.Шаблон:Citation needed
See also
- Independence of Smith-dominated alternatives
- Luce's choice axiom
- Sure-thing principle
- Menu dependence
References
Bibliography
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite journal Discusses and deduces the not always recognized differences between various formulations of IIA.
Further reading
