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]}

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.

Artificial intelligence

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