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Arrow's impossibility theorem

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Arrow's impossibility theorem is a key result in social choice theory, showing that no ranking-based decision rule can satisfy the requirements of rational choice theory.[1] Most notably, Arrow showed that no such rule can satisfy independence of irrelevant alternatives, the principle that a choice between two alternatives A and B should not depend on the quality of some third, unrelated option C.[2][3][4]

The result is most often cited in discussions of voting rules,[5] where it shows that no ranked voting rule can eliminate the spoiler effect.[6][7][8] However, Arrow's theorem is substantially broader, and can be applied to methods of social decision-making other than voting. It therefore generalizes Condorcet's voting paradox, and shows similar problems exist for every collective decision-making procedure based on relative comparisons.[1]

Plurality-rule methods like first-past-the-post and ranked-choice (instant-runoff) voting are highly sensitive to spoilers,[9][10] particularly in situations where they are not forced.[11][12] By contrast, majority-rule (Condorcet) methods of ranked voting uniquely minimize the number of spoiled elections[12] by restricting them to rare[13][14] situations called cyclic ties.[11] Under some idealized models of voter behavior (e.g. Black's left-right spectrum), spoiler effects can disappear entirely for these methods.[15][16]

Rated voting rules, where voters assign a separate grade to each candidate, are not affected by Arrow's theorem.[6][7][17] 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.[18] However, he and other authors would later recognize this as a mistake,[19][20] with Arrow admitting rules based on cardinal utilities (such as score and approval voting) are not subject to his theorem.[21][22]

Background

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When Kenneth Arrow proved his theorem in 1950, it inaugurated the modern field of social choice theory, a branch of welfare economics that deals with aggregating preferences and beliefs to make optimal decisions.[20] The goal of social choice theory 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.[2] Such a procedure can be a market, voting system, constitution, or even a moral or ethical framework.[1] Ideally, such a procedure should satisfy the properties of rational choice and avoid any kind of self-contradiction.[2]

Axioms of voting systems

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Preferences

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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

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Arrow's theorem assumes as background that any non-degenerate (i.e. actually usable) social choice rule will satisfy:[23]

  • Non-dictatorship — the system does not depend on only one voter's ballot.[3]
    • This weakens anonymity (one vote, one value) to allow rules that treat voters unequally.
    • It essentially defines social choices as those depending on more than one person's input.[3]
  • Non-imposition — the system does not ignore the voters entirely when choosing between some pairs of candidates.[4][24]
    • In other words, it is possible for any candidate to defeat any other candidate, given some combination of votes.[4][24][25]
    • This is often replaced with the stronger Pareto efficiency axiom: if every voter prefers A over B, then A should defeat B. However, the weaker non-imposition condition is sufficient.[4]

Arrow's original statement of the theorem included non-negative responsiveness as a condition, i.e. that increasing the rank of an outcome should not make them lose—in other words, that a voting rule shouldn’t penalize a candidate for being more popular.[2] However, this assumption is not needed or used in his proof (except to derive the weaker condition of Pareto efficiency), and so is not related to the paradox.[3] While Arrow considered it an obvious requirement of any proposed social choice rule, ranked-choice voting (RCV) fails this condition.[26] Arrow later corrected his statement of the theorem to include runoffs and other voting rules vulnerable to the additional support paradox.[3][26]

Rationality

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Among the most important axioms of rational choice is independence of irrelevant alternatives (IIA), which says that when deciding between A and B, one's opinion about a third option C should not affect their decision.[2]

IIA is sometimes illustrated with a short joke by philosopher Sidney Morgenbesser:[27]

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.[27]

Theorem

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Intuitive argument

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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.[28] 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.[28] 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.

Formal statement

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Let be a set of alternatives. A voter's preferences over are 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 or . Denote the set of all preferences on by . Let be a positive integer. An ordinal (ranked) social welfare function is a function[2]

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:[29]

Pareto efficiency
If alternative is preferred to for all orderings , then is preferred to by .[2]
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 .[2]
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 .[2]

Formal proof

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Proof by decisive coalition

Arrow's proof used the concept of decisive coalitions.[3]

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[30] and Ariel Rubinstein.[31] 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 showing there is only one pivotal voter

Proofs using the concept of the pivotal voter originated from Salvador Barberá in 1980.[32] The proof given here is a simplified version based on two proofs published in Economic Theory.[29][33]

Setup

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Assume there are n voters. We assign all of these voters an arbitrary ID number, ranging from 1 through n, which we can use to keep track of each voter's identity as we consider what happens when they change their votes. Without loss of generality, we can say there are three candidates who we call A, B, and C. (Because of IIA, including more than 3 candidates does not affect the proof.)

We will prove that any social choice rule respecting unanimity and independence of irrelevant alternatives (IIA) is a dictatorship. The proof is in three parts:

  1. We identify a pivotal voter for each individual contest (A vs. B, B vs. C, and A vs. C). Their ballot swings the societal outcome.
  2. We prove this voter is a partial dictator. In other words, they get to decide whether A or B is ranked higher in the outcome.
  3. We prove this voter is the same person, hence this voter is a dictator.

Part one: There is a pivotal voter for A vs. B

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Part one: Successively move B from the bottom to the top of voters' ballots. The voter whose change results in B being ranked over A is the pivotal voter for B over A.

Consider the situation where everyone prefers A to B, and everyone also prefers C to B. By unanimity, society must also prefer both A and C to B. Call this situation profile[0, x].

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

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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.

Part two: Switching A and B on the ballot of voter k causes the same switch to the societal outcome, by part one of the argument. Making any or all of the indicated switches to the other ballots has no effect on the outcome.

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

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Part three: Since voter k is the dictator for B over C, the pivotal voter for B over C must appear among the first k voters. That is, outside of segment two. Likewise, the pivotal voter for C over B must appear among voters k through N. That is, outside of Segment One.

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/AkC/B.

Now repeating the entire argument above with B and C switched, we also have

kC/BkB/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

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Arrow's impossibility theorem still holds if Pareto efficiency is weakened to the following condition:[4]

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

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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."[8][34]

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.[27]

Minimizing IIA failures: Majority-rule methods

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An example of a Condorcet cycle, where some candidate must cause a spoiler effect

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. (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.[12]) 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.[28]

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.[35] 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.[28]

Unlike pluralitarian rules such as ranked-choice runoff (RCV) or first-preference plurality,[9] 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.[14][clarification needed] Spatial voting models also suggest such paradoxes are likely to be infrequent[36][13] or even non-existent.[15]

Left-right spectrum

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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.[15][37]

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.[15][37][12]

The rule does not fully generalize from the political spectrum to the political compass, a result related to the McKelvey-Schofield chaos theorem.[15][38] However, a well-defined Condorcet winner does exist if the distribution of voters is rotationally symmetric or otherwise has a uniquely-defined median.[39][40] In most realistic situations, where voters' opinions follow a roughly-normal distribution or can be accurately summarized by one or two dimensions, Condorcet cycles are rare (though not unheard of).[36][11]

Generalized stability theorems

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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.[12] 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.[12]

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.[41]

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).[11]

Eliminating IIA failures: Rated voting

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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.[34] 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).[42][page needed]

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),[43] so the informal dictum that "no voting system is perfect" still has some mathematical basis.[44]

Meaningfulness of cardinal information

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Arrow's framework assumed individual and social preferences are orderings or rankings, i.e. statements about which outcomes are better or worse than others.[45] Taking inspiration from the strict behaviorism popular in psychology, some philosophers and economists rejected the idea of comparing internal human experiences of well-being.[46][27] 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.[47]

Arrow originally agreed with these positions and rejected cardinal utility, leading him to focus his theorem on preference rankings;[46][48] 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.[27] As a result, Arrow initially interpreted his theorem as a kind of mathematical proof for nihilism or egoism.[27][45][2] However, he later reversed this opinion, admitting cardinal methods can provide useful information that allows them to evade his theorem.[19][49] 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.[47]

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.[42][page needed]

John Harsanyi noted Arrow's theorem could be considered a weaker version of his own theorem[50] and other Utility representation theorems like the VNM theorem, which generally show that rational behavior requires consistent cardinal utilities.[51] Harsanyi[50] and Vickrey[52] each independently derived results showing such interpersonal comparisons of utility could be rigorously defined as individual preferences over the lottery of birth.[53][54]

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,[27] and Hildreth pointed out that "any procedure that extends the partial ordering of [Pareto efficiency] must involve interpersonal comparisons of utility."[55] 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.[56]

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.[57][58] Research has consistently found cardinal rating scales (e.g. Likert scales) provide more information than rankings alone.[58][59] 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".[60] 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.[60] 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".[60]

Nonstandard spoilers

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Behavioral economists have shown individual irrationality involves violations of IIA (e.g. with decoy effects),[61] suggesting human behavior can cause IIA failures even if the voting method itself does not.[62] However, past research has typically found such effects to be fairly small,[63] 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.[42][page needed] Similar techniques are often discussed in the context of contingent valuation.[49]

Esoteric solutions

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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

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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.[64]

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).[65] 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.[65]

Infinite populations

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Fishburn shows all of Arrow's conditions can be satisfied for uncountably infinite sets of voters given the axiom of choice;[66] 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".[67]

Common misconceptions

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Arrow's theorem is not related to strategic voting, which does not appear in his framework,[3][1] though the theorem does have important implications for strategic voting (being used as a lemma to prove Gibbard's theorem[23]). The Arrovian framework of social welfare assumes all voter preferences are known and the only issue is in aggregating them.[1]

Monotonicity (called positive association by Arrow) is not a condition of Arrow's theorem.[3] 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.[2] Dropping the assumption does not allow for constructing a social welfare function that meets his other conditions.[3]

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.[1][68]

See also

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References

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  1. ^ a b c d e f Morreau, Michael (2019). "Arrow's Theorem". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
  2. ^ a b c d e f g h i j k Arrow, Kenneth J. (1950). "A Difficulty in the Concept of Social Welfare" (PDF). Journal of Political Economy. 58 (4): 328–346. doi:10.1086/256963. JSTOR 1828886. S2CID 13923619. Archived from the original (PDF) on 2011-07-20.
  3. ^ a b c d e f g h i j Arrow, Kenneth Joseph (1963). Social Choice and Individual Values (PDF). Yale University Press. ISBN 978-0300013641. Archived (PDF) from the original on 2022-10-09.
  4. ^ a b c d e Wilson, Robert (December 1972). "Social choice theory without the Pareto Principle". Journal of Economic Theory. 5 (3): 478–486. doi:10.1016/0022-0531(72)90051-8. ISSN 0022-0531.
  5. ^ Borgers, Christoph (2010-01-01). Mathematics of Social Choice: Voting, Compensation, and Division. SIAM. ISBN 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
  6. ^ a b Ng, 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. ISSN 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.
  7. ^ a b 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. doi:10.2307/138144. ISSN 0315-4890. JSTOR 138144. Retrieved 2020-03-20. The abandonment of Condition 3 makes it possible to formulate a procedure for arriving at a social choice. Such a procedure is described below
  8. ^ a b 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.
  9. ^ a b 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. doi:10.2307/3088418. ISSN 0092-5853. JSTOR 3088418. As with simple plurality elections, it is apparent the outcome will be highly sensitive to the distribution of candidates.
  10. ^ Borgers, Christoph (2010-01-01). Mathematics of Social Choice: Voting, Compensation, and Division. SIAM. ISBN 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, although it unquestionably makes it less likely to occur in practice.
  11. ^ a b c d Holliday, Wesley H.; Pacuit, Eric (2023-03-14). "Stable Voting". Constitutional Political Economy. 34 (3): 421–433. arXiv:2108.00542. doi:10.1007/s10602-022-09383-9. ISSN 1572-9966. 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.
  12. ^ a b c d e f g Campbell, D. E.; Kelly, J. S. (2000). "A simple characterization of majority rule". Economic Theory. 15 (3): 689–700. doi:10.1007/s001990050318. JSTOR 25055296. S2CID 122290254.
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    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.
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    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.

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    CES: Now, you mention that your theorem applies to preferential systems or ranking systems.
    Dr. Arrow: Yes.
    CES: But the system that you're just referring to, approval voting, falls within a class called cardinal systems. So not within ranking systems.
    Dr. Arrow: And as I said, that in effect implies more information.
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Further reading

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