Independence of irrelevant alternatives

The independence of irrelevant alternatives (IIA), also known as binary independence[1] 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 (alpha): 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.[2] 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 ("votes") satisfying IIA in addition to certain other reasonable conditions. Arrow defines IIA thus:

The social preferences between alternatives x and y depend only on the individual preferences between x and y.[3]

Another expression of the principle:

If A is preferred to B out of the choice set {A,B}, introducing a third option X, 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.

In social choice theory, IIA is also defined as:

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 if only preferences for X change, the voting rule must not lead to B's being selected over A.

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.

Voting theory

In voting systems, independence from irrelevant alternatives is often interpreted as, if one candidate (X) would win an election, and if a new candidate (Y) 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:

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 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).[4] LIIA requires that both of the following conditions always hold:

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.

Criticism of IIA

IIA is too strong to be satisfied by any voting method capable of majority rule when there are only two alternatives.

Consider a scenario in which there are three candidates A, B, & C, and the voters' preferences are as follows:

25% of the voters prefer A over B, and B over C. (A>B>C)
40% of the voters prefer B over C, and C over A. (B>C>A)
35% of the voters prefer C over A, and A over B. (C>A>B)

(These are preferences, not votes, and thus are independent of the voting method.)

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:

To show failure, it is only assumed at least possible that enough voters in the majority might cast a minimally positive vote for their preferred candidate when there are only two candidates, rather than abstain. Most ranked ballot methods and Plurality voting satisfy the Majority Criterion, and therefore fail IIA automatically by the example above. Meanwhile, passage of IIA by Approval and Range voting requires in certain cases that voters in the majority are necessarily excluded from voting (they are assumed to necessarily abstain in a two candidate race, despite having a meaningful preference between the alternatives).

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 (see Condorcet's Jury Theorem). 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 (who prefer A over B) 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,[5] each "voter" i in the society has an ordering Ri 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, ...,Rn) 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.[6] For example, suppose an aggregation rule ranks a above b at the profile given by

(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:

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.[7]

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 (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

Borda count

Main article: 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]. They change their preferences only over the pairs [B, D], [B, E] and [D, E].

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

The social choice has changed the ranking of [B, A] and [B, C]. 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:

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 from 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.

Copeland

Main article: Copeland's method

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:

Pairwise Preferences
X
A B C D
Y A [X] 2
[Y] 4		 [X] 2
[Y] 4		 [X] 4
[Y] 2
B [X] 4
[Y] 2		 [X] 3
[Y] 3		 [X] 2
[Y] 4
C [X] 4
[Y] 2		 [X] 3
[Y] 3		 [X] 2
[Y] 4
D [X] 2
[Y] 4		 [X] 4
[Y] 2		 [X] 4
[Y] 2
Pairwise election results (won-tied-lost): 2-0-1 1-1-1 1-1-1 1-0-2

Result: A has two wins and one defeat, while no other candidate has more wins than defeats. Thus, A is elected Copeland winner.

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:

Pairwise Preferences
X
A B C D
Y A [X] 2
[Y] 4		 [X] 2
[Y] 4		 [X] 4
[Y] 2
B [X] 4
[Y] 2		 [X] 3
[Y] 3		 [X] 6
[Y] 0
C [X] 4
[Y] 2		 [X] 3
[Y] 3		 [X] 6
[Y] 0
D [X] 2
[Y] 4		 [X] 0
[Y] 6		 [X] 0
[Y] 6
Pairwise election results (won-tied-lost): 2-0-1 0-1-2 0-1-2 3-0-0

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

Main article: 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]. They change only their preferences over B and C.

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

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

Kemeny–Young method

Main article: 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:

All possible pairs
of choice names 		Number of votes with indicated preference
Prefer X over Y Equal preference Prefer Y over X
X = A Y = B 5 0 2
X = A Y = C 3 0 4
X = B Y = C 5 0 2

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 (marked bold) 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:

All possible pairs
of choice names 		Number of votes with indicated preference
Prefer X over Y Equal preference Prefer Y over X
X = A Y = B 5 0 2
X = A Y = C 3 0 4
X = B Y = C 3 0 4

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

Main article: Minimax Condorcet

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 (no equals are present), all three Minimax methods (winning votes, margins and pairwise opposite) elect the same winners.

The results would be tabulated as follows:

Pairwise election results
X
A B C
Y A [X] 5
[Y] 8		 [X] 7
[Y] 6
B [X] 8
[Y] 5		 [X] 4
[Y] 9
C [X] 6
[Y] 7		 [X] 9
[Y] 4
Pairwise election results (won-tied-lost): 1-0-1 1-0-1 1-0-1
worst pairwise defeat (winning votes): 7 8 9
worst pairwise defeat (margins): 1 3 5
worst pairwise opposition: 7 8 9

Result: A has the closest biggest defeat. Thus, A is elected Minimax winner.

Change of irrelevant preferences

Now, assume the two voters (marked bold) 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:

Pairwise election results
X
A B C
Y A [X] 5
[Y] 8		 [X] 9
[Y] 4
B [X] 8
[Y] 5		 [X] 4
[Y] 9
C [X] 4
[Y] 9		 [X] 9
[Y] 4
Pairwise election results (won-tied-lost): 1-0-1 1-0-1 1-0-1
worst pairwise defeat (winning votes): 9 8 9
worst pairwise defeat (margins): 5 3 5
worst pairwise opposition: 9 8 9

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 (A, B, C).

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.

Ranked pairs

Main article: 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:

Pairwise election results
X
A B C
Y A [X] 2
[Y] 5		 [X] 4
[Y] 3
B [X] 5
[Y] 2		 [X] 2
[Y] 5
C [X] 3
[Y] 4		 [X] 5
[Y] 2
Pairwise election results (won-tied-lost): 1-0-1 1-0-1 1-0-1

The sorted list of victories would be:

Pair Winner
A (5) vs. B (2) A 5
B (5) vs. C (2) B 5
A (3) vs. C (4) C 4

Result: A > B and B > C are locked in (and C > A cannot be locked in after that), so the full ranking is A > B > C. Thus, A is elected Ranked pairs winner.

Change of irrelevant preferences

Now, assume the two voters (marked bold) 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:

Pairwise election results
X
A B C
Y A [X] 2
[Y] 5		 [X] 4
[Y] 3
B [X] 5
[Y] 2		 [X] 4
[Y] 3
C [X] 3
[Y] 4		 [X] 3
[Y] 4
Pairwise election results (won-tied-lost): 1-0-1 0-0-2 2-0-0

The sorted list of victories would be:

Pair Winner
A (5) vs. B (2) A 5
B (3) vs. C (4) C 4
A (3) vs. C (4) 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

Main article: 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:

Matrix of pairwise preferences
d[*,A] d[*,B] d[*,C] d[*,D]
d[A,*] 9 6 4
d[B,*] 3 7 6
d[C,*] 6 5 9
d[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 (which is nullified, since it is a tie).

Strengths of the strongest paths
d[*,A] d[*,B] d[*,C] d[*,D]
d[A,*] 9 7 7
d[B,*] 7 7 7
d[C,*] 8 8 9
d[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 (marked bold) 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:

Matrix of pairwise preferences
d[*,A] d[*,B] d[*,C] d[*,D]
d[A,*] 9 6 4
d[B,*] 3 9 6
d[C,*] 6 3 9
d[D,*] 8 6 3

Now, the strongest paths have to be identified:

Strengths of the strongest paths
d[*,A] d[*,B] d[*,C] d[*,D]
d[A,*] 9 9 9
d[B,*] 8 9 9
d[C,*] 8 8 9
d[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

Main article: Two-round system

A probable example of the two-round system 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 election David Duke finished second and entered the runoff instead of Roemer, which Edwards won by a large margin. Thus, the presence of Duke in the election changed which of the non-Duke 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.[8]

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, the new probabilities must be car 0.33; red bus 0.33; blue bus 0.33.[9] 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".

In econometrics

IIA is a property assumed by the multinomial logit and the conditional logit models in econometrics. If these models are used in situations which in fact violate independence (such as multicandidate elections in which preferences exhibit cycling or situations mimicking the Red Bus/Blue Bus example given above) then these estimators become invalid.

Many modeling advances have been motivated by a desire to alleviate the concerns raised by IIA. Generalized extreme value,[10] multinomial probit (also called conditional 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. The multinomial probit model has as a disadvantage that it makes calculation of maximum likelihood infeasible for more than five options as it involves multiple integrals. IIA can be relaxed by specifying a hierarchical model, ranking the choice alternatives. The most popular of these is the nested logit model.[11]

Generalized extreme value and multinomial probit models possess another property, the Invariant Proportion of Substitution,[12] 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 , then for any and ,

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 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.

In nature

Natural selection can favor animals' non-IIA-type choices, thought to be due to occasional availability of foodstuffs, according to a study published in January 2014.[13]

See also

Footnotes

  1. ↑ Saari, Donald G. (2001). Decisions and elections : explaining the unexpected (1. publ. ed.). Cambridge [u.a.]: Cambridge Univ. Press. p. 39. ISBN 0-521-00404-7.
  2. ↑ Sen, 1970, page 17.
  3. ↑ Arrow 1963, p. 28.
  4. ↑ Young, H. Peyton (1995). Equity: In Theory and Practice. Princeton University Press. ISBN 0-691-04464-3.
  5. ↑ Arrow 1951, pp. 15, 23, 27.
  6. ↑ More formally, an aggregation rule (social welfare function) f is pairwise independent if for any profiles , of preferences and for any alternatives x, y, if for all i, then . This is the definition of Arrow's IIA adopted in the context 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). This formulation does not consider addition or deletion of options, since the set of options is fixed, and this is a condition involving two profiles.
  7. ↑ Ray 1973.
  8. ↑ Beethoven/Debussy (Debreu 1960; Tversky 1972), Bicycle/Pony (Luce and Suppes 1965), and Red Bus/Blue Bus (McFadden 1974)
  9. ↑ Wooldridge 2002, pp. 501-2
  10. ↑ McFadden 1978
  11. ↑ McFadden 1984
  12. ↑ Steenburgh 2008
  13. ↑ McNamara, J. M.; Trimmer, P. C.; Houston, A. I. (2014). "Natural selection can favour 'irrational' behaviour" (PDF). Biology Letters. 10 (1). doi:10.1098/rsbl.2013.0935.

References

Further reading

This article is issued from Wikipedia - version of the 10/18/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.