Social choice theory

In social choice theory and the theory of electoral systems, IIA is defined by Arrow as the assumption that "The social preferences between alternatives x and y depend only on the individual preferences between x and y."^[2]

Concretely, suppose Charlie (an irrelevant alternative) enters a race between Alice and Bob, where Alice (leader) is set to defeat Bob. IIA says that in a rational system of voting, if Charlie joins the race but loses, Alice should not suddenly lose the election to Bob. In this context, violating IIA is commonly referred to as the "spoiler effect": support for Charlie "spoils" the election for Alice.

In this form, IIA is one of the conditions in Arrow's impossibility theorem, which shows that it is impossible for any ranked-choice electoral system (including the commonly-used plurality) to behave rationally (i.e. satisfy both IIA and Pareto efficiency).

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 ${\displaystyle \,L\prec M}$, then for any ${\displaystyle \,N}$ and ${\displaystyle \,p\in (0,1]}$,

${\displaystyle \,pL+(1-p)N\prec pM+(1-p)N,}$

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 ${\displaystyle \,L\prec M}$ 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 models in econometrics, meaning such models cannot precisely describe situations where consumers violate IIA.

Local independence

A criterion weaker than IIA proposed by H. Peyton Young and A. Levenglick is called local independence from irrelevant alternatives (LIIA).^[3] LIIA requires that both of the following conditions always hold:

• If the option that finished in last place is deleted from all the votes, then the order of finish of the remaining options must not change. (The winner must not change.)
• If the winning option is deleted from all the votes, the order of finish of the remaining options must not change. (The option that finished in second place must become the winner.)

An equivalent way to express LIIA is that if a subset of the options are in consecutive positions in the order of finish, then their relative order of finish must not change if all other options are deleted from the votes. For example, if all options except those in 3rd, 4th and 5th place are deleted, the option that finished 3rd must win, the 4th must finish second, and 5th must finish 3rd.

Another equivalent way to express LIIA is that if two options are consecutive in the order of finish, the one that finished higher must win if all options except those two are deleted from the votes.

LIIA is weaker than IIA because satisfaction of IIA implies satisfaction of LIIA, but not vice versa.

Despite being a weaker criterion (i.e. easier to satisfy) than IIA, LIIA is satisfied by very few voting methods. These include Kemeny-Young and ranked pairs, but not Schulze. Just as with IIA, LIIA compliance for rating methods such as approval voting, range voting, and majority judgment require the assumption that voters rate each alternative individually and independently of knowing any other alternatives, on an absolute scale (calibrated prior to the election), even when this assumption implies that voters having meaningful preferences in a two candidate election will necessarily abstain.

In social choice

From Kenneth Arrow,^[4] each "voter" i in the society has an ordering R_i that ranks the (conceivable) objects of social choice—x, y, and z in simplest case—from high to low. An aggregation rule (voting rule) in turn maps each profile or tuple (R₁, ...,R_n) of voter preferences (orderings) to a social ordering R that determines the social preference (ranking) of x, y, and z.

Arrow's IIA requires that whenever a pair of alternatives is ranked the same way in two preference profiles (over the same choice set), then the aggregation rule must order these alternatives identically across the two profiles.^[5] For example, suppose an aggregation rule ranks a above b at the profile given by

• (acbd, dbac),

(i.e., the first individual prefers a first, c second, b third, d last; the second individual prefers d first, ..., and c last). Then, if it satisfies IIA, it must rank a above b at the following three profiles:

• (abcd, bdca)
• (abcd, bacd)
• (acdb, bcda).

The last two forms of profiles (placing the two at the top; and placing the two at the top and bottom) are especially useful in the proofs of theorems involving IIA.

Arrow's IIA does not imply an IIA similar to those different from this at the top of this article nor conversely.^[6]

In the first edition of his book, Arrow misinterpreted IIA by considering the removal of a choice from the consideration set. Among the objects of choice, he distinguished those that by hypothesis are specified as feasible and infeasible. Consider two possible sets of voter orderings (${\displaystyle R_{1}}$, ...,${\displaystyle R_{n}}$ ) and (${\displaystyle R_{1}'}$, ...,${\displaystyle R_{n}'}$) such that the ranking of X and Y for each voter i is the same for ${\displaystyle R_{i}}$ and ${\displaystyle R_{i}'}$. The voting rule generates corresponding social orderings R and R'. Now suppose that X and Y are feasible but Z is infeasible (say, the candidate is not on the ballot or the social state is outside the production possibility curve). Arrow required that the voting rule that R and R' select the same (top-ranked) social choice from the feasible set (X, Y), and that this requirement holds no matter what the ranking is of infeasible Z relative to X and Y in the two sets of orderings. IIA does not allow "removing" an alternative from the available set (a candidate from the ballot), and it says nothing about what would happen in such a case: all options are assumed to be "feasible."

Examples

Bibliography

• Arrow, Kenneth Joseph (1951). Social Choice and Individual Values (1st ed.). Wiley.
• 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

• Callander, Steven; Wilson, Catherine H. (July 2006). "Context-dependent voting". Quarterly Journal of Political Science. Now Publishing Inc. 1 (3): 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.