Independence of irrelevant alternatives

Independence of irrelevant alternatives (IIA) is an axiom of decision theory and various social sciences. The word is used in different meanings in different contexts. Although they all attempt to provide a rational account of individual behavior or aggregation of individual preferences, the exact formulations differ from context to context.

In individual choice theory, the name "IIA" is sometimes used to refer to Chernoff's condition or Sen's property α (alpha): if an alternative x chosen from a set T is an element of a subset S of T, then x must be chosen from S.^[1]

In social choice theory, Arrow's IIA is well known as one of the conditions in Arrow's impossibility theorem: the social preferences between alternatives x and y depend only on the individual preferences between x and y.^[2] Kenneth Arrow (1951) shows the impossibility of aggregating individual rank-order preferences ("votes") satisfying IIA and certain other reasonable conditions.

There are other requirements that go by the name of "IIA".

One such requirement is as follows: If A is preferred to B out of the choice set {A,B}, then introducing a third alternative X, thus expanding the choice set to {A,B,X}, must not make B preferable to A. In other words, preferences for A or B should not be changed by the inclusion of X, i.e., X is irrelevant to the choice between A and B. This formulation appears in bargaining theory, theories of individual choice, and voting theory. Some theorists find it too strict an axiom; experiments by Amos Tversky, Daniel Kahneman, and others have shown that human behavior rarely adheres to this axiom.

A distinct formulation of IIA is found in social choice theory: If A is selected over B out of the choice set {A,B} by a voting rule for given voter preferences of A, B, and an unavailable third alternative X, then B must not be selected over A by the voting rule if only preferences for X change. In other words, whether A or B is selected should not be affected by a change in the vote for an unavailable X, which is thus irrelevant to the choice between A and B.

Voting theory

In voting systems, independence of irrelevant alternatives is often interpreted as, if one candidate (X) wins the election, and a new candidate (Y) is added to the ballot, only X or Y will win the election.

Approval voting and range voting satisfy the independence of irrelevant alternatives criterion. Another cardinal system, cumulative voting, does not satisfy the criterion.

An anecdote which illustrates a violation of this property has been attributed to Sidney Morgenbesser:

After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie."

All voting systems have some degree of inherent susceptibility to strategic nomination considerations. Some regard these considerations as less serious unless the voting system specifically fails the (easier to satisfy) independence of clones criterion.

Local independence

A related criterion proposed by H. P. Young and A. Levenglick is called local independence of irrelevant alternatives. It says that if one candidate (X) would win an election, and a new alternative (Y) is added, X would still win if Y is not in the Smith set. In other words, the outcome of the election is independent of alternatives which are not in the Smith set. Note that this neither implies nor is implied by IIA; in fact, the two are mutually exclusive.

No deterministic ranked methods satisfy IIA, but local IIA is satisfied by some methods which always elect from the Smith set, such as ranked pairs and the Schulze method.

Criticism

IIA may be too strong a criterion for rank-order voting systems (e.g. the majority criterion). Consider

7 votes for A > B > C

6 votes for B > C > A

5 votes for C > A > B

then the net preference of the group under a majority criterion is that A wins over B, B wins over C, and C wins over A: these yield rock-paper-scissors preferences for any pairwise comparison. For example, if B drops out of the race, the remaining votes will be:

7 votes for A > C

11 votes for C > A

Thus if a voting system satisfying the majority criterion selects any one of these, eliminating one of the candidates reverses the decision. For example if the system chooses A, then if B drops out of the race, C will now win under the majority criterion, even though the change (B dropping out) concerned an "irrelevant" alternative candidate who did not win in the original circumstance. This is an example of the spoiler effect.

In social choice

From Kenneth Arrow,^[3] 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 two alternatives identically across the two profiles.^[4] 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, ..., 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 (the one placing the two at the top; the other 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.^[5]

Historical Remark. In the first edition of his book, Arrow misinterpreted the 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 (R₁, ...,R_n ) and (R₁', ...,R_n') such that the ranking of X and Y for each voter i is the same for R_i and 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. In fact, the IIA axiom does not allow "removing" an alternative from the available set (a candidate from the ballot). It says nothing about what would happen in such a case. All alternatives are assumed "feasible."

Examples

Borda count

In a Borda count election, 5 voters rank 5 alternatives [A, B, C, D, E].

3 voters rank [A>B>C>D>E]. 1 voter ranks [C>D>E>B>A]. 1 voter ranks [E>C>D>B>A].

Borda count (a=0, b=1): C=13, A=12, B=11, D=8, E=6. C wins.

Now, the voter who ranks [C>D>E>B>A] instead ranks [C>B>E>D>A]; and the voter who ranks [E>C>D>B>A] instead ranks [E>C>B>D>A]. Note that they change their preferences only over the pairs [B, D] and [B, E].

The new Borda count: B=14, C=13, A=12, E=6, D=5. B wins.

Note that the social choice has changed the ranking of [B, A], [B, C] and [D, E]. The changes in the social choice ranking are dependent on irrelevant changes in the preference profile. In particular, B now wins instead of C, even though no voter changed their preference over [B, C].

Borda count and strategic voting

Consider an election in which there are three candidates, A, B, and C, and only two voters. Each voter ranks the candidates in order of preference. The highest ranked candidate in a voter's preference is given 2 points, the second highest 1, and the lowest ranked 0; the overall ranking of a candidate is determined by the total score it gets; the highest ranked candidate wins.

We consider two profiles:

• In profiles 1 and 2, the first voter casts his votes in the order BAC; so B receives 2 points, A receives 1, and C receives 0 from this voter.
• In profile 1, the second voter votes ACB, so A will win outright (the total scores: A 3, B 2, C 1).
• In profile 2, the second voter votes ABC, so A and B will tie (the total scores: A 3, B 3, C 0).

Thus, if the second voter wishes A to be elected, he had better vote ACB regardless of his actual opinion of C and B. This violates the idea of "independence of irrelevant alternatives" because the voter's comparative opinion of C and B affects whether A is elected or not. In both profiles, the rankings of A relative to B are the same for each voter, but the social rankings of A relative to B are different.

Instant-runoff voting

In an instant-runoff election, 5 voters rank 3 alternatives [A, B, C].

2 voters rank [A>B>C]. 2 voters rank [C>B>A]. 1 voter ranks [B>A>C].

Round 1: A=2, B=1, C=2; B eliminated. Round 2: A=3, C=2; A wins.

Now, the two voters who rank [C>B>A] instead rank [B>C>A]. Note that they only change their preferences over B and C.

Round 1: A=2, B=3, C=0; C eliminated. Round 2: A=2, B=3; B wins.

Note that the social choice ranking of [A, B] is dependent on preferences over the irrelevant alternatives [B, C].

Plurality voting system

In a plurality voting system 7 voters rank 3 alternatives (A, B, C).

• 3 voters rank (A>B>C)
• 2 voters rank (B>A>C)
• 2 voters rank (C>B>A)

In an election, initially only A and B run: B wins with 4 votes to A's 3, but the entry of C into the race makes A the new winner.

The relative positions of A and B are reversed by the introduction of C, an "irrelevant" alternative.

Two-round system

A probable example of the two-round system's failing this criterion was the 1991 Louisiana gubernatorial election. Polls leading up to the election suggested that, had the runoff been Edwin Edwards v Buddy Roemer, Roemer would have won. However, in the actual election, David Duke managed to finish second and make the runoff instead of Roemer, a runoff which Edwards then won by a large margin. Thus, the presence of Duke in the election changed which of the non-Duke candidates won.

In econometrics

IIA is a property of the multinomial logit and the conditional logit models in econometrics; outcomes that could theoretically violate this IIA (such as the outcome of multicandidate elections or any choice made by humans) may make multinomial logit and conditional logit invalid estimators.

IIA implies that adding another alternative or changing the characteristics of a third alternative does not affect the relative odds between the two alternatives considered. This implication is not realistic for applications with similar alternatives. Many examples have been constructed to illustrate this problem.^[6]

Consider the Red Bus/Blue Bus example. Commuters initially face a decision between two modes of transportation: car and red bus. Suppose that a consumer chooses between these two options with equal probability, 0.5, so that the odds ratio equals 1. Now suppose a third mode, blue bus, is added. Assuming bus commuters do not care about the color of the bus, consumers are expected to choose between bus and car still with equal probability, so the probability of car is still 0.5, while the probabilities of each of the two bus types is 0.25. But IIA implies that this is not the case: for the odds ratio between car and red bus to be preserved, the new probabilities must be: car 0.33; red bus 0.33; blue bus 0.33.^[7] In intuitive terms, the problem with the IIA axiom is that it leads to a failure to take account of the fact that red bus and blue bus are very similar, and are "perfect substitutes".

Many modeling advances have been motivated by a desire to alleviate the concerns raised by IIA. Generalized extreme value,^[8] multinomial probit (also called conditional probit) and mixed logit are alternative models for nominal outcomes which relax IIA, but these models often have assumptions of their own that may be difficult to meet or are computationally infeasible. The multinomial probit model has as a disadvantage that it makes calculation of maximum likelihood infeasible for more than five alternatives as it involves multiple integrals. IIA can also be relaxed by specifying a hierarchical model, ranking the choice alternatives. The most popular of these is called the nested logit model.^[9]

Generalized extreme value and multinomial probit models possess another property, the Invariant Proportion of Substitution^[10] which suggests similarly counterintuitive individual choice behavior.

Choice under uncertainty

In the expected utility theory of von Neumann and Morgenstern, four axioms together imply that individuals act in situations of risk as if they maximize the expected value of a utility function. One of the axioms is a version of the IIA axiom:

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

$\,pL+(1-p)N \prec pM+(1-p)N.\,$

where p is a probability and $\,L\prec M\,$ means that M is preferred over L. This axiom says that if one outcome (or lottery ticket) L is considered to be not as good as another (M), then having a chance 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.

See also

• Luce's choice axiom
• Menu dependence
• Monty Hall problem, in which an seemingly-unrelated piece of information makes a difference to a choice

References

• Kenneth J. Arrow (1963), Social Choice and Individual Values
• Paramesh Ray (1973). "Independence of Irrelevant Alternatives," Econometrica, Vol. 41, No. 5, p p. 987-991. Discusses and deduces the (not always recognized) differences between various formulations of IIA.
• Peter Kennedy (2003), A Guide to Econometrics, 5th ed.
• G.S. Maddala (1983). Limited-dependent and Qualitative Variables in Econometrics

External links

• Steven Callander and Catherine H.Wilson, "Context-dependent Voting," Quarterly Journal of Political Science, 2006, 1: 227–254

• Thomas J. Steenburgh, (2008) "Invariant Proportion of Substitution Property (IPS) of Discrete-Choice Models," Marketing Science, Vol. 27, No. 2, pp. 300–307.

Footnotes

1. ^ Sen, 1970, page 17.
2. ^ Arrow, 1963, page 28.
3. ^ Arrow, 1951, pp. 15, 23, 27
4. ^ More formally, an aggregation rule (social welfare function) f is Pairwise Independent if for any profiles $p=(R_1, \ldots, R_n)$, $p'=(R'_1, \ldots, R'_n)$ of preferences and for any alternatives x, y, if $R_i\cap\{x,y\}^2=R'_i\cap \{x,y\}^2$ for all i, then $f(p)\cap\{x,y\}^2=f(p')\cap \{x,y\}^2$. This is the definition of Arrow's IIA adopted in the contxt of Arrow's theorem in most textbooks and surveys (Austen-Smith and Banks, 1999, page 27; Campbell and Kelly, 2002, in Handbook of SCW, page 43; Feldman and Serrano, 2005, Section 13.3.5; Gaertner, 2009, page 20; Mas-Colell, Whinston, Green, 1995, page 794; Nitzan, 2010, page 40; Tayor, 2005, page 18; see also Arrow, 1963, page 28 and Sen, 1970, page 37). Observe that this formulation does not consider addition or deletion of alternatives, since the set of alternatives is fixed. Also, note that this is a condition involving two profiles.
5. ^ Paramesh Ray, "Independence of Irrelevant Alternatives," Econometrica, Vol. 41, No. 5, pp. 987-991.
6. ^ Beethoven/Debussy (Debreu 1960; Tversky 1972), Bicycle/Pony (Luce and Suppes 1965), and Red Bus/Blue Bus (McFadden 1974)
7. ^ Wooldridge 2002, pp. 501-2
8. ^ McFadden 1978
9. ^ McFadden 1984
10. ^ Steenburgh 2008