Independence of irrelevant alternatives
The independence of irrelevant alternatives, also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used with different meanings in different contexts; although they all attempt to provide an account of rational individual behavior or aggregation of individual preferences, the exact formulations differ from context to context.
In individual choice theory, IIA sometimes refers to Chernoff's condition or Sen's property α :
if an alternative x is chosen from a set T, and x is also an element of a subset S of T, then x must be chosen from S. That is, eliminating some of the unchosen alternatives shouldn't affect the selection of x as the best option.
In social choice theory, Arrow's IIA is one of the conditions in Arrow's impossibility theorem, which states that it is impossible to aggregate individual rank-order preferences satisfying IIA in addition to certain other reasonable conditions. Arrow defines IIA thus:
Another expression of the principle:
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 have shown that human behavior rarely adheres to this axiom.
In social choice theory, IIA is also defined as:
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 irrelevant to the choice between A and B. The results of violating IIA is commonly referred to as the "Spoiler Effect" because support for X "spoils" the election for A.
Voting theory
In voting systems, independence from irrelevant alternatives is often interpreted as, if one candidate would win an election, and if a new candidate were added to the ballot, then either X or Y would win the election.Approval voting, range voting, and majority judgment satisfy the IIA criterion if it is assumed that voters rate candidates individually and independently of knowing the available alternatives in the election, using their own absolute scale. This assumption implies that some voters having meaningful preferences in an election with only two alternatives will necessarily cast a vote which has little or no voting power, or necessarily abstain. If it is assumed to be at least possible that any voter having preferences might not abstain, or vote their favorite and least favorite candidates at the top and bottom ratings respectively, then these systems fail IIA. Allowing either of these conditions alone causes failure. Another cardinal system, cumulative voting, does not satisfy the criterion regardless of either assumption.
An anecdote that illustrates a violation of IIA has been attributed to Sidney Morgenbesser:
All voting systems have some degree of inherent susceptibility to strategic nomination considerations. Some regard these considerations as less serious unless the voting system fails the easier-to-satisfy independence of clones criterion.
Local independence
A criterion weaker than IIA proposed by H. Peyton Young and A. Levenglick is called local independence from irrelevant alternatives.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.
- If the winning option is deleted from all the votes, the order of finish of the remaining options must not change.
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 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, even when this assumption implies that voters having meaningful preferences in a two candidate election will necessarily abstain.
Criticism of IIA
IIA is largely incompatible with the majority criterion unless there are only two alternatives.Consider a scenario in which there are three candidates A, B, & C, and the voters' preferences are as follows:
75% prefer C over A, 65% prefer B over C, and 60% prefer A over B. The presence of this societal intransitivity is the voting paradox. Regardless of the voting method and the actual votes, there are only three cases to consider:
- Case 1: A is elected. IIA is violated because the 75% who prefer C over A would elect C if B were not a candidate.
- Case 2: B is elected. IIA is violated because the 60% who prefer A over B would elect A if C were not a candidate.
- Case 3: C is elected. IIA is violated because the 65% who prefer B over C would elect B if A were not a candidate.
So even if IIA is desirable, requiring its satisfaction seems to allow only voting methods that are undesirable in some other way, such as treating one of the voters as a dictator. Thus the goal must be to find which voting methods are best, rather than which are perfect.
An argument can be made that IIA is itself undesirable. IIA assumes that when deciding whether A is likely to be better than B, information about voters' preferences regarding C is irrelevant and should not make a difference. However, the heuristic that leads to majority rule when there are only two options is that the larger the number of people who think one option is better than the other, the greater the likelihood that it is better, all else being equal. A majority is more likely than the opposing minority to be right about which of the two candidates is better, all else being equal, hence the use of majority rule.
The same heuristic implies that the larger the majority, the more likely it is that they are right. It would seem to also imply that when there is more than one majority, larger majorities are more likely to be right than smaller majorities. Assuming this is so, the 75% who prefer C over A and the 65% who prefer B over C are more likely to be right than the 60% who prefer A over B, and since it is not possible for all three majorities to be right, the smaller majority are more likely to be wrong, and less likely than their opposing minority to be right. Rather than being irrelevant to whether A is better than B, the additional information about the voters' preferences regarding C provides a strong hint that this is a situation where all else is not equal.
In social choice
From Kenneth Arrow, each "voter" i in the society has an ordering Ri that ranks the objects of social choice—x, y, and z in simplest case—from high to low.An aggregation rule in turn maps each profile or tuple of voter preferences
to a social ordering R that determines the social preference of x, y, and z.
Arrow's IIA requires that whenever a pair of alternatives is ranked the same way in two preference profiles, then the aggregation rule must order these alternatives identically across the two profiles.
For example, suppose an aggregation rule ranks a above b at the profile given by
- ,
*
*
- .
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.
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 and such that the ranking of X and Y for each voter i is the same for ' and '. The voting rule generates corresponding social orderings R and R'. Now suppose that X and Y are feasible but Z is infeasible. Arrow required that the voting rule that R and R' select the same social choice from the feasible set, 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, and it says nothing about what would happen in such a case: all options are assumed to be "feasible."
Examples
Borda count
In a Borda count election, 5 voters rank 5 alternatives .3 voters rank .
1 voter ranks .
1 voter ranks .
Borda count : C=13, A=12, B=11, D=8, E=6. C wins.
Now, the voter who ranks instead ranks ; and the voter who ranks instead ranks . They change their preferences only over the pairs , and .
The new Borda count: B=14, C=13, A=12, E=6, D=5. B wins.
The social choice has changed the ranking of and . 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 .
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.Considering 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.
- In profile 2, the second voter votes ABC, so A and B will tie.
Copeland
This example shows that Copeland's method violates IIA. Assume four candidates A, B, C and D with 6 voters with the following preferences:| # of voters | Preferences |
| 1 | A > B > C > D |
| 1 | A > C > B > D |
| 2 | B > D > A > C |
| 2 | C > D > A > B |
The results would be tabulated as follows:
- indicates voters who preferred the candidate in the column caption to the one in the row caption
- indicates voters who preferred the candidate in the row caption to the one in the column caption
Change of irrelevant preferences
Now, assume all voters would raise D over B and C without changing the order of A and D. The preferences of the voters would now be:| # of voters | Preferences |
| 1 | A > D > B > C |
| 1 | A > D > C > B |
| 2 | D > B > A > C |
| 2 | D > C > A > B |
The results would be tabulated as follows:
Result: D wins against all three opponents. Thus, D is elected Copeland winner.
Conclusion
The voters changed only their preference orders over B, C and D. As a result, the outcome order of D and A changed. A turned from winner to loser without any change of the voters' preferences regarding A. Thus, Copeland's method fails the IIA criterion.Instant-runoff voting
In an instant-runoff election, 5 voters rank 3 alternatives .2 voters rank .
2 voters rank .
1 voter ranks .
Round 1: A=2, B=1, C=2; B eliminated.
Round 2: A=3, C=2; A wins.
Now, the two voters who rank instead rank . They change only their preferences over B and C.
Round 1: A=2, B=3, C=0; B wins with a majority of the vote.
The social choice ranking of is dependent on preferences over the irrelevant alternatives .
Kemeny–Young method
This example shows that the Kemeny–Young method violates the IIA criterion. Assume three candidates A, B and C with 7 voters and the following preferences:| # of voters | Preferences |
| 3 | A > B > C |
| 2 | B > C > A |
| 2 | C > A > B |
The Kemeny–Young method arranges the pairwise comparison counts in the following tally table:
The ranking scores of all possible rankings are:
| Preferences | 1. vs 2. | 1. vs 3. | 2. vs 3. | Total |
| A > B > C | 5 | 3 | 5 | 13 |
| A > C > B | 3 | 5 | 2 | 10 |
| B > A > C | 2 | 5 | 3 | 10 |
| B > C > A | 5 | 2 | 4 | 11 |
| C > A > B | 4 | 2 | 5 | 11 |
| C > B > A | 2 | 4 | 2 | 8 |
Result: The ranking A > B > C has the highest ranking score. Thus, A wins ahead of B and C.
Change of irrelevant preferences
Now, assume the two voters with preferences B > C > A would change their preferences over the pair B and C. The preferences of the voters would then be in total:| # of voters | Preferences |
| 3 | A > B > C |
| 2 | C > B > A |
| 2 | C > A > B |
The Kemeny–Young method arranges the pairwise comparison counts in the following tally table:
The ranking scores of all possible rankings are:
| Preferences | 1. vs 2. | 1. vs 3. | 2. vs 3. | Total |
| A > B > C | 5 | 3 | 3 | 11 |
| A > C > B | 3 | 5 | 4 | 12 |
| B > A > C | 2 | 3 | 3 | 8 |
| B > C > A | 3 | 2 | 4 | 9 |
| C > A > B | 4 | 4 | 5 | 13 |
| C > B > A | 4 | 4 | 2 | 10 |
Result: The ranking C > A > B has the highest ranking score. Thus, C wins ahead of A and B.
Conclusion
The two voters changed only their preferences over B and C, but this resulted in a change of the order of A and C in the result, turning A from winner to loser without any change of the voters' preferences regarding A. Thus, the Kemeny-Young method fails the IIA criterion.Minimax
This example shows that the Minimax method violates the IIA criterion. Assume four candidates A, B and C and 13 voters with the following preferences:| # of voters | Preferences |
| 2 | B > A > C |
| 4 | A > B > C |
| 3 | B > C > A |
| 4 | C > A > B |
Since all preferences are strict rankings, all three Minimax methods elect the same winners.
The results would be tabulated as follows:
- indicates voters who preferred the candidate in the column caption to the one in the row caption
- indicates voters who preferred the candidate in the row caption to the one in the column caption
Change of irrelevant preferences
Now, assume the two voters with preferences B > A > C change the preferences over the pair A and C. The preferences of the voters would then be in total:| # of voters | Preferences |
| 4 | A > B > C |
| 5 | B > C > A |
| 4 | C > A > B |
The results would be tabulated as follows:
Result: Now, B has the closest biggest defeat. Thus, B is elected Minimax winner.
Conclusion
So, by changing the order of A and C in the preferences of some voters, the order of A and B in the result changed. B is turned from loser to winner without any change of the voters' preferences regarding B. Thus, the Minimax method fails the IIA criterion.Plurality voting system
In a plurality voting system 7 voters rank 3 alternatives.- 3 voters rank
- 2 voters rank
- 2 voters rank
The relative positions of A and B are reversed by the introduction of C, an "irrelevant" alternative.
Ranked pairs
This example shows that the Ranked pairs method violates the IIA criterion. Assume three candidates A, B and C and 7 voters with the following preferences:| # of voters | Preferences |
| 3 | A > B > C |
| 2 | B > C > A |
| 2 | C > A > B |
The results would be tabulated as follows:
The sorted list of victories would be:
| Pair | Winner |
| A vs. B | A 5 |
| B vs. C | B 5 |
| A vs. C | C 4 |
Result: A > B and B > C are locked in, so the full ranking is A > B > C. Thus, A is elected Ranked pairs winner.
Change of irrelevant preferences
Now, assume the two voters with preferences B > C > A change their preferences over the pair B and C. The preferences of the voters would then be in total:| # of voters | Preferences |
| 3 | A > B > C |
| 2 | C > B > A |
| 2 | C > A > B |
The results would be tabulated as follows:
The sorted list of victories would be:
| Pair | Winner |
| A vs. B | A 5 |
| B vs. C | C 4 |
| A vs. C | C 4 |
Result: All three duels are locked in, so the full ranking is C > A > B. Thus, the Condorcet winner C is elected Ranked pairs winner.
Conclusion
So, by changing their preferences over B and C, the two voters changed the order of A and C in the result, turning A from winner to loser without any change of the voters' preferences regarding A. Thus, the Ranked pairs method fails the IIA criterion.Schulze method
This example shows that the Schulze method violates the IIA criterion. Assume four candidates A, B, C and D and 12 voters with the following preferences:| # of voters | Preferences |
| 4 | A > B > C > D |
| 2 | C > B > D > A |
| 3 | C > D > A > B |
| 2 | D > A > B > C |
| 1 | D > B > C > A |
The pairwise preferences would be tabulated as follows:
| d | d | d | d | |
| d | 9 | 6 | 4 | |
| d | 3 | 7 | 6 | |
| d | 6 | 5 | 9 | |
| d | 8 | 6 | 3 | - |
Now, the strongest paths have to be identified, e.g. the path D > A > B is stronger than the direct path D > B.
| d | d | d | d | |
| d | 9 | 7 | 7 | |
| d | 7 | 7 | 7 | |
| d | 8 | 8 | 9 | |
| d | 8 | 8 | 7 | - |
Result: The full ranking is C > D > A > B. Thus, C is elected Schulze winner and D is preferred over A.
Change of irrelevant preferences
Now, assume the two voters with preferences C > B > D > A change their preferences over the pair B and C. The preferences of the voters would then be in total:| # of voters | Preferences |
| 4 | A > B > C > D |
| 2 | B > C > D > A |
| 3 | C > D > A > B |
| 2 | D > A > B > C |
| 1 | D > B > C > A |
Hence, the pairwise preferences would be tabulated as follows:
| d | d | d | d | |
| d | 9 | 6 | 4 | |
| d | 3 | 9 | 6 | |
| d | 6 | 3 | 9 | |
| d | 8 | 6 | 3 | - |
Now, the strongest paths have to be identified:
| d | d | d | d | |
| d | 9 | 9 | 9 | |
| d | 8 | 9 | 9 | |
| d | 8 | 8 | 9 | |
| d | 8 | 8 | 8 | - |
Result: Now, the full ranking is A > B > C > D. Thus, A is elected Schulze winner and is preferred over D.
Conclusion
So, by changing their preferences over B and C, the two voters changed the order of A and D in the result, turning A from loser to winner without any change of the voters' preferences regarding A. Thus, the Schulze method fails the IIA criterion.Two-round system
A probable example of the two-round system failing this criterion was the 2002 French presidential election. Polls leading up to the election have suggested a runoff between centre-right candidate Jacques Chirac and centre-left candidate Lionel Jospin, in which Jospin has been expected to win. However, the first round was contested by an unprecedented 16 candidates, including left-wing candidates who intended to support Jospin in the runoff, eventually resulting in the far-right candidate, Jean-Marie Le Pen, finishing second and entering the runoff instead of Jospin, which Chirac won by a large margin. Thus, the presence of many candidates who did not intend to win in the election changed which of the candidates won.Criticisms of the IIA assumption
IIA implies that adding another option or changing the characteristics of a third option does not affect the relative odds between the two options considered. This implication is not realistic for applications with similar options. Many examples have been constructed to illustrate this problem.Consider the Red Bus/Blue Bus example. Commuters face a decision between car and red bus. Suppose that a commuter chooses between these two options with equal probability, 0.5, so that the odds ratio equals 1:1. Now suppose a third mode, blue bus, is added. Assuming bus commuters do not care about the color of the bus, they are expected to choose between bus and car still with equal probability, so the probability of car is still 0.5, while the probability 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, and the odds of red and blue bus to be equal, the new probabilities must be car 0.33; red bus 0.33; blue bus 0.33. The blue bus is of course not irrelevant if it is chosen, but it must be treated as irrelevant when it is not chosen, leading to a decreased overall probability of car travel, which does not make sense for a commuter who does not care about colors. 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".
The failure of this assumption has also been observed in practice, for example in the opinion polling for the 2019 European Elections held in the United Kingdom. In one survey, 21% of potential voters expressed support for the Labour Party under the scenario where there were three smaller Anti-Brexit parties to choose from, but under a scenario where two of those three parties did not stand candidates, the support for Labour dropped to 18%. This means at least 3% of potential voters stopped supporting their preferred party when a less preferred party dropped out.
In econometrics
IIA is a direct consequence of the assumptions underlying the multinomial logit and the conditional logit models in econometrics. If these models are used in situations which in fact violate independence then these estimators become invalid.Many modeling advances have been motivated by a desire to alleviate the concerns raised by IIA. Generalized extreme value, multinomial probit and mixed logit are models for nominal outcomes that relax IIA, but they often have assumptions of their own that may be difficult to meet or are computationally infeasible. IIA can be relaxed by specifying a hierarchical model, ranking the choice alternatives. The most popular of these is the nested logit model.
Generalized extreme value and multinomial probit models possess another property, the Invariant Proportion of Substitution, 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 an independence axiom analogous to the IIA axiom:where p is a probability, pL+N means a gamble with probability p of yielding L and probability of yielding N, and means that M is preferred over L. This axiom says that if one outcome L is considered to be not as good as another, 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.