Independence of irrelevant alternatives

Summary

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 edit

In rational choice theory and economics edit

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 edit

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.

See also edit

References edit

  1. ^ Saari, Donald G. (2001). Decisions and elections : explaining the unexpected (1. publ. ed.). Cambridge [u.a.]: Cambridge Univ. Press. pp. 39. ISBN 0-521-00404-7.

Bibliography edit

  • Arrow, Kenneth Joseph (1963). Social Choice and Individual Values (2nd ed.). Wiley.
  • Kennedy, Peter (2003). A Guide to Econometrics (5th ed.). MIT Press. ISBN 978-0-262-61183-1.
  • Maddala, G. S. (1983). Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press. ISBN 978-1-107-78241-9.
  • Ray, Paramesh (1973). "Independence from Irrelevant Alternatives". Econometrica. 41 (5): 987–991. doi:10.2307/1913820. JSTOR 1913820. Discusses and deduces the not always recognized differences between various formulations of IIA.

Further reading edit

  • Callander, Steven; Wilson, Catherine H. (July 2006). "Context-dependent voting". Quarterly Journal of Political Science. 1 (3). Now Publishing Inc.: 227–254. doi:10.1561/100.00000007.
  • Steenburgh, Thomas J. (2008). "The Invariant Proportion of Substitution Property (IPS) of Discrete-Choice Models" (PDF). Marketing Science. 27 (2): 300–307. doi:10.1287/mksc.1070.0301. S2CID 207229327. Archived from the original (PDF) on 2010-06-15.