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Arrow's impossibility theorem is a key result insocial choice theory showing that noranked-choice procedure for group decision-making can satisfy the requirements ofrational choice.[1] Specifically,Arrow showed no such rule can satisfyindependence of irrelevant alternatives, the principle that a choice between two alternativesA andB should not depend on the quality of some third, unrelated option,C.[2][3][4]
The result is often cited in discussions ofvoting rules,[5] where it shows noranked voting rule can eliminate thespoiler effect.[6][7][8] This result was first shown by theMarquis de Condorcet, whosevoting paradox showed the impossibility of logically-consistentmajority rule; Arrow's theoremgeneralizes Condorcet's findings to include non-majoritarian rules likecollective leadership orconsensus decision-making.[1]
While the impossibility theorem shows all ranked voting rules must have spoilers, the frequency of spoilers differs dramatically by rule.Plurality-rule methods likechoose-one andranked-choice (instant-runoff) voting are highly sensitive to spoilers,[9][10] creating them even in some situations where they are notmathematically necessary (e.g. incenter squeezes).[11][12] In contrast,majority-rule (Condorcet) methods ofranked voting uniquelyminimize the number of spoiled elections[12] by restricting them tovoting cycles,[11] which are rare in ideologically-driven elections.[13][14] Under somemodels of voter preferences (like the left-right spectrum assumed in themedian voter theorem), spoilers 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.[17][18][19] 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.[20] However, Arrow would later describe this as a mistake,[21][22] admitting rules based oncardinal utilities (such asscore andapproval voting) are not subject to his theorem.[23][24]
WhenKenneth Arrow proved his theorem in 1950, it inaugurated the modern field ofsocial choice theory, a branch ofwelfare economics studying mechanisms to aggregatepreferences andbeliefs across a society.[25] Such a mechanism of study can be amarket,voting system,constitution, or even amoral orethical framework.[1]
In the context of Arrow's theorem, citizens are assumed to haveordinal preferences, i.e.orderings of candidates. IfA andB are different candidates or alternatives, then meansA is preferred toB. Individual preferences (or ballots) are required to satisfy intuitive properties of orderings, e.g. they must betransitive—if and, then. The social choice function is then amathematical function that maps the individual orderings to a new ordering that represents the preferences of all of society.
Arrow's theorem assumes as background that anynon-degenerate social choice rule will satisfy:[26]
Arrow's original statement of the theorem includednon-negative responsiveness as a condition, i.e., thatincreasing the rank of an outcome should not make themlose—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 Arrow later corrected his statement of the theorem to remove the inclusion of this condition.[3][29]
A commonly-considered axiom ofrational choice isindependence of irrelevant alternatives (IIA), which says that when deciding betweenA andB, one's opinion about a third optionC should not affect their decision.[2]
IIA is sometimes illustrated with a short joke by philosopherSidney Morgenbesser:[30]
Arrow's theorem shows that if a society wishes to make decisions while always avoiding such self-contradictions, it cannot use ranked information alone.[30]
Condorcet's example is already enough to see the impossibility of a fairranked voting system, given stronger conditions for fairness than Arrow's theorem assumes.[31] 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 creditCondorcet with having given an intuitive argument that presents the core of Arrow's theorem.[31] However, Arrow's theorem is substantially more general; it applies to methods of making decisions other than one-person-one-vote elections, such asmarkets orweighted voting, based onranked ballots.
Let be a set ofalternatives. A voter'spreferences over are acomplete andtransitivebinary relation on (sometimes called atotal preorder), that is, a subset of satisfying:
The element being in is interpreted to mean that alternative is preferred to or indifferent to alternative. This situation is often denoted or. The symmetric part of yields the indifference relation. This is written as or if and only if and are both in. The asymmetric part of yields the (strict) preference relation. This is written as or if and only if is in and is not in. In the following, preference of one alternative over another denotes strict preference.
Denote the set of all preferences on by. Equivalently, is the set of rankings of the alternatives in from top to bottom, with ties allowed. Let be a positive integer. Anordinal (ranked)social welfare function is a function[2]
which aggregates voters' preferences into a single preference on. An-tuple of voters' preferences is called apreference 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:[32]
Proof by decisive coalition |
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Arrow's proof used the concept ofdecisive coalitions.[3] Definition:
Our goal is to prove that thedecisive coalition contains only one voter, who controls the outcome—in other words, adictator. The following proof is a simplification taken fromAmartya Sen[33] andAriel Rubinstein.[34] The simplified proof uses an additional concept:
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 |
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Proofs using the concept of thepivotal voter originated from Salvador Barberá in 1980.[35] The proof given here is a simplified version based on two proofs published inEconomic Theory.[32][36] Setup[edit]Assume there aren voters. We assign all of these voters an arbitrary ID number, ranging from1 throughn, 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 callA,B, andC. (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:
Part one: There is a pivotal voter for A vs. B[edit] Consider the situation where everyone prefersA toB, and everyone also prefersC toB. By unanimity, society must also prefer bothA andC toB. Call this situationprofile[0, x]. On the other hand, if everyone preferredB to everything else, then society would have to preferB to everything else by unanimity. Now arrange all the voters in some arbitrary but fixed order, and for eachi letprofile i be the same asprofile 0, but moveB to the top of the ballots for voters 1 throughi. Soprofile 1 hasB at the top of the ballot for voter 1, but not for any of the others.Profile 2 hasB at the top for voters 1 and 2, but no others, and so on. SinceB eventually moves to the top of the societal preference as the profile number increases, there must be some profile, numberk, for whichBfirst movesaboveA in the societal rank. We call the voterk whose ballot change causes this to happen thepivotal voter forB overA. Note that the pivotal voter forB overA is not,a priori, the same as the pivotal voter forA overB. 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 ifprofile 0 is any profile in whichA is ranked aboveB by every voter, and the pivotal voter forB overA will still be voterk. We will use this observation below. Part two: The pivotal voter for B over A is a dictator for B over C[edit]In this part of the argument we refer to voterk, the pivotal voter forB overA, as thepivotal voter for simplicity. We will show that the pivotal voter dictates society's decision forB overC. That is, we show that no matter how the rest of society votes, ifpivotal voter ranksB overC, then that is the societal outcome. Note again that the dictator forB overC is not a priori the same as that forC overB. 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 throughk − 1,segment one, and votersk + 1 throughN,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 rankA aboveB. This is because, except for a repositioning ofC, this profile is the same asprofile k − 1 from part one. Furthermore, by unanimity the societal outcome must rankB aboveC. Therefore, we know the outcome in this case completely. Now suppose that pivotal voter movesB aboveA, but keepsC in the same position and imagine that any number (even all!) of the other voters change their ballots to moveB belowC, without changing the position ofA. Then aside from a repositioning ofC this is the same asprofile k from part one and hence the societal outcome ranksB aboveA. Furthermore, by IIA the societal outcome must rankA aboveC, as in the previous case. In particular, the societal outcome ranksB aboveC, even though Pivotal Voter may have been theonly voter to rankB aboveC.By IIA, this conclusion holds independently of howA is positioned on the ballots, so pivotal voter is a dictator forB overC. Part three: There exists a dictator[edit] 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 forB overC must appear earlier (or at the same position) in the line than the dictator forB overC: As we consider the argument of part one applied toB andC, successively movingB to the top of voters' ballots, the pivot point where society ranksB aboveC must come at or before we reach the dictator forB overC. Likewise, reversing the roles ofB andC, the pivotal voter forC overB must be at or later in line than the dictator forB overC. In short, ifkX/Y denotes the position of the pivotal voter forX overY (for any two candidatesX andY), then we have shown
Now repeating the entire argument above withB andC switched, we also have
Therefore, we have
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. |
Arrow's impossibility theorem still holds if Pareto efficiency is weakened to the following condition:[4]
Arrow's theorem establishes that no ranked voting rule canalways 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."[37][38]
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 one or more of his assumptions, such as by focusing onrated voting rules.[30]
The first set of methods studied by economists are themajority-rule, orCondorcet, methods. These rules limit spoilers to situations where majority rule is self-contradictory, calledCondorcet 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, thenCondorcet method will adhere to Arrow's criteria.[12]) Condorcet believed voting rules should satisfy both independence of irrelevant alternatives and themajority rule principle, i.e. if most voters rankAlice ahead ofBob,Alice should defeatBob in the election.[31]
Unfortunately, as Condorcet proved, this rule can be intransitive on some preference profiles.[39] 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.[31]
Unlike pluralitarian rules such asranked-choice runoff (RCV) orfirst-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, suggesting they may be of limited practical concern.[14]Spatial voting models also suggest such paradoxes are likely to be infrequent[40][13] or even non-existent.[15]
Soon after Arrow published his theorem,Duncan Black showed his own remarkable result, themedian voter theorem. The theorem proves that if voters and candidates are arranged on aleft-right spectrum, Arrow's conditions are all fully compatible, and all will be met by any rule satisfyingCondorcet's majority-rule principle.[15][16]
More formally, Black's theorem assumes preferences aresingle-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][16][12]
The rule does not fully generalize from the political spectrum to the political compass, a result related to theMcKelvey-Schofield chaos theorem.[15][41] However, a well-defined Condorcet winner does exist if thedistribution of voters isrotationally symmetric or otherwise has auniquely-defined median.[42][43] 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).[40][11]
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 andEitan Muller gave a full characterization of domain restrictions admitting a nondictatorial andstrategyproof social welfare function. These correspond to preferences for which there is a Condorcet winner.[44]
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 failingvote positivity (though at a much lower rate than seen ininstant-runoff voting).[11][clarification needed]
As shown above, the proof of Arrow's theorem relies crucially on the assumption ofranked voting, and is not applicable torated voting systems. 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).[45]: 4–5 This opens up the possibility of finding another social choice procedure that satisfies independence of irrelevant alternatives.[46] Arrow's theorem can thus be considered a special case ofHarsanyi's utilitarian theorem and otherutility representation theorems like theVNM theorem, which showrational behavior requires consistentcardinal utilities.[47][48]
While Arrow's theorem does not apply to graded systems,Gibbard's theorem still does: no voting game can bestraightforward (i.e. have a single, clear, always-best strategy).[49]
Arrow's framework assumed individual and social preferences areorderings orrankings, i.e. statements about which outcomes are better or worse than others.[50] Taking inspiration from thebehavioralist approach, some philosophers and economists rejected the idea of comparing internal human experiences ofwell-being.[51][30] 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 theGreat Fire of Rome was good or bad, because despite killing thousands of Romans, it had the positive effect of lettingNero expand his palace.[52]
Arrow originally agreed with these position, rejecting the meaningfulness ofcardinal utilities,[3][51] thus interpreting his theorem as a kind of proof fornihilism oregoism.[30][50] However, he later stated that cardinal methods can provide additional useful information, and that his theorem is not applicable to them.[37][53] 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 they would only require "rather limited levels of partial comparability" to hold in practice.[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 choice procedure, with cardinal voting rules making these comparisons explicit.David Pearce identified Arrow's original nihilist interpretation with a kind ofcircular reasoning,[55] with Hildreth pointing out that "any procedure that extends the partial ordering of [Pareto efficiency] must involve interpersonal comparisons of utility."[56] Similar observations have led toimplicit utilitarian voting approaches, which attempt to make the assumptions of ranked procedures more explicit by modeling them as approximations of theutilitarian rule (orscore voting).[57]
Inpsychometrics, there is a general consensus that self-reported ratings (e.g.Likert scales) are meaningful and provide more information than pure rankings, as well as showing highervalidity andreliability.[58] Cardinalrating scales (e.g.Likert scales) provide more information than rankings alone.[59][60] A review by Kaiser and Oswald found that ratings were more predictive of important decisions (such as international migration and divorce) than even standardsocioeconomic predictors like income and demographics,[61] writing that "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".[61]
Behavioral economists have shown individualirrationality involves violations of IIA (e.g. withdecoy effects),[62] suggesting human behavior can cause IIA failures even if the voting method itself does not.[63] However, past research has typically found such effects to be fairly small,[64] and such psychological spoilers can appear regardless of electoral system.Balinski andLaraki discuss techniques ofballot design derived frompsychometrics 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.[45] Similar techniques are often discussed in the context ofcontingent valuation.[53]
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 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 producevoting paradoxes.[65]
Inspatial (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).[66] 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.[66]
Fishburn shows all of Arrow's conditions can be satisfied foruncountably infinite sets of voters given theaxiom of choice;[67] however, Kirman and Sondermann demonstrated this requires disenfranchisingalmost all members of a society (eligible voters form a set ofmeasure 0), leading them to refer to such societies as "invisible dictatorships".[68]
Arrow's theorem is not related tostrategic 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 proveGibbard's theorem[26]). The Arrovian framework ofsocial welfare assumes all voter preferences are known and the only issue is in aggregating them.[1]
Monotonicity (calledpositive 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 ofranked-choice voting systems, rather than voting systems as a whole.[1][69]
{{cite book}}:ISBN / Date incompatibility (help)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 ... doesnot do away with the spoiler problem entirely
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.
The abandonment of Condition 3 makes it possible to formulate a procedure for arriving at a social choice. Such a procedure is described below
As with simple plurality elections, it is apparent the outcome will be highly sensitive to the distribution of candidates.
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 ... doesnot do away with the spoiler problem entirely, although it unquestionably makes it less likely to occur in practice.
This is a kind of stability property of Condorcet winners: you cannot dislodge a Condorcet winnerA by adding a new candidateB 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.
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.
The abandonment of Condition 3 makes it possible to formulate a procedure for arriving at a social choice. Such a procedure is described below
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
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 isunavailable in Arrow's original framework.
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.
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 isunavailable in Arrow's original framework.
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.
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.
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
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