Arrow's impossibility theorem explained

Arrow's impossibility theorem is a key result in social choice showing that no rank-order method for collective decision-making can satisfy the requirements of rational choice.[1] Specifically, any such rule violates independence of irrelevant alternatives, the principle that a choice between

A

and

B

should not depend on the quality of a third, unrelated option

C

.[2] [3]

The result is most often cited in election science and voting theory, where

C

is called a spoiler candidate.[4] In this context, Arrow's theorem can be restated as showing that no ranked voting rule can eliminate the spoiler effect.[5] [6] [7] Arrow's theorem therefore generalizes Nicolas de Condorcet's voting paradox, and shows similar problems exist for any possible collective decision-making procedure relying only on how people order different options.

Despite this, some ranked methods are much more susceptible to spoilers than others. Plurality-rule methods like first-past-the-post and ranked-choice voting (RCV) in particular are highly sensitive to spoilers,[8] [9] manufacturing them even in center squeezes (where they are not forced).[10] [11] By contrast, majority-rule methods uniquely minimize the possibility of spoilers,[12] limiting them to rare[13] [14] situations called Condorcet paradoxes. Under plausible models of voter behavior, such as the left-right political spectrum assumed by the median voter theorem, the spoiler effect can vanish entirely for Condorcet methods,[15] though not for most other systems. As a result, the practical consequences of the theorem are debatable, with Arrow noting "Most [ranked] systems are not going to work badly all of the time. All I proved is that all can work badly at times."[16] [17]

Rated methods are not affected by Arrow's theorem or IIA failures at all.[18] Arrow initially asserted the information provided by these systems was meaningless,[19] and therefore could not be used to prevent paradoxes, leading him to overlook them. However, he and other authors would later recognize this to have been a mistake,[20] [21] with Arrow admitting rules based on cardinal utilities (such as score and approval voting) are not subject to his theorem.[22] [23] [24]

Background

See main article: Social welfare function, Voting systems and Social choice theory. Arrow's theorem falls under the branch of welfare economics called social choice theory, which deals with aggregating preferences and beliefs to make optimal decisions. The goal of social choice is to identify a social choice rule, a mathematical function that determines which of two outcomes or options is better, according to all members of a society. Such a procedure can be a market, voting system, constitution, or even a moral or ethical framework.[25] Ideally, such a procedure should satisfy the properties of rational choice, avoiding any kind of self-contradiction.

Axioms of voting systems

Preferences

In social choice theory, preferences are modeled as binary relations. If and are different candidates or alternatives, then

A\succB

means is preferred to . Individual preferences (or ballots) are required to be:
  1. Transitive—if

A\succeqB

and

B\succeqC

, then

A\succeqC

.
  1. Complete—at least one of

A\succeqB

or

B\succeqA

must be true.

The social choice function is then a mathematical function mapping a collection of individual orderings to a new ordering, which represents the preferences of society as a whole.

Basic assumptions

Arrow's theorem assumes as background that non-degenerate ranked social choice rules satisfy:[26]

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.[30] 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. While Arrow considered it an obvious requirement of any proposed social choice rule, ranked-choice runoff (RCV) fails this condition.[31] Arrow later corrected his statement of the theorem to include runoffs and other perverse voting rules.

Rationality

Among the most important axioms of rational choice is independence of irrelevant alternatives, which says that when deciding between and, one's opinion about a third option should not affect their decision.

A\succB

to

B\succA

if voters change their preference about whether

A\succC

.

IIA is sometimes illustrated with a short joke by philosopher Sidney Morgenbesser:[33]

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.[34]

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.[35] Suppose we have three candidates, A, B, and C, and that there are three voters with preferences 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 C is chosen as the winner, it can be argued that any fair voting system would say B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, 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: A is preferred over B which is preferred over C which is preferred over A.

Because of this example, some authors credit Condorcet with having given an intuitive argument that presents the core of Arrow's theorem. However, Arrow's theorem is substantially more general and even applies to many "unfair" systems for making decisions, which give some voters more influence than others or favor some candidates over others.

Formal statement

Let be a set of alternatives. A preference on is a complete and transitive binary relation on (sometimes called a total preorder), that is, a subset of satisfying:

  1. (Transitivity) If is in and is in, then is in,
  2. (Completeness) At least one of or must be in .

The element being in is interpreted to mean that alternative is preferred to alternative . This situation is often denoted

a\succb

or . Denote the set of all preferences on by .

Let be a positive integer. An ordinal (ranked) social welfare function is a function[36]

F:\Pi(A)N\to\Pi(A)

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:

Pareto efficiency
  • If alternative is preferred to for all orderings, then is preferred to by .
    Non-dictatorship
  • 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 .
    Independence of irrelevant alternatives
  • 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 .

    Formal proof

    Arrow's proof used the concept of decisive coalitions.

    Definition:

    (x,y)

    if, when everyone in the coalition ranks

    x\succiy

    , society overall will always rank

    x\succy

    .

    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[37] and Ariel Rubinstein.[38] The simplified proof uses an additional concept:

    (x,y)

    if and only if when every voter

    i

    in the coalition ranks

    x\succiy

    , and every voter

    j

    outside the coalition ranks

    y\succjx

    , then

    x\succy

    .

    Thenceforth assume that the social choice system satisfies unrestricted domain, Pareto efficiency, and IIA. Also assume that there are at least 3 distinct outcomes.

    By Pareto, the entire set of voters is decisive. Thus by the group contraction lemma, there is a size-one decisive coalition—a dictator.

    Proofs using the concept of the pivotal voter originated from Salvador Barberá in 1980.[39] The proof given here is a simplified version based on two proofs published in Economic Theory.[40] [41]

    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 B first moves above A 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:

    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:[42]

    Non-imposition
  • For any two alternatives a and b, there exists some preference profile such that is preferred to by .

    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."[43] [44]

    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.[45]

    Minimizing IIA failures: Majority-rule methods

    See main article: Condorcet cycle. 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.[46] 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.[47]

    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.[48] 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.

    Unlike pluralitarian rules such as ranked-choice runoff (RCV) or first-preference plurality,[49] 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.[50] Spatial voting models also suggest such paradoxes are likely to be infrequent[51] [52] or even non-existent.[53]

    Left-right spectrum

    See main article: Median voter theorem. Soon after Arrow published his theorem, Duncan Black showed his own remarkable result, the median voter theorem. The theorem proves that if voters and candidates are arranged on a left-right spectrum, Arrow's conditions are all fully compatible, and all will be met by any rule satisfying Condorcet's majority-rule principle.

    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.[54]

    The rule does not fully generalize from the political spectrum to the political compass, a result related to the McKelvey-Schofield Chaos Theorem.[55] However, a well-defined Condorcet winner does exist if the distribution of voters is rotationally symmetric or otherwise has a uniquely-defined median.[56] In realistic cases, where voters' opinions often follow a roughly-normal distribution or can be accurately summarized in one or two dimensions, Condorcet cycles tend to be rare.[57]

    Generalized stability theorems

    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. 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 never cause a new one.

    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.[58]

    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).

    Eliminating IIA failures: Rated voting

    As shown above, the proof of Arrow's theorem relies crucially on the assumption of ranked voting, and is not applicable to rated voting systems. As a result, systems like score voting and graduated majority judgment pass independence of irrelevant alternatives. These systems ask voters to rate candidates on a numerical scale (e.g. from 0–10), and then elect the candidate with the highest average (for score voting) or median (graduated majority judgment).[59]

    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),[60] so the informal dictum that "no voting system is perfect" still has some mathematical basis.[61]

    Meaningfulness of cardinal information

    See main article: Cardinal utility. Arrow's framework assumed individual and social preferences are orderings or rankings, i.e. statements about which outcomes are better or worse than others.[62] Taking inspiration from the strict behaviorism popular in psychology, some philosophers and economists rejected the idea of comparing internal human experiences of well-being.[63] 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.[64]

    Arrow originally agreed with these positions and rejected cardinal utility, leading him to focus his theorem on preference rankings;[65] 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. As a result, Arrow initially interpreted his theorem as a kind of mathematical proof of nihilism or egoism.[66] However, he later reversed this opinion, admitting cardinal methods can provide useful information that allows them to evade his theorem.[67] [68] 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.

    Balinski and Laraki dispute the necessity of any genuinely cardinal information for rated voting methods 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.

    John Harsanyi noted Arrow's theorem could be considered a weaker version of his own theorem[69] and other utility representation theorems like the VNM theorem, which generally show that rational behavior requires consistent cardinal utilities.[70] Harsanyi and Vickrey[71] each independently derived results showing such interpersonal comparisons of utility could be rigorously defined as individual preferences over the lottery of birth.[72] [73]

    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 referred to Arrow's original interpretation of the theorem as a mathematical proof of nihilism or egoism is effectively circular reasoning, and Hildreth pointed out that "any procedure that extends the partial ordering of [<nowiki/>[[Pareto efficiency]]] must involve interpersonal comparisons of utility."[74] 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),[75] making such implicit comparisons more explicit.

    In psychometrics, there is a near-universal scientific consensus for the usefulness and meaningfulness of self-reported ratings, which empirically show greater validity and reliability than rankings in measuring human opinions.[76] [77] Research has consistently found cardinal rating scales (e.g. Likert scales) provide more information than rankings alone.[78] 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".[79] 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), with larger effect sizes than standard socioeconomic predictors like income and demographics. 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".

    Nonstandard spoilers

    Behavioral economists have shown individual irrationality involves violations of IIA (e.g. with decoy effects),[80] suggesting human behavior might cause IIA failures even if the voting method itself does not.[81] However, past research on similar effects has found their effects are typically small,[82] and such psychological spoiler effects can occur regardless of the electoral system in use. Balinski and Laraki discuss techniques of ballot design based on 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.

    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

    2/3

    majority for ordering 3 outcomes,

    3/4

    for 4, etc. does not produce voting paradoxes.[83]

    In spatial (n-dimensional ideology) models of voting, this can be relaxed to require only

    1-e-1

    (roughly 64%) of the vote to prevent cycles, so long as the distribution of voters is well-behaved (quasiconcave). 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.[84]

    Uncountable voter sets

    Fishburn shows all of Arrow's conditions can be satisfied for uncountable sets of voters given the axiom of choice;[85] however, Kirman and Sondermann showed 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".[86]

    Common misconceptions

    Arrow's theorem is not related to strategic voting, which does not appear in his framework,[87] [88] though the theorem does have important implications for strategic voting (being used as a lemma to prove Gibbard's theorem[89]). The Arrovian framework of social welfare assumes all voter preferences are known and the only issue is in aggregating them.

    Monotonicity (sometimes called non-perversity) is not a condition of Arrow's theorem (contrary to a mistake by Arrow himself, who included the axiom in his original statement of the theorem but did not use it). Dropping the assumption does not allow for constructing a social welfare function that meets his other conditions.[90]

    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]

    See also

    References

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    8. McGann . Anthony J. . Koetzle . William . Grofman . Bernard . 2002 . How an Ideologically Concentrated Minority Can Trump a Dispersed Majority: Nonmedian Voter Results for Plurality, Run-off, and Sequential Elimination Elections . American Journal of Political Science . 46 . 1 . 134–147 . 10.2307/3088418 . 0092-5853 . 3088418 . "As with simple plurality elections, it is apparent the outcome will be highly sensitive to the distribution of candidates.".
    9. Book: Borgers, Christoph . Mathematics of Social Choice: Voting, Compensation, and Division . 2010-01-01 . SIAM . 9780898716955 . 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.
    10. . "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."
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    12. Indeed, 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 . . 15 . 3 . 689–700 . 10.1007/s001990050318 . 25055296 . 122290254.
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    19. "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
    20. Web site: Hamlin . Aaron . 2012-10-06 . Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow . dead . https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ . 2023-06-05 . The Center for Election Science.
    21. Harsanyi . John C. . 1979-09-01 . Bayesian decision theory, rule utilitarianism, and Arrow's impossibility theorem . Theory and Decision . 11 . 3 . 289–317 . 10.1007/BF00126382 . 1573-7187 . It is shown that the utilitarian welfare function satisfies all of Arrow's social choice postulates — avoiding the celebrated impossibility theorem by making use of information which is unavailable in Arrow's original framework. . 2020-03-20.
    22. Web site: Hamlin . Aaron . 2012-10-06 . Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow . dead . https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ . 2023-06-05 . The Center for Election Science. Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best.
    23. Web site: Hamlin . Aaron . 2012-10-06 . Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow . dead . https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ . 2023-06-05 . The Center for Election Science. 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.
    24. Web site: Hamlin . Aaron . 2012-10-06 . Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow . dead . https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ . 2023-06-05 . The Center for Election Science.
    25. Book: The Stanford Encyclopedia of Philosophy . 2019 . Metaphysics Research Lab, Stanford University . Arrow's Theorem . https://plato.stanford.edu/entries/arrows-theorem/#Rei.
    26. Gibbard . Allan . 1973 . Manipulation of Voting Schemes: A General Result . Econometrica . 41 . 4 . 587–601 . 10.2307/1914083 . 1914083 . 0012-9682.
    27. Campbell . D.E. . Kelly . J.S. . 2000 . A simple characterization of majority rule . . 15 . 3 . 689–700 . 10.1007/s001990050318 . 25055296 . 122290254.
    28. Book: Arrow, Kenneth Joseph Arrow . Social Choice and Individual Values . 1963 . Yale University Press . 978-0300013641 . https://ghostarchive.org/archive/20221009/http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf . 2022-10-09 . live.
    29. Quesada . Antonio . 2002 . From social choice functions to dictatorial social welfare functions . Economics Bulletin . 4 . 16 . 1–7.
    30. Arrow . Kenneth J. . Kenneth Arrow . 1950 . A Difficulty in the Concept of Social Welfare . dead . . 58 . 4 . 328–346 . 10.1086/256963 . 1828886 . 13923619 . https://web.archive.org/web/20110720090207/http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf . 2011-07-20.
    31. Doron . Gideon . Kronick . Richard . 1977 . Single Transferrable Vote: An Example of a Perverse Social Choice Function . American Journal of Political Science . 21 . 2 . 303–311 . 10.2307/2110496 . 2110496 . 0092-5853.
    32. Quesada . Antonio . 2002 . From social choice functions to dictatorial social welfare functions . Economics Bulletin . 4 . 16 . 1–7.
    33. Pearce . David . Individual and social welfare: a Bayesian perspective . Frisch Lecture Delivered to the World Congress of the Econometric Society.
    34. Pearce . David . Individual and social welfare: a Bayesian perspective . Frisch Lecture Delivered to the World Congress of the Econometric Society.
    35. McLean . Iain . 1995-10-01 . Independence of irrelevant alternatives before Arrow . Mathematical Social Sciences . 30 . 2 . 107–126 . 10.1016/0165-4896(95)00784-J . 0165-4896.
    36. Arrow . Kenneth J. . Kenneth Arrow . 1950 . A Difficulty in the Concept of Social Welfare . dead . . 58 . 4 . 328–346 . 10.1086/256963 . 1828886 . 13923619 . https://web.archive.org/web/20110720090207/http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf . 2011-07-20.
    37. Book: Sen, Amartya . The Arrow Impossibility Theorem . 2014-07-22 . Columbia University Press . 978-0-231-52686-9 . 29–42 . Arrow and the Impossibility Theorem . 10.7312/mask15328-003.
    38. Book: Rubinstein, Ariel . Lecture Notes in Microeconomic Theory: The Economic Agent . Princeton University Press . 2012 . 978-1-4008-4246-9 . 2nd . Problem 9.5 . 29649010M.
    39. Barberá . Salvador . January 1980 . Pivotal voters: A new proof of arrow's theorem . Economics Letters . 6 . 1 . 13–16 . 10.1016/0165-1765(80)90050-6 . 0165-1765.
    40. Geanakoplos . John . 2005 . Three Brief Proofs of Arrow's Impossibility Theorem . live . . 26 . 1 . 211–215 . 10.1.1.193.6817 . 10.1007/s00199-004-0556-7 . 25055941 . 17101545 . https://ghostarchive.org/archive/20221009/https://cowles.yale.edu/sites/default/files/files/pub/d11/d1123-r4.pdf . 2022-10-09.
    41. Yu . Ning Neil . 2012 . A one-shot proof of Arrow's theorem . . 50 . 2 . 523–525 . 10.1007/s00199-012-0693-3 . 41486021 . 121998270.
    42. Wilson . Robert . December 1972 . Social choice theory without the Pareto Principle . Journal of Economic Theory . 5 . 3 . 478–486 . 10.1016/0022-0531(72)90051-8 . 0022-0531.
    43. Web site: Hamlin . Aaron . 25 May 2015 . CES Podcast with Dr Arrow . dead . https://web.archive.org/web/20181027170517/https://electology.org/podcasts/2012-10-06_kenneth_arrow . 27 October 2018 . 9 March 2023 . Center for Election Science . CES.
    44. McKenna . Phil . 12 April 2008 . Vote of no confidence . New Scientist . 198 . 2651 . 30–33 . 10.1016/S0262-4079(08)60914-8.
    45. Pearce . David . Individual and social welfare: a Bayesian perspective . Frisch Lecture Delivered to the World Congress of the Econometric Society.
    46. Indeed, 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 . . 15 . 3 . 689–700 . 10.1007/s001990050318 . 25055296 . 122290254.
    47. McLean . Iain . 1995-10-01 . Independence of irrelevant alternatives before Arrow . Mathematical Social Sciences . 30 . 2 . 107–126 . 10.1016/0165-4896(95)00784-J . 0165-4896.
    48. Gehrlein . William V. . 1983-06-01 . Condorcet's paradox . Theory and Decision . en . 15 . 2 . 161–197 . 10.1007/BF00143070 . 1573-7187.
    49. McGann . Anthony J. . Koetzle . William . Grofman . Bernard . 2002 . How an Ideologically Concentrated Minority Can Trump a Dispersed Majority: Nonmedian Voter Results for Plurality, Run-off, and Sequential Elimination Elections . American Journal of Political Science . 46 . 1 . 134–147 . 10.2307/3088418 . 0092-5853 . 3088418 . "As with simple plurality elections, it is apparent the outcome will be highly sensitive to the distribution of candidates.".
    50. Van Deemen . Adrian . 2014-03-01 . On the empirical relevance of Condorcet's paradox . Public Choice . 158 . 3 . 311–330 . 10.1007/s11127-013-0133-3 . 1573-7101.
    51. Wolk . Sara . Quinn . Jameson . Ogren . Marcus . 2023-09-01 . STAR Voting, equality of voice, and voter satisfaction: considerations for voting method reform . Constitutional Political Economy . 34 . 3 . 310–334 . 10.1007/s10602-022-09389-3 . 1572-9966.
    52. Gehrlein . William V. . 2002-03-01 . Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences* . Theory and Decision . 52 . 2 . 171–199 . 10.1023/A:1015551010381 . 1573-7187.
    53. Book: Black, Duncan . Duncan Black . The theory of committees and elections . University Press . 1968 . 978-0-89838-189-4 . Cambridge, Eng..
    54. Campbell . D.E. . Kelly . J.S. . 2000 . A simple characterization of majority rule . . 15 . 3 . 689–700 . 10.1007/s001990050318 . 25055296 . 122290254.
    55. McKelvey . Richard D. . Richard McKelvey . 1976 . Intransitivities in multidimensional voting models and some implications for agenda control . Journal of Economic Theory . 12 . 3 . 472–482 . 10.1016/0022-0531(76)90040-5.
    56. Multidimensional voting models: theory and applications . UCL (University College London) . 2016-09-28 . Doctoral . V. . Dotti.
    57. Holliday . Wesley H. . Pacuit . Eric . 2023-09-01 . Stable Voting . Constitutional Political Economy . 34 . 3 . 421–433 . 10.1007/s10602-022-09383-9 . 1572-9966 . free.
    58. Kalai . Ehud . Muller . Eitan . 1977 . Characterization of Domains Admitting Nondictatorial Social Welfare Functions and Nonmanipulable Voting Procedures . Journal of Economic Theory . 16 . 2 . 457–469 . 10.1016/0022-0531(77)90019-9.
    59. Book: Balinski . M. L. . Majority judgment: measuring, ranking, and electing . Laraki . Rida . 2010 . MIT Press . 9780262545716 . Cambridge, Mass.
    60. Book: Poundstone, William . Gaming the Vote: Why Elections Are not Fair (and What We Can Do About It) . 2009-02-17 . Macmillan . 9780809048922.
    61. Web site: Cockrell . Jeff . 2016-03-08 . What economists think about voting . dead . https://web.archive.org/web/20160326065043/http://www.chicagobooth.edu/capideas/blog/2016/march/what-economists-think-about-voting . 2016-03-26 . 2016-09-05 . Capital Ideas . Chicago Booth . Is there such a thing as a perfect voting system? The respondents were unanimous in their insistence that there is not..
    62. Lützen . Jesper . 2019-02-01 . How mathematical impossibility changed welfare economics: A history of Arrow's impossibility theorem . Historia Mathematica . 46 . 56–87 . 10.1016/j.hm.2018.11.001 . 0315-0860.
    63. "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
    64. Sen . Amartya . 1999 . The Possibility of Social Choice . American Economic Review . 89 . 3 . 349–378 . 10.1257/aer.89.3.349.
    65. Book: Arrow, Kenneth Joseph Arrow . Social Choice and Individual Values . 1963 . Yale University Press . 978-0300013641 . 31–33 . III. The Social Welfare Function . https://ghostarchive.org/archive/20221009/http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf . 2022-10-09 . live.
    66. Arrow . Kenneth J. . Kenneth Arrow . 1950 . A Difficulty in the Concept of Social Welfare . dead . . 58 . 4 . 328–346 . 10.1086/256963 . 1828886 . 13923619 . https://web.archive.org/web/20110720090207/http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf . 2011-07-20.
    67. Web site: Hamlin . Aaron . 2012-10-06 . Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow . dead . https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ . 2023-06-05 . The Center for Election Science.
    68. Arrow, Kenneth et al. 1993. Report of the NOAA panel on Contingent Valuation.
    69. Harsanyi . John C. . 1955 . Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility . Journal of Political Economy . 63 . 4 . 309–321 . 10.1086/257678 . 1827128 . 222434288.
    70. [John von Neumann|Neumann, John von]
    71. Vickrey . William . 1945 . Measuring Marginal Utility by Reactions to Risk . Econometrica . 13 . 4 . 319–333 . 10.2307/1906925 . 1906925.
    72. Mongin . Philippe . October 2001 . The impartial observer theorem of social ethics . Economics & Philosophy . 17 . 2 . 147–179 . 10.1017/S0266267101000219 . 2024-08-02 . 1474-0028.
    73. Book: Arrow and the Foundations of the Theory of Economic Policy . 1987 . Springer . 9781349073573 . Feiwel . George . 92 . "...the fictitious notion of 'original position' [was] developed by Vickery (1945), Harsanyi (1955), and Rawls (1971).".
    74. Hildreth . Clifford . 1953 . Alternative Conditions for Social Orderings . Econometrica . 21 . 1 . 81–94 . 10.2307/1906944 . 0012-9682.
    75. Book: Procaccia . Ariel D. . Cooperative Information Agents X . Rosenschein . Jeffrey S. . 2006 . 978-3-540-38569-1 . Lecture Notes in Computer Science . 4149 . 317–331 . The Distortion of Cardinal Preferences in Voting . 10.1.1.113.2486 . 10.1007/11839354_23.
    76. Moore . Michael . 1 July 1975 . Rating versus ranking in the Rokeach Value Survey: An Israeli comparison . European Journal of Social Psychology . en . 5 . 3 . 405–408 . 10.1002/ejsp.2420050313 . 1099-0992 . 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..
    77. Maio . 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 . 10.1207/s15324834basp1802_4 . 0197-3533 . 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..
    78. Conklin . E. S. . Sutherland . J. W. . 1 February 1923 . A Comparison of the Scale of Values Method with the Order-of-Merit Method. . Journal of Experimental Psychology . en . 6 . 1 . 44–57 . 10.1037/h0074763 . 0022-1015 . the scale-of-values method can be used for approximately the same purposes as the order-of-merit method, but that the scale-of-values method is a better means of obtaining a record of judgments.
    79. Kaiser . Caspar . Oswald . Andrew J. . 18 October 2022 . The scientific value of numerical measures of human feelings . Proceedings of the National Academy of Sciences . 119 . 42 . e2210412119 . 2022PNAS..11910412K . 10.1073/pnas.2210412119 . 0027-8424 . 9586273 . 36191179 . free.
    80. 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 . 10.1086/208899 . 120998684.
    81. Ohtsubo . Yohsuke . Watanabe . Yoriko . September 2003 . Contrast Effects and Approval Voting: An Illustration of a Systematic Violation of the Independence of Irrelevant Alternatives Condition . Political Psychology . en . 24 . 3 . 549–559 . 10.1111/0162-895X.00340 . 0162-895X.
    82. 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 . 10.1509/jmr.14.0208 . 0022-2437 . 143974563.
    83. Moulin . Hervé . 1985-02-01 . From social welfare ordering to acyclic aggregation of preferences . Mathematical Social Sciences . 9 . 1 . 1–17 . 10.1016/0165-4896(85)90002-2 . 0165-4896.
    84. Caplin . Andrew . Nalebuff . Barry . 1988 . On 64%-Majority Rule . Econometrica . 56 . 4 . 787–814 . 10.2307/1912699 . 0012-9682 . 1912699.
    85. Fishburn . Peter Clingerman . 1970 . Arrow's impossibility theorem: concise proof and infinite voters . Journal of Economic Theory . 2 . 1 . 103–106 . 10.1016/0022-0531(70)90015-3.
    86. See Chapter 6 of Book: Taylor, Alan D. . Social choice and the mathematics of manipulation . Cambridge University Press . 2005 . 978-0-521-00883-9 . New York . none. for a concise discussion of social choice for infinite societies.
    87. Book: Arrow, Kenneth Joseph Arrow . Social Choice and Individual Values . 1963 . Yale University Press . 978-0300013641 . https://ghostarchive.org/archive/20221009/http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf . 2022-10-09 . live.
    88. Book: The Stanford Encyclopedia of Philosophy . 2019 . Metaphysics Research Lab, Stanford University . Arrow's Theorem . https://plato.stanford.edu/entries/arrows-theorem/#Rei.
    89. Gibbard . Allan . 1973 . Manipulation of Voting Schemes: A General Result . Econometrica . 41 . 4 . 587–601 . 10.2307/1914083 . 0012-9682 . 1914083.
    90. Book: Arrow, Kenneth Joseph Arrow . Social Choice and Individual Values . 1963 . Yale University Press . 978-0300013641 . https://ghostarchive.org/archive/20221009/http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf . 2022-10-09 . live.
    91. Web site: Hamlin . Aaron . March 2017 . Remembering Kenneth Arrow and His Impossibility Theorem . 5 May 2024 . Center for Election Science.

    Further reading

    External links

    1. Arrow . Kenneth J. . Kenneth Arrow . 1950 . A Difficulty in the Concept of Social Welfare . dead . . 58 . 4 . 328–346 . 10.1086/256963 . 1828886 . 13923619 . https://web.archive.org/web/20110720090207/http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf . 2011-07-20.
    2. Book: Arrow, Kenneth Joseph Arrow . Social Choice and Individual Values . 1963 . Yale University Press . 978-0300013641 . https://ghostarchive.org/archive/20221009/http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf . 2022-10-09 . live.
    3. Wilson . Robert . December 1972 . Social choice theory without the Pareto Principle . Journal of Economic Theory . 5 . 3 . 478–486 . 10.1016/0022-0531(72)90051-8 . 0022-0531.
    4. Book: Borgers, Christoph . Mathematics of Social Choice: Voting, Compensation, and Division . 2010-01-01 . SIAM . 9780898716955 . 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.
    5. Ng . Y. K. . November 1971 . The Possibility of a Paretian Liberal: Impossibility Theorems and Cardinal Utility . Journal of Political Economy . 79 . 6 . 1397–1402 . 10.1086/259845 . 0022-3808 . "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.".
    6. Web site: Hamlin . Aaron . 25 May 2015 . CES Podcast with Dr Arrow . dead . https://web.archive.org/web/20181027170517/https://electology.org/podcasts/2012-10-06_kenneth_arrow . 27 October 2018 . 9 March 2023 . Center for Election Science . CES.
    7. Kemp . Murray . Asimakopulos . A. . 1952-05-01 . A Note on "Social Welfare Functions" and Cardinal Utility* . Canadian Journal of Economics and Political Science . 18 . 2 . 195–200 . 10.2307/138144 . 138144 . 0315-4890 . The abandonment of Condition 3 makes it possible to formulate a procedure for arriving at a social choice. Such a procedure is described below . 2020-03-20.
    8. McGann . Anthony J. . Koetzle . William . Grofman . Bernard . 2002 . How an Ideologically Concentrated Minority Can Trump a Dispersed Majority: Nonmedian Voter Results for Plurality, Run-off, and Sequential Elimination Elections . American Journal of Political Science . 46 . 1 . 134–147 . 10.2307/3088418 . 0092-5853 . 3088418 . "As with simple plurality elections, it is apparent the outcome will be highly sensitive to the distribution of candidates.".
    9. Book: Borgers, Christoph . Mathematics of Social Choice: Voting, Compensation, and Division . 2010-01-01 . SIAM . 9780898716955 . 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.
    10. . "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."
    11. Campbell . D.E. . Kelly . J.S. . 2000 . A simple characterization of majority rule . . 15 . 3 . 689–700 . 10.1007/s001990050318 . 25055296 . 122290254.
    12. Indeed, 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 . . 15 . 3 . 689–700 . 10.1007/s001990050318 . 25055296 . 122290254.
    13. Gehrlein . William V. . 2002-03-01 . Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences* . Theory and Decision . 52 . 2 . 171–199 . 10.1023/A:1015551010381 . 1573-7187.
    14. Van Deemen . Adrian . 2014-03-01 . On the empirical relevance of Condorcet's paradox . Public Choice . 158 . 3 . 311–330 . 10.1007/s11127-013-0133-3 . 1573-7101.
    15. Book: Black, Duncan . Duncan Black . The theory of committees and elections . University Press . 1968 . 978-0-89838-189-4 . Cambridge, Eng..
    16. Web site: Hamlin . Aaron . 25 May 2015 . CES Podcast with Dr Arrow . dead . https://web.archive.org/web/20181027170517/https://electology.org/podcasts/2012-10-06_kenneth_arrow . 27 October 2018 . 9 March 2023 . Center for Election Science . CES.
    17. McKenna . Phil . 12 April 2008 . Vote of no confidence . New Scientist . 198 . 2651 . 30–33 . 10.1016/S0262-4079(08)60914-8.
    18. Book: Poundstone, William. . Gaming the vote : why elections aren't fair (and what we can do about it) . 2013 . Farrar, Straus and Giroux . 9781429957649 . 168, 197, 234 . 872601019 . 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.
    19. "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
    20. Web site: Hamlin . Aaron . 2012-10-06 . Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow . dead . https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ . 2023-06-05 . The Center for Election Science.
    21. Harsanyi . John C. . 1979-09-01 . Bayesian decision theory, rule utilitarianism, and Arrow's impossibility theorem . Theory and Decision . 11 . 3 . 289–317 . 10.1007/BF00126382 . 1573-7187 . It is shown that the utilitarian welfare function satisfies all of Arrow's social choice postulates — avoiding the celebrated impossibility theorem by making use of information which is unavailable in Arrow's original framework. . 2020-03-20.
    22. Web site: Hamlin . Aaron . 2012-10-06 . Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow . dead . https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ . 2023-06-05 . The Center for Election Science. Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best.
    23. Web site: Hamlin . Aaron . 2012-10-06 . Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow . dead . https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ . 2023-06-05 . The Center for Election Science. 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.
    24. Web site: Hamlin . Aaron . 2012-10-06 . Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow . dead . https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ . 2023-06-05 . The Center for Election Science.
    25. Book: The Stanford Encyclopedia of Philosophy . 2019 . Metaphysics Research Lab, Stanford University . Arrow's Theorem . https://plato.stanford.edu/entries/arrows-theorem/#Rei.
    26. Gibbard . Allan . 1973 . Manipulation of Voting Schemes: A General Result . Econometrica . 41 . 4 . 587–601 . 10.2307/1914083 . 1914083 . 0012-9682.
    27. Campbell . D.E. . Kelly . J.S. . 2000 . A simple characterization of majority rule . . 15 . 3 . 689–700 . 10.1007/s001990050318 . 25055296 . 122290254.
    28. Book: Arrow, Kenneth Joseph Arrow . Social Choice and Individual Values . 1963 . Yale University Press . 978-0300013641 . https://ghostarchive.org/archive/20221009/http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf . 2022-10-09 . live.
    29. Quesada . Antonio . 2002 . From social choice functions to dictatorial social welfare functions . Economics Bulletin . 4 . 16 . 1–7.
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    33. Pearce . David . Individual and social welfare: a Bayesian perspective . Frisch Lecture Delivered to the World Congress of the Econometric Society.
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    44. McKenna . Phil . 12 April 2008 . Vote of no confidence . New Scientist . 198 . 2651 . 30–33 . 10.1016/S0262-4079(08)60914-8.
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    46. Indeed, 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 . . 15 . 3 . 689–700 . 10.1007/s001990050318 . 25055296 . 122290254.
    47. McLean . Iain . 1995-10-01 . Independence of irrelevant alternatives before Arrow . Mathematical Social Sciences . 30 . 2 . 107–126 . 10.1016/0165-4896(95)00784-J . 0165-4896.
    48. Gehrlein . William V. . 1983-06-01 . Condorcet's paradox . Theory and Decision . en . 15 . 2 . 161–197 . 10.1007/BF00143070 . 1573-7187.
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    50. Van Deemen . Adrian . 2014-03-01 . On the empirical relevance of Condorcet's paradox . Public Choice . 158 . 3 . 311–330 . 10.1007/s11127-013-0133-3 . 1573-7101.
    51. Wolk . Sara . Quinn . Jameson . Ogren . Marcus . 2023-09-01 . STAR Voting, equality of voice, and voter satisfaction: considerations for voting method reform . Constitutional Political Economy . 34 . 3 . 310–334 . 10.1007/s10602-022-09389-3 . 1572-9966.
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    56. Multidimensional voting models: theory and applications . UCL (University College London) . 2016-09-28 . Doctoral . V. . Dotti.
    57. Holliday . Wesley H. . Pacuit . Eric . 2023-09-01 . Stable Voting . Constitutional Political Economy . 34 . 3 . 421–433 . 10.1007/s10602-022-09383-9 . 1572-9966 . free.
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    59. Book: Balinski . M. L. . Majority judgment: measuring, ranking, and electing . Laraki . Rida . 2010 . MIT Press . 9780262545716 . Cambridge, Mass.
    60. Book: Poundstone, William . Gaming the Vote: Why Elections Are not Fair (and What We Can Do About It) . 2009-02-17 . Macmillan . 9780809048922.
    61. Web site: Cockrell . Jeff . 2016-03-08 . What economists think about voting . dead . https://web.archive.org/web/20160326065043/http://www.chicagobooth.edu/capideas/blog/2016/march/what-economists-think-about-voting . 2016-03-26 . 2016-09-05 . Capital Ideas . Chicago Booth . Is there such a thing as a perfect voting system? The respondents were unanimous in their insistence that there is not..
    62. Lützen . Jesper . 2019-02-01 . How mathematical impossibility changed welfare economics: A history of Arrow's impossibility theorem . Historia Mathematica . 46 . 56–87 . 10.1016/j.hm.2018.11.001 . 0315-0860.
    63. "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
    64. Sen . Amartya . 1999 . The Possibility of Social Choice . American Economic Review . 89 . 3 . 349–378 . 10.1257/aer.89.3.349.
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    66. Arrow . Kenneth J. . Kenneth Arrow . 1950 . A Difficulty in the Concept of Social Welfare . dead . . 58 . 4 . 328–346 . 10.1086/256963 . 1828886 . 13923619 . https://web.archive.org/web/20110720090207/http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf . 2011-07-20.
    67. Web site: Hamlin . Aaron . 2012-10-06 . Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow . dead . https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ . 2023-06-05 . The Center for Election Science.
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    76. Moore . Michael . 1 July 1975 . Rating versus ranking in the Rokeach Value Survey: An Israeli comparison . European Journal of Social Psychology . en . 5 . 3 . 405–408 . 10.1002/ejsp.2420050313 . 1099-0992 . 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..
    77. Maio . 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 . 10.1207/s15324834basp1802_4 . 0197-3533 . 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..
    78. Conklin . E. S. . Sutherland . J. W. . 1 February 1923 . A Comparison of the Scale of Values Method with the Order-of-Merit Method. . Journal of Experimental Psychology . en . 6 . 1 . 44–57 . 10.1037/h0074763 . 0022-1015 . the scale-of-values method can be used for approximately the same purposes as the order-of-merit method, but that the scale-of-values method is a better means of obtaining a record of judgments.
    79. Kaiser . Caspar . Oswald . Andrew J. . 18 October 2022 . The scientific value of numerical measures of human feelings . Proceedings of the National Academy of Sciences . 119 . 42 . e2210412119 . 2022PNAS..11910412K . 10.1073/pnas.2210412119 . 0027-8424 . 9586273 . 36191179 . free.
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    81. Ohtsubo . Yohsuke . Watanabe . Yoriko . September 2003 . Contrast Effects and Approval Voting: An Illustration of a Systematic Violation of the Independence of Irrelevant Alternatives Condition . Political Psychology . en . 24 . 3 . 549–559 . 10.1111/0162-895X.00340 . 0162-895X.
    82. 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 . 10.1509/jmr.14.0208 . 0022-2437 . 143974563.
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    86. See Chapter 6 of Book: Taylor, Alan D. . Social choice and the mathematics of manipulation . Cambridge University Press . 2005 . 978-0-521-00883-9 . New York . none. for a concise discussion of social choice for infinite societies.
    87. Book: Arrow, Kenneth Joseph Arrow . Social Choice and Individual Values . 1963 . Yale University Press . 978-0300013641 . https://ghostarchive.org/archive/20221009/http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf . 2022-10-09 . live.
    88. Book: The Stanford Encyclopedia of Philosophy . 2019 . Metaphysics Research Lab, Stanford University . Arrow's Theorem . https://plato.stanford.edu/entries/arrows-theorem/#Rei.
    89. Gibbard . Allan . 1973 . Manipulation of Voting Schemes: A General Result . Econometrica . 41 . 4 . 587–601 . 10.2307/1914083 . 0012-9682 . 1914083.
    90. Book: Arrow, Kenneth Joseph Arrow . Social Choice and Individual Values . 1963 . Yale University Press . 978-0300013641 . https://ghostarchive.org/archive/20221009/http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf . 2022-10-09 . live.
    91. Web site: Hamlin . Aaron . March 2017 . Remembering Kenneth Arrow and His Impossibility Theorem . 5 May 2024 . Center for Election Science.

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