Rated voting rules, where voters assign a separate grade to each candidate, are not affected by Arrow's theorem (allowing some to satisfy IIA).[5][6][20]Arrow initially asserted the information provided by these systems was meaningless and therefore could not be used to prevent paradoxes, leading him to overlook them.[21] However, he and other authors would later recognize this as a mistake,[22][23] with Arrow admitting rules based on cardinal utilities (such as score and approval voting) are not subject to his theorem.[24][25]
In the context of Arrow's theorem, citizens are assumed to have ordinal preferences, i.e. orderings of candidates. If A and B are different candidates or alternatives, then means A is preferred to B. Individual preferences (or ballots) are required to satisfy intuitive properties of orderings, e.g. they must be transitive—if and , then . The social choice function is then a mathematical function that maps the individual orderings to a new ordering that represents the preferences of all of society.
Basic assumptions
Arrow's theorem assumes as background that non-degenerate ranked social choice rules satisfy:[26]
This assumption defines social choices as those depending on more than one person's input.[28]
Non-imposition — the system does not ignore the voters entirely when choosing between some pairs of candidates.[29][30]
In other words, it is possible for any candidate to defeat any other candidate, given some combination of votes.[29][30][31]
This is typically replaced with the stronger Pareto efficiency: if voters unanimously support candidate A over candidate B, then candidate A should beat candidate B.[29]
Arrow's original statement of the theorem included the assumption of nonperversity, i.e. increasing the rank of an outcome should not make them lose.[32] However, this assumption is not needed or used in his proof, except to derive the weaker Pareto efficiency axiom, and as a result is not related to the paradox.[28] While Arrow considered it an obvious requirement of any proposed social choice rule, ranked-choice runoff (RCV) fails this condition.[33] Arrow later corrected his statement of the theorem to include runoffs and other perverse voting rules.[28][33]
Independence of irrelevant alternatives (IIA) — the social preference between candidate A and candidate B should only depend on the individual preferences between A and B.
In other words, the social preference should not change from to if voters change their preference about whether .[28]
Morgenbesser, ordering dessert, is told by a waitress that he can choose between blueberry or apple pie. He orders apple. Soon the waitress comes back and explains cherry pie is also an option. Morgenbesser replies "In that case, I'll have blueberry."
Arrow's theorem shows that if a society wishes to make decisions while avoiding such self-contradictions, it cannot use methods that discard cardinal information.[36]
Theorem
Intuitive argument
Condorcet's example is already enough to see the impossibility of a fair ranked voting system, given stronger conditions for fairness than Arrow's theorem assumes.[37] Suppose we have three candidates (, , and ) and three voters whose preferences are as follows:
Voter
First preference
Second preference
Third preference
Voter 1
A
B
C
Voter 2
B
C
A
Voter 3
C
A
B
If is chosen as the winner, it can be argued any fair voting system would say should win instead, since two voters (1 and 2) prefer to and only one voter (3) prefers to . However, by the same argument is preferred to , and is preferred to , by a margin of two to one on each occasion. Thus, even though each individual voter has consistent preferences, the preferences of society are contradictory: is preferred over which is preferred over which is preferred over .
Because of this example, some authors credit Condorcet with having given an intuitive argument that presents the core of Arrow's theorem.[37] However, Arrow's theorem is substantially more general; it applies to methods of making decisions other than one-man-one-vote elections, such as markets or weighted voting, based on ranked ballots.
The element being in is interpreted to mean that alternative is preferred to alternative . This situation is often denoted or . Denote the set of all preferences on by .
Let be a positive integer. An ordinal (ranked)social welfare function is a function[38]
which aggregates voters' preferences into a single preference on . An -tuple of voters' preferences is called a preference profile.
Arrow's impossibility theorem: If there are at least three alternatives, then there is no social welfare function satisfying all three of the conditions listed below:[39]
There is no individual whose preferences always prevail. That is, there is no such that for all and all and , when is preferred to by then is preferred to by .[38]
For two preference profiles and such that for all individuals , alternatives and have the same order in as in , alternatives and have the same order in as in .[38]
Formal proof
Proof by decisive coalition
Arrow's proof used the concept of decisive coalitions.[40]
Definition:
A subset of voters is a coalition.
A coalition is decisive over an ordered pair if, when everyone in the coalition ranks , society overall will always rank .
A coalition is decisive if and only if it is decisive over all ordered pairs.
Our goal is to prove that the decisive coalition contains only one voter, who controls the outcome—in other words, a dictator.
The following proof is a simplification taken from Amartya Sen[41] and Ariel Rubinstein.[42] The simplified proof uses an additional concept:
A coalition is weakly decisive over if and only if when every voter in the coalition ranks , and every voter outside the coalition ranks , then .
Thenceforth assume that the social choice system satisfies unrestricted domain, Pareto efficiency, and IIA. Also assume that there are at least 3 distinct outcomes.
Field expansion lemma — if a coalition is weakly decisive over for some , then it is decisive.
Proof
Let be an outcome distinct from .
Claim: is decisive over .
Let everyone in vote over . By IIA, changing the votes on does not matter for . So change the votes such that in and and outside of .
By Pareto, . By coalition weak-decisiveness over , . Thus .
Similarly, is decisive over .
By iterating the above two claims (note that decisiveness implies weak-decisiveness), we find that is decisive over all ordered pairs in . Then iterating that, we find that is decisive over all ordered pairs in .
Group contraction lemma — If a coalition is decisive, and has size , then it has a proper subset that is also decisive.
Proof
Let be a coalition with size . Partition the coalition into nonempty subsets .
Fix distinct . Design the following voting pattern (notice that it is the cyclic voting pattern which causes the Condorcet paradox):
(Items other than are not relevant.)
Since is decisive, we have . So at least one is true: or .
If , then is weakly decisive over . If , then is weakly decisive over . Now apply the field expansion lemma.
By Pareto, the entire set of voters is decisive. Thus by the group contraction lemma, there is a size-one decisive coalition—a dictator.
Proof by pivotal voter
Proofs using the concept of the pivotal voter originated from Salvador Barberá in 1980.[43] The proof given here is a simplified version based on two proofs published in Economic Theory.[39][44]
We will prove that any social choice system respecting unrestricted domain, unanimity, and independence of irrelevant alternatives (IIA) is a dictatorship. The key idea is to identify a pivotal voter whose ballot swings the societal outcome. We then prove that this voter is a partial dictator (in a specific technical sense, described below). Finally we conclude by showing that all of the partial dictators are the same person, hence this voter is a dictator.
For simplicity we have presented all rankings as if there are no ties. A complete proof taking possible ties into account is not essentially different from the one given here, except that one ought to say "not above" instead of "below" or "not below" instead of "above" in some cases. Full details are given in the original articles.
Part one: There is a "pivotal" voter for B over A
Say there are three choices for society, call them A, B, and C. Suppose first that everyone prefers option B the least: everyone prefers A to B, and everyone prefers C to B. By unanimity, society must also prefer both A and C to B. Call this situation profile 0.
On the other hand, if everyone preferred B to everything else, then society would have to prefer B to everything else by unanimity. Now arrange all the voters in some arbitrary but fixed order, and for each i let profile i be the same as profile 0, but move B to the top of the ballots for voters 1 through i. So profile 1 has B at the top of the ballot for voter 1, but not for any of the others. Profile 2 has B at the top for voters 1 and 2, but no others, and so on.
Since B eventually moves to the top of the societal preference as the profile number increases, there must be some profile, number k, for which Bfirst moves aboveA in the societal rank. We call the voter k whose ballot change causes this to happen the pivotal voter for B over A. Note that the pivotal voter for B over A is not, a priori, the same as the pivotal voter for A over B. In part three of the proof we will show that these do turn out to be the same.
Also note that by IIA the same argument applies if profile 0 is any profile in which A is ranked above B by every voter, and the pivotal voter for B over A will still be voter k. We will use this observation below.
Part two: The pivotal voter for B over A is a dictator for B over C
In this part of the argument we refer to voter k, the pivotal voter for B over A, as the pivotal voter for simplicity. We will show that the pivotal voter dictates society's decision for B over C. That is, we show that no matter how the rest of society votes, if pivotal voter ranks B over C, then that is the societal outcome. Note again that the dictator for B over C is not a priori the same as that for C over B. In part three of the proof we will see that these turn out to be the same too.
In the following, we call voters 1 through k − 1, segment one, and voters k + 1 through N, segment two. To begin, suppose that the ballots are as follows:
Every voter in segment one ranks B above C and C above A.
Pivotal voter ranks A above B and B above C.
Every voter in segment two ranks A above B and B above C.
Then by the argument in part one (and the last observation in that part), the societal outcome must rank A above B. This is because, except for a repositioning of C, this profile is the same as profile k − 1 from part one. Furthermore, by unanimity the societal outcome must rank B above C. Therefore, we know the outcome in this case completely.
Now suppose that pivotal voter moves B above A, but keeps C in the same position and imagine that any number (even all!) of the other voters change their ballots to move B below C, without changing the position of A. Then aside from a repositioning of C this is the same as profile k from part one and hence the societal outcome ranks B above A. Furthermore, by IIA the societal outcome must rank A above C, as in the previous case. In particular, the societal outcome ranks B above C, even though Pivotal Voter may have been the only voter to rank B above C. By IIA, this conclusion holds independently of how A is positioned on the ballots, so pivotal voter is a dictator for B over C.
Part three: There exists a dictator
In this part of the argument we refer back to the original ordering of voters, and compare the positions of the different pivotal voters (identified by applying parts one and two to the other pairs of candidates). First, the pivotal voter for B over C must appear earlier (or at the same position) in the line than the dictator for B over C: As we consider the argument of part one applied to B and C, successively moving B to the top of voters' ballots, the pivot point where society ranks B above C must come at or before we reach the dictator for B over C. Likewise, reversing the roles of B and C, the pivotal voter for C over B must be at or later in line than the dictator for B over C. In short, if kX/Y denotes the position of the pivotal voter for X over Y (for any two candidates X and Y), then we have shown
kB/C ≤ kB/A ≤ kC/B.
Now repeating the entire argument above with B and C switched, we also have
kC/B ≤ kB/C.
Therefore, we have
kB/C = kB/A = kC/B
and the same argument for other pairs shows that all the pivotal voters (and hence all the dictators) occur at the same position in the list of voters. This voter is the dictator for the whole election.
Generalizations
Arrow's impossibility theorem still holds if Pareto efficiency is weakened to the following condition:[45]
Non-imposition
For any two alternatives a and b, there exists some preference profile R1 , …, RN such that a is preferred to b by F(R1, R2, …, RN).
Interpretation and practical solutions
Arrow's theorem establishes that no ranked voting rule can always satisfy independence of irrelevant alternatives, but it says nothing about the frequency of spoilers. This led Arrow to remark that "Most systems are not going to work badly all of the time. All I proved is that all can work badly at times."[46][47]
Attempts at dealing with the effects of Arrow's theorem take one of two approaches: either accepting his rule and searching for the least spoiler-prone methods, or dropping his assumption of ranked voting to focus on studying rated voting rules.[48]
The first set of methods studied by economists are the majority-rule, or Condorcet, methods. These rules limit spoilers to situations where majority rule is self-contradictory, called Condorcet cycles, and as a result uniquely minimize the possibility of a spoiler effect among ranked rules.[49] Condorcet believed voting rules should satisfy both independence of irrelevant alternatives and the majority rule principle, i.e. if most voters rank Alice ahead of Bob, Alice should defeat Bob in the election.[50]
Unfortunately, as Condorcet proved, this rule can be self-contradictory (intransitive), because there can be a rock-paper-scissors cycle with three or more candidates defeating each other in a circle.[51] Thus, Condorcet proved a weaker form of Arrow's impossibility theorem long before Arrow, under the stronger assumption that a voting system in the two-candidate case will agree with a simple majority vote.[50]
Unlike pluralitarian rules such as ranked-choice runoff (RCV) or first-preference plurality,[52]Condorcet methods avoid the spoiler effect in non-cyclic elections, where candidates can be chosen by majority rule. Political scientists have found such cycles to be fairly rare, likely in the range of a few percent, suggesting they may be of limited practical concern.[53]Spatial voting models also suggest such paradoxes are likely to be infrequent[54][55] or even non-existent.[18]
More formally, Black's theorem assumes preferences are single-peaked: a voter's happiness with a candidate goes up and then down as the candidate moves along some spectrum. For example, in a group of friends choosing a volume setting for music, each friend would likely have their own ideal volume; as the volume gets progressively too loud or too quiet, they would be increasingly dissatisfied. If the domain is restricted to profiles where every individual has a single-peaked preference with respect to the linear ordering, then social preferences are acyclic. In this situation, Condorcet methods satisfy a wide variety of highly-desirable properties, including being fully spoilerproof.[18][56][57]
The Campbell-Kelly theorem shows that Condorcet methods are the most spoiler-resistant class of ranked voting systems: whenever it is possible for some ranked voting system to avoid a spoiler effect, a Condorcet method will do so.[49] In other words, replacing a ranked method with its Condorcet variant (i.e. elect a Condorcet winner if they exist, and otherwise run the method) will sometimes prevent a spoiler effect, but can never create a new one.[49]
In 1977, Ehud Kalai and Eitan Muller gave a full characterization of domain restrictions admitting a nondictatorial and strategyproof social welfare function. These correspond to preferences for which there is a Condorcet winner.[62]
Holliday and Pacuit devised a voting system that provably minimizes the number of candidates who are capable of spoiling an election, albeit at the cost of occasionally failing vote positivity (though at a much lower rate than seen in instant-runoff voting).[61]
While Arrow's theorem does not apply to graded systems, Gibbard's theorem still does: no voting game can be straightforward (i.e. have a single, clear, always-best strategy),[64] so the informal dictum that "no voting system is perfect" still has some mathematical basis.[65]
Arrow's framework assumed individual and social preferences are orderings or rankings, i.e. statements about which outcomes are better or worse than others.[66] Taking inspiration from the strict behaviorism popular in psychology, some philosophers and economists rejected the idea of comparing internal human experiences of well-being.[67][48] Such philosophers claimed it was impossible to compare the strength of preferences across people who disagreed; Sen gives as an example that it would be impossible to know whether the Great Fire of Rome was good or bad, because despite killing thousands of Romans, it had the positive effect of letting Nero expand his palace.[68]
Arrow originally agreed with these positions and rejected cardinal utility, leading him to focus his theorem on preference rankings;[67][69] his goal in adding the independence axiom was, in part, to prevent from the social choice function from "sneaking in" cardinal information by attempting to infer it from the rankings.[36] As a result, Arrow initially interpreted his theorem as a kind of mathematical proof for nihilism or egoism.[48][66][70] However, he later reversed this opinion, admitting cardinal methods can provide useful information that allows them to evade his theorem.[71][72] Similarly, Amartya Sen first claimed interpersonal comparability is necessary for IIA, but later came to argue in favor of cardinal methods for assessing social choice, arguing it would only require "rather limited levels of partial comparability" to hold in practice.[68]
Balinski and Laraki disputed that any interpersonal comparisons are required for rated voting rules to pass IIA. They argue the availability of a common language with verbal grades is sufficient for IIA by allowing voters to give consistent responses to questions about candidate quality. In other words, they argue most voters will not change their beliefs about whether a candidate is "good", "bad", or "neutral" simply because another candidate joins or drops out of a race.[63]
Other scholars have noted that interpersonal comparisons of utility are not unique to cardinal voting, but are instead a necessity of any non-dictatorial (or non-egoist) choice procedure, with cardinal voting rules simply making these comparisons explicit. David Pearce identified Arrow's original interpretation of the theorem as a mathematical proof of nihilism or egoism with a kind of circular reasoning,[48] and Hildreth pointed out that "any procedure that extends the partial ordering of [Pareto efficiency] must involve interpersonal comparisons of utility."[78] These observations have led to the rise of implicit utilitarian voting, which identifies ranked procedures with approximations of the utilitarian rule (i.e. score voting), helping to make them more explicit.[79]
In psychometrics, there is a near-universal scientific consensus for the usefulness and meaningfulness of self-reported ratings, which empirically show higher validity and reliability than rankings in measuring human opinions.[80][81] Research has consistently found cardinal rating scales (e.g. Likert scales) provide more information than rankings alone.[81][82] Kaiser and Oswald conducted an empirical review of four decades of research including over 700,000 participants who provided self-reported ratings of utility, with the goal of identifying whether people "have a sense of an actual underlying scale for their innermost feelings".[83] They found responses to these questions were consistent with all expectations of a well-specified quantitative measure. Furthermore, such ratings were highly predictive of important decisions (such as international migration and divorce) and had larger effect sizes than standard socioeconomic predictors like income and demographics.[83] Ultimately, the authors concluded "this feelings-to-actions relationship takes a generic form, is consistently replicable, and is fairly close to linear in structure. Therefore, it seems that human beings can successfully operationalize an integer scale for feelings".[83]
Nonstandard spoilers
Behavioral economists have shown individual irrationality involves violations of IIA (e.g. with decoy effects),[84] suggesting human behavior can cause IIA failures even if the voting method itself does not.[85] However, past research has typically found such effects to be fairly small,[86] and such psychological spoilers can appear regardless of electoral system. Balinski and Laraki discuss techniques of ballot design derived from psychometrics that minimize these psychological effects, such as asking voters to give each candidate a verbal grade (e.g. "bad", "neutral", "good", "excellent") and issuing instructions to voters that refer to their ballots as judgments of individual candidates.[63] Similar techniques are often discussed in the context of contingent valuation.[72]
Esoteric solutions
In addition to the above practical resolutions, there exist unusual (less-than-practical) situations where Arrow's requirement of IIA can be satisfied.
Supermajority rules
Supermajority rules can avoid Arrow's theorem at the cost of being poorly-decisive (i.e. frequently failing to return a result). In this case, a threshold that requires a majority for ordering 3 outcomes, for 4, etc. does not produce voting paradoxes.[87]
In spatial (n-dimensional ideology) models of voting, this can be relaxed to require only (roughly 64%) of the vote to prevent cycles, so long as the distribution of voters is well-behaved (quasiconcave).[88] These results provide some justification for the common requirement of a two-thirds majority for constitutional amendments, which is sufficient to prevent cyclic preferences in most situations.[88]
Infinite populations
Fishburn shows all of Arrow's conditions can be satisfied for uncountably infinite sets of voters given the axiom of choice;[89] however, Kirman and Sondermann demonstrated this requires disenfranchising almost all members of a society (eligible voters form a set of measure 0), leading them to refer to such societies as "invisible dictatorships".[90]
Common misconceptions
Arrow's theorem is not related to strategic voting, which does not appear in his framework,[40][91] though the theorem does have important implications for strategic voting (being used as a lemma to prove Gibbard's theorem[92]). The Arrovian framework of social welfare assumes all voter preferences are known and the only issue is in aggregating them.[91]
Non-perversity (called positive association by Arrow) is not a condition of Arrow's theorem.[2] This misconception is caused by a mistake by Arrow himself, who included the axiom in his original statement of the theorem but did not use it.[70] Dropping the assumption does not allow for constructing a social welfare function that meets his other conditions.[2]
Contrary to a common misconception, Arrow's theorem deals with the limited class of ranked-choice voting systems, rather than voting systems as a whole.[91][93]
^Borgers, Christoph (2010-01-01). Mathematics of Social Choice: Voting, Compensation, and Division. SIAM. ISBN9780898716955. Candidates C and D spoiled the election for B ... With them in the running, A won, whereas without them in the running, B would have won. ... Instant runoff voting ... does not do away with the spoiler problem entirely
^ abNg, Y. K. (November 1971). "The Possibility of a Paretian Liberal: Impossibility Theorems and Cardinal Utility". Journal of Political Economy. 79 (6): 1397–1402. doi:10.1086/259845. ISSN0022-3808. In the present stage of the discussion on the problem of social choice, it should be common knowledge that the General Impossibility Theorem holds because only the ordinal preferences is or can be taken into account. If the intensity of preference or cardinal utility can be known or is reflected in social choice, the paradox of social choice can be solved.
^Hamlin, Aaron (25 May 2015). "CES Podcast with Dr Arrow". Center for Election Science. CES. Archived from the original on 27 October 2018. Retrieved 9 March 2023.
^ ab"Arrow's Theorem". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 2019.
^Borgers, Christoph (2010-01-01). Mathematics of Social Choice: Voting, Compensation, and Division. SIAM. ISBN9780898716955. Candidates C and D spoiled the election for B ... With them in the running, A won, whereas without them in the running, B would have won. ... Instant runoff voting ... does not do away with the spoiler problem entirely, although it unquestionably makes it less likely to occur in practice.
^Holliday, Wesley H.; Pacuit, Eric (2023-02-11), Stable Voting, arXiv:2108.00542, This is a kind of stability property of Condorcet winners: you cannot dislodge a Condorcet winner A by adding a new candidate B to the election if A beats B in a head-to-head majority vote. For example, although the 2000 U.S. Presidential Election in Florida did not use ranked ballots, it is plausible (see Magee 2003) that Al Gore (A) would have won without Ralph Nader (B) in the election, and Gore would have beaten Nader head-to-head. Thus, Gore should still have won with Nader included in the election.
^Holliday, Wesley H.; Pacuit, Eric (2023-02-11), Stable Voting, arXiv:2108.00542. "This is a kind of stability property of Condorcet winners: you cannot dislodge a Condorcet winner A by adding a new candidate B to the election if A beats B in a head-to-head majority vote. For example, although the 2000 U.S. Presidential Election in Florida did not use ranked ballots, it is plausible (see Magee 2003) that Al Gore (A) would have won without Ralph Nader (B) in the election, and Gore would have beaten Nader head-to-head. Thus, Gore should still have won with Nader included in the election."
^Poundstone, William. (2013). Gaming the vote : why elections aren't fair (and what we can do about it). Farrar, Straus and Giroux. pp. 168, 197, 234. ISBN9781429957649. OCLC872601019. IRV is subject to something called the "center squeeze." A popular moderate can receive relatively few first-place votes through no fault of her own but because of vote splitting from candidates to the right and left. [...] Approval voting thus appears to solve the problem of vote splitting simply and elegantly. [...] Range voting solves the problems of spoilers and vote splitting
^"Modern economic theory has insisted on the ordinal concept of utility; that is, only orderings can be observed, and therefore no measurement of utility independent of these orderings has any significance. In the field of consumer's demand theory the ordinalist position turned out to create no problems; cardinal utility had no explanatory power above and beyond ordinal. Leibniz' Principle of the identity of indiscernibles demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on p. 33 by Racnchetti, Fabio (2002), "Choice without utility? Some reflections on the loose foundations of standard consumer theory", in Bianchi, Marina (ed.), The Active Consumer: Novelty and Surprise in Consumer Choice, Routledge Frontiers of Political Economy, vol. 20, Routledge, pp. 21–45
Dr. Arrow: Now there’s another possible way of thinking about it, which is not included in my theorem. But we have some idea how strongly people feel. In other words, you might do something like saying each voter does not just give a ranking. But says, this is good. And this is not good[...] So this gives more information than simply what I have asked for.
Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes (in spite of what I said about manipulation) is probably the best.[...] And some of these studies have been made. In France, [Michel] Balinski has done some studies of this kind which seem to give some support to these scoring methods.
^Hamlin, Aaron (25 May 2015). "CES Podcast with Dr Arrow". Center for Election Science. CES. Archived from the original on 27 October 2018. Retrieved 9 March 2023.
^ abcIndeed, many different social welfare functions can meet Arrow's conditions under such restrictions of the domain. It has been proven, however, that under any such restriction, if there exists any social welfare function that adheres to Arrow's criteria, then Condorcet method will adhere to Arrow's criteria. See Campbell, D. E.; Kelly, J. S. (2000). "A simple characterization of majority rule". Economic Theory. 15 (3): 689–700. doi:10.1007/s001990050318. JSTOR25055296. S2CID122290254.
^Cockrell, Jeff (2016-03-08). "What economists think about voting". Capital Ideas. Chicago Booth. Archived from the original on 2016-03-26. Retrieved 2016-09-05. Is there such a thing as a perfect voting system? The respondents were unanimous in their insistence that there is not.
^ ab"Modern economic theory has insisted on the ordinal concept of utility; that is, only orderings can be observed, and therefore no measurement of utility independent of these orderings has any significance. In the field of consumer's demand theory the ordinalist position turned out to create no problems; cardinal utility had no explanatory power above and beyond ordinal. Leibniz' Principle of the identity of indiscernibles demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on p. 33 by Racnchetti, Fabio (2002), "Choice without utility? Some reflections on the loose foundations of standard consumer theory", in Bianchi, Marina (ed.), The Active Consumer: Novelty and Surprise in Consumer Choice, Routledge Frontiers of Political Economy, vol. 20, Routledge, pp. 21–45
Dr. Arrow: Now there’s another possible way of thinking about it, which is not included in my theorem. But we have some idea how strongly people feel. In other words, you might do something like saying each voter does not just give a ranking. But says, this is good. And this is not good[...] So this gives more information than simply what I have asked for.
^ abArrow, Kenneth et al. 1993. Report of the NOAA panel on Contingent Valuation.
^ abHarsanyi, John C. (1955). "Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility". Journal of Political Economy. 63 (4): 309–321. doi:10.1086/257678. JSTOR1827128. S2CID222434288.
^Procaccia, Ariel D.; Rosenschein, Jeffrey S. (2006). "The Distortion of Cardinal Preferences in Voting". Cooperative Information Agents X. Lecture Notes in Computer Science. Vol. 4149. pp. 317–331. CiteSeerX10.1.1.113.2486. doi:10.1007/11839354_23. ISBN978-3-540-38569-1.
^Moore, Michael (1 July 1975). "Rating versus ranking in the Rokeach Value Survey: An Israeli comparison". European Journal of Social Psychology. 5 (3): 405–408. doi:10.1002/ejsp.2420050313. ISSN1099-0992. The extremely high degree of correspondence found between ranking and rating averages ... does not leave any doubt about the preferability of the rating method for group description purposes. The obvious advantage of rating is that while its results are virtually identical to what is obtained by ranking, it supplies more information than ranking does.
^ abMaio, Gregory R.; Roese, Neal J.; Seligman, Clive; Katz, Albert (1 June 1996). "Rankings, Ratings, and the Measurement of Values: Evidence for the Superior Validity of Ratings". Basic and Applied Social Psychology. 18 (2): 171–181. doi:10.1207/s15324834basp1802_4. ISSN0197-3533. Many value researchers have assumed that rankings of values are more valid than ratings of values because rankings force participants to differentiate more incisively between similarly regarded values ... Results indicated that ratings tended to evidence greater validity than rankings within moderate and low-differentiating participants. In addition, the validity of ratings was greater than rankings overall.
^Huber, Joel; Payne, John W.; Puto, Christopher (1982). "Adding Asymmetrically Dominated Alternatives: Violations of Regularity and the Similarity Hypothesis". Journal of Consumer Research. 9 (1): 90–98. doi:10.1086/208899. S2CID120998684.
^Huber, Joel; Payne, John W.; Puto, Christopher P. (2014). "Let's Be Honest About the Attraction Effect". Journal of Marketing Research. 51 (4): 520–525. doi:10.1509/jmr.14.0208. ISSN0022-2437. S2CID143974563.
^Fishburn, Peter Clingerman (1970). "Arrow's impossibility theorem: concise proof and infinite voters". Journal of Economic Theory. 2 (1): 103–106. doi:10.1016/0022-0531(70)90015-3.
^See Chapter 6 of Taylor, Alan D. (2005). Social choice and the mathematics of manipulation. New York: Cambridge University Press. ISBN978-0-521-00883-9 for a concise discussion of social choice for infinite societies.
^ abc"Arrow's Theorem". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 2019.
Hansen, Paul (2002). "Another Graphical Proof of Arrow's Impossibility Theorem". The Journal of Economic Education. 33 (3): 217–235. doi:10.1080/00220480209595188. S2CID145127710.
Hunt, Earl (2007). The Mathematics of Behavior. Cambridge University Press. ISBN9780521850124.. The chapter "Defining Rationality: Personal and Group Decision Making" has a detailed discussion of the Arrow Theorem, with proof.
Lewis, Harold W. (1997). Why flip a coin? : The art and science of good decisions. John Wiley. ISBN0-471-29645-7. Gives explicit examples of preference rankings and apparently anomalous results under different electoral system. States but does not prove Arrow's theorem.
^in social choice, ranked rules include First-preference plurality and all other rules where a candidate's score can be determined from a ranking of candidates.