Independence of irrelevant alternatives

For a discussion of independence in the context of elections and social choice, see spoiler effect.

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 ${\displaystyle A}$ and ${\displaystyle B}$ should not depend on the quality of a third, unrelated outcome ${\textstyle C}$.

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 are called menu effects or menu dependence.

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

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

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

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

In other words, the probabilities involving ${\displaystyle N}$ cancel out, because the probability of ${\displaystyle N}$ 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.^{[ citation needed ]}

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

Main article: Spoiler effect

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.

Artificial intelligence

Main article: Luce's choice axiom

See also

• Spoiler effect
• Luce's choice axiom
• Sure-thing principle
• Menu dependence
  • Decoy effect
• Decision theory

Bibliography

• 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.

• 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.
• Sen, Amartya (1994). "The Formulation of Rational Choice". The American Economic Review. 84 (2): 385–390. JSTOR   2117864.
• Sen, Amartya (July 1997). "Maximization and the Act of Choice". Econometrica. 65 (4): 745–779. doi:10.2307/2171939. JSTOR   2171939.
• Sen, Amartya (2002). Rationality and Freedom. Harvard University Press. ISBN   978-0-674-01351-3.
• Sniderman, Paul M.; Bullock, John (2018). "A Consistency Theory of Public Opinion and Political Choice: The Hypothesis of Menu Dependence". In Saris, Willem E.; Sniderman, Paul M. (eds.). Studies in Public Opinion: Attitudes, Nonattitudes, Measurement Error, and Change. Princeton University Press. pp. 337–358. doi:10.2307/j.ctv346px8.16. ISBN   978-0-691-18838-6. JSTOR   j.ctv346px8.16.
• Saini, Ritesh (2008). Menu dependence in risky choice (Thesis). OCLC   857236573. CiteSeer^x:  a51c1c0b707a028be4337c348c95c52b548db0e3.