Arrow's impossibility theorem
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Arrow's impossibility theorem is a key result in social choice, discovered by Kenneth Arrow, showing that no ranked voting rule^{[note 1]} can behave rationally.^{[1]} Specifically, any such rule violates independence of irrelevant alternatives (IIA), the idea that a choice between and should not depend on the quality of a third, unrelated option .^{[2]}^{[3]} The result is most often cited in election science and voting theory, where 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]}
However, Arrow's theorem is substantially broader, and can be applied to other methods of social decisionmaking besides voting. It therefore generalizes Nicolas de Condorcet's voting paradox, and shows similar problems will exist for any collective decisionmaking procedure based on relative comparisons.^{[8]} It can be generalized in turn by the doctrinal paradox.^{[9]}
Pluralityrule methods like firstpastthepost and rankedchoice (instantrunoff) voting are highly sensitive to spoilers,^{[10]}^{[11]} and can manufacture them in situations where they are not forced.^{[12]}^{[13]} By contrast, majorityrule (Condorcet) methods of ranked voting uniquely minimize the number of spoiled elections^{[14]} by restricting them to rare^{[15]}^{[16]} situations called cyclic ties.^{[17]} Under some idealized models of voter behavior (e.g. Duncan Black's leftright spectrum), spoiler effects can disappear entirely for such rules.^{[18]}^{[19]}
Rated voting rules, where voters assign a separate grade to each candidate, are not affected by Arrow's theorem (allowing some to satisfy IIA).^{[5]}^{[7]}^{[20]} Arrow initially asserted the information provided by these systems was meaningless and therefore could not be used to prevent paradoxes, leading him to overlook them.^{[21]} However, he and other authors would later recognize this as a mistake,^{[22]}^{[23]} with Arrow admitting rules based on cardinal utilities (such as score and approval voting) are not subject to his theorem.^{[24]}^{[25]}
Background
[edit]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.^{[23]} 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.^{[1]} Such a procedure can be a market, voting system, constitution, or even a moral or ethical framework.^{[8]} Ideally, such a procedure should satisfy the properties of rational choice and avoid any kind of selfcontradiction.^{[1]}
Axioms of voting systems
[edit]Preferences
[edit]In social choice theory, ordinal preferences are modeled as 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
[edit]Arrow's theorem assumes as background that nondegenerate ranked social choice rules satisfy:^{[26]}
 Universal domain — the social choice function is a total function over the domain of all possible orderings of outcomes, not just a partial function.
 In other words, the system must always make some choice, and cannot simply "give up" when the voters have unusual opinions.
 Without this assumption, majority rule satisfies Arrow's axioms by "giving up" whenever there is a Condorcet cycle.^{[27]}
 Nondictatorship — the system does not depend on only one voter's ballot.^{[28]}
 This weakens anonymity (one vote, one value) to allow rules that treat voters unequally.
 This assumption defines social choices as those depending on more than one person's input.^{[28]}
 Nonimposition — the system does not ignore the voters entirely when choosing between some pairs of candidates.^{[29]}^{[30]}
 In other words, it is possible for any candidate to defeat any other candidate, given some combination of votes.^{[29]}^{[30]}^{[31]}
 This is typically replaced with the stronger Pareto efficiency: if voters unanimously support candidate A over candidate B, then candidate A should beat candidate B.^{[29]}
Arrow's original statement of the theorem included the assumption of nonperversity, i.e. increasing the rank of an outcome should not make them lose.^{[32]} However, this assumption is not needed or used in his proof, except to derive the weaker Pareto efficiency axiom, and as a result is not related to the paradox.^{[28]} While Arrow considered it an obvious requirement of any proposed social choice rule, rankedchoice runoff (RCV) fails this condition.^{[33]} Arrow later corrected his statement of the theorem to include runoffs and other perverse voting rules.^{[28]}^{[33]}
Rationality
[edit]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.^{[34]}
 Independence of irrelevant alternatives (IIA) — the social preference between candidate A and candidate B should only depend on the individual preferences between A and B.
 In other words, the social preference should not change from to if voters change their preference about whether .^{[28]}
 This is equivalent to the claim about independence of spoiler candidates when using the standard construction of a placement function.^{[35]}
IIA is sometimes illustrated with a short joke by philosopher Sidney Morgenbesser:^{[36]}
 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 selfcontradictions, it cannot use methods that discard cardinal information.^{[36]}
Theorem
[edit]Intuitive argument
[edit]Condorcet's example is already enough to see the impossibility of a fair ranked voting system, given stronger conditions for fairness than Arrow's theorem assumes.^{[37]} Suppose we have three candidates (, , and ) and three voters whose preferences are as follows:
Voter  First preference  Second preference  Third preference 

Voter 1  A  B  C 
Voter 2  B  C  A 
Voter 3  C  A  B 
If is chosen as the winner, it can be argued any fair voting system would say should win instead, since two voters (1 and 2) prefer to and only one voter (3) prefers to . However, by the same argument is preferred to , and is preferred to , by a margin of two to one on each occasion. Thus, even though each individual voter has consistent preferences, the preferences of society are contradictory: is preferred over which is preferred over which is preferred over .
Because of this example, some authors credit Condorcet with having given an intuitive argument that presents the core of Arrow's theorem.^{[37]} However, Arrow's theorem is substantially more general; it applies to methods of making decisions other than onemanonevote elections, such as markets or weighted voting, based on ranked ballots.
Formal statement
[edit]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:
 (Transitivity) If is in and is in , then is in ,
 (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^{[38]}
which aggregates voters' preferences into a single preference on . An tuple of voters' preferences is called a preference profile.
Arrow's impossibility theorem: If there are at least three alternatives, then there is no social welfare function satisfying all three of the conditions listed below:^{[39]}
 Pareto efficiency
 If alternative is preferred to for all orderings , then is preferred to by .^{[38]}
 Nondictatorship
 There is no individual whose preferences always prevail. That is, there is no such that for all and all and , when is preferred to by then is preferred to by .^{[38]}
 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 .^{[38]}
Formal proof
[edit]Proof by decisive coalition


Arrow's proof used the concept of decisive coalitions.^{[40]} Definition:
Our goal is to prove that the decisive coalition contains only one voter, who controls the outcome—in other words, a dictator. The following proof is a simplification taken from Amartya Sen^{[41]} and Ariel Rubinstein.^{[42]} The simplified proof uses an additional concept:
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 weakdecisiveness over , . Thus . Similarly, is decisive over . By iterating the above two claims (note that decisiveness implies weakdecisiveness), 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 sizeone decisive coalition—a dictator. 
Proof by pivotal voter


Proofs using the concept of the pivotal voter originated from Salvador Barberá in 1980.^{[43]} The proof given here is a simplified version based on two proofs published in Economic Theory.^{[39]}^{[44]} We will prove that any social choice system respecting unrestricted domain, unanimity, and independence of irrelevant alternatives (IIA) is a dictatorship. The key idea is to identify a pivotal voter whose ballot swings the societal outcome. We then prove that this voter is a partial dictator (in a specific technical sense, described below). Finally we conclude by showing that all of the partial dictators are the same person, hence this voter is a dictator. For simplicity we have presented all rankings as if there are no ties. A complete proof taking possible ties into account is not essentially different from the one given here, except that one ought to say "not above" instead of "below" or "not below" instead of "above" in some cases. Full details are given in the original articles. Part one: There is a "pivotal" voter for B over A[edit]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[edit]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[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 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 k_{X/Y} denotes the position of the pivotal voter for X over Y (for any two candidates X and Y), then we have shown
Now repeating the entire argument above with B and C 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. 
Generalizations
[edit]Arrow's impossibility theorem still holds if Pareto efficiency is weakened to the following condition:^{[45]}
 Nonimposition
 For any two alternatives a and b, there exists some preference profile R_{1} , …, R_{N} such that a is preferred to b by F(R_{1}, R_{2}, …, R_{N}).
Interpretation and practical solutions
[edit]Arrow's theorem establishes that no ranked voting rule can always satisfy independence of irrelevant alternatives, but it says nothing about the frequency of spoilers. This led Arrow to remark that "Most systems are not going to work badly all of the time. All I proved is that all can work badly at times."^{[46]}^{[47]}
Attempts at dealing with the effects of Arrow's theorem take one of two approaches: either accepting his rule and searching for the least spoilerprone methods, or dropping his assumption of ranked voting to focus on studying rated voting rules.^{[48]}
Minimizing IIA failures: Majorityrule methods
[edit]The first set of methods studied by economists are the majorityrule, or Condorcet, methods. These rules limit spoilers to situations where majority rule is selfcontradictory, called Condorcet cycles, and as a result uniquely minimize the possibility of a spoiler effect among ranked rules.^{[49]} Condorcet believed voting rules should satisfy both independence of irrelevant alternatives and the majority rule principle, i.e. if most voters rank Alice ahead of Bob, Alice should defeat Bob in the election.^{[50]}
Unfortunately, as Condorcet proved, this rule can be selfcontradictory (intransitive), because there can be a rockpaperscissors cycle with three or more candidates defeating each other in a circle.^{[51]} Thus, Condorcet proved a weaker form of Arrow's impossibility theorem long before Arrow, under the stronger assumption that a voting system in the twocandidate case will agree with a simple majority vote.^{[50]}
Unlike pluralitarian rules such as rankedchoice runoff (RCV) or firstpreference plurality,^{[52]} Condorcet methods avoid the spoiler effect in noncyclic elections, where candidates can be chosen by majority rule. Political scientists have found such cycles to be fairly rare, likely in the range of a few percent, suggesting they may be of limited practical concern.^{[53]} Spatial voting models also suggest such paradoxes are likely to be infrequent^{[54]}^{[55]} or even nonexistent.^{[18]}
Leftright spectrum
[edit]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 leftright spectrum, Arrow's conditions are all fully compatible, and all will be met by any rule satisfying Condorcet's majorityrule principle.^{[18]}^{[56]}
More formally, Black's theorem assumes preferences are singlepeaked: 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 singlepeaked preference with respect to the linear ordering, then social preferences are acyclic. In this situation, Condorcet methods satisfy a wide variety of highlydesirable properties, including being fully spoilerproof.^{[18]}^{[56]}^{[57]}
The rule does not fully generalize from the political spectrum to the political compass, a result related to the McKelveySchofield Chaos Theorem.^{[18]}^{[58]} However, a welldefined Condorcet winner does exist if the distribution of voters is rotationally symmetric or otherwise has a uniquelydefined median.^{[59]}^{[60]} In most realistic situations, where voters' opinions follow a roughlynormal distribution or can be accurately summarized by one or two dimensions, Condorcet cycles are rare (though not unheard of).^{[54]}^{[61]}
Generalized stability theorems
[edit]The CampbellKelly theorem shows that Condorcet methods are the most spoilerresistant class of ranked voting systems: whenever it is possible for some ranked voting system to avoid a spoiler effect, a Condorcet method will do so.^{[49]} In other words, replacing a ranked method with its Condorcet variant (i.e. elect a Condorcet winner if they exist, and otherwise run the method) will sometimes prevent a spoiler effect, but never cause a new one.^{[49]}
In 1977, Ehud Kalai and Eitan Muller gave a full characterization of domain restrictions admitting a nondictatorial and strategyproof social welfare function. These correspond to preferences for which there is a Condorcet winner.^{[62]}
Holliday and Pacuit devised a voting system that provably minimizes the number of candidates who are capable of spoiling an election, albeit at the cost of occasionally failing vote positivity (though at a much lower rate than seen in instantrunoff voting).^{[61]}
Eliminating IIA failures: Rated voting
[edit]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.^{[47]} 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).^{[63]}
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, alwaysbest strategy),^{[64]} so the informal dictum that "no voting system is perfect" still has some mathematical basis.^{[65]}
Meaningfulness of cardinal information
[edit]Arrow's framework assumed individual and social preferences are orderings or rankings, i.e. statements about which outcomes are better or worse than others.^{[66]} Taking inspiration from the strict behaviorism popular in psychology, some philosophers and economists rejected the idea of comparing internal human experiences of wellbeing.^{[67]}^{[48]} Such philosophers claimed it was impossible to compare the strength of preferences across people who disagreed; Sen gives as an example that it would be impossible to know whether the Great Fire of Rome was good or bad, because despite killing thousands of Romans, it had the positive effect of letting Nero expand his palace.^{[68]}
Arrow originally agreed with these positions and rejected cardinal utility, leading him to focus his theorem on preference rankings;^{[67]}^{[69]} his goal in adding the independence axiom was, in part, to prevent from the social choice function from "sneaking in" cardinal information by attempting to infer it from the rankings.^{[36]} As a result, Arrow initially interpreted his theorem as a kind of mathematical proof for nihilism or egoism.^{[48]}^{[66]}^{[70]} However, he later reversed this opinion, admitting cardinal methods can provide useful information that allows them to evade his theorem.^{[71]}^{[72]} Similarly, Amartya Sen first claimed interpersonal comparability is necessary for IIA, but later came to argue in favor of cardinal methods for assessing social choice, arguing it would only require "rather limited levels of partial comparability" to hold in practice.^{[68]}
Balinski and Laraki disputed that any interpersonal comparisons are required for rated voting rules to pass IIA. They argue the availability of a common language with verbal grades is sufficient for IIA by allowing voters to give consistent responses to questions about candidate quality. In other words, they argue most voters will not change their beliefs about whether a candidate is "good", "bad", or "neutral" simply because another candidate joins or drops out of a race.^{[63]}
John Harsanyi noted Arrow's theorem could be considered a weaker version of his own theorem^{[73]} and other utility representation theorems like the VNM theorem, which generally show that rational behavior requires consistent cardinal utilities.^{[74]} Harsanyi^{[73]} and Vickrey^{[75]} each independently derived results showing such interpersonal comparisons of utility could be rigorously defined as individual preferences over the lottery of birth.^{[76]}^{[77]}
Other scholars have noted that interpersonal comparisons of utility are not unique to cardinal voting, but are instead a necessity of any nondictatorial (or nonegoist) choice procedure, with cardinal voting rules simply making these comparisons explicit. David Pearce identified Arrow's original interpretation of the theorem as a mathematical proof of nihilism or egoism with a kind of circular reasoning,^{[48]} and Hildreth pointed out that "any procedure that extends the partial ordering of [Pareto efficiency] must involve interpersonal comparisons of utility."^{[78]} These observations have led to the rise of implicit utilitarian voting, which identifies ranked procedures with approximations of the utilitarian rule (i.e. score voting), helping to make them more explicit.^{[79]}
In psychometrics, there is a nearuniversal scientific consensus for the usefulness and meaningfulness of selfreported ratings, which empirically show higher validity and reliability than rankings in measuring human opinions.^{[80]}^{[81]} Research has consistently found cardinal rating scales (e.g. Likert scales) provide more information than rankings alone.^{[81]}^{[82]} Kaiser and Oswald conducted an empirical review of four decades of research including over 700,000 participants who provided selfreported ratings of utility, with the goal of identifying whether people "have a sense of an actual underlying scale for their innermost feelings".^{[83]} They found responses to these questions were consistent with all expectations of a wellspecified quantitative measure. Furthermore, such ratings were highly predictive of important decisions (such as international migration and divorce) and had larger effect sizes than standard socioeconomic predictors like income and demographics.^{[83]} Ultimately, the authors concluded "this feelingstoactions relationship takes a generic form, is consistently replicable, and is fairly close to linear in structure. Therefore, it seems that human beings can successfully operationalize an integer scale for feelings".^{[83]}
Nonstandard spoilers
[edit]Behavioral economists have shown individual irrationality involves violations of IIA (e.g. with decoy effects),^{[84]} suggesting human behavior might cause IIA failures even if the voting method itself does not.^{[85]} However, past research on similar effects has found their effects are typically small,^{[86]} and such psychological spoiler effects can occur regardless of the electoral system in use.^{[85]} Balinski and Laraki discuss techniques of ballot design derived from psychometrics that minimize these psychological effects, such as asking voters to give each candidate a verbal grade (e.g. "bad", "neutral", "good", "excellent") and issuing instructions to voters that refer to their ballots as judgments of individual candidates.^{[63]} Similar techniques are often discussed in the context of contingent valuation.^{[72]}
Esoteric solutions
[edit]In addition to the above practical resolutions, there exist unusual (lessthanpractical) situations where Arrow's requirement of IIA can be satisfied.
Supermajority rules
[edit]Supermajority rules can avoid Arrow's theorem at the cost of being poorlydecisive (i.e. frequently failing to return a result). In this case, a threshold that requires a majority for ordering 3 outcomes, for 4, etc. does not produce voting paradoxes.^{[87]}
In spatial (ndimensional 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 wellbehaved (quasiconcave).^{[88]} These results provide some justification for the common requirement of a twothirds majority for constitutional amendments, which is sufficient to prevent cyclic preferences in most situations.^{[88]}
Uncountable voter sets
[edit]Fishburn shows all of Arrow's conditions can be satisfied for uncountably infinite sets of voters given the axiom of choice;^{[89]} however, Kirman and Sondermann demonstrated this requires disenfranchising almost all members of a society (eligible voters form a set of measure 0), leading them to refer to such societies as "invisible dictatorships".^{[90]}
Common misconceptions
[edit]Arrow's theorem is not related to strategic voting, which does not appear in his framework,^{[40]}^{[91]} though the theorem does have important implications for strategic voting (being used as a lemma to prove Gibbard's theorem^{[92]}). The Arrovian framework of social welfare assumes all voter preferences are known and the only issue is in aggregating them.^{[91]}
Monotonicity (sometimes called nonperversity) 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^{[70]}). Dropping the assumption does not allow for constructing a social welfare function that meets his other conditions.^{[93]}
Contrary to a common misconception, Arrow's theorem deals with the limited class of rankedchoice voting systems, rather than voting systems as a whole.^{[91]}^{[94]}
See also
[edit] May's theorem
 Gibbard–Satterthwaite theorem
 Gibbard's theorem
 Holmström's theorem
 Market failure
 Condorcet paradox
 Comparison of electoral systems
References
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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
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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 headtohead 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 headtohead. Thus, Gore should still have won with Nader included in the election.
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 ^ Holliday, Wesley H.; Pacuit, Eric (20230211), Stable Voting, arXiv:2108.00542. "This is a kind of stability property of Condorcet winners: you cannot dislodge a Condorcet winner A by adding a new candidate B to the election if A beats B in a headtohead 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 headtohead. Thus, Gore should still have won with Nader included in the election."
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 ^ Black, Duncan (1968). The theory of committees and elections. Cambridge, Eng.: University Press. ISBN 9780898381894.
 ^ Poundstone, William. (2013). Gaming the vote : why elections aren't fair (and what we can do about it). Farrar, Straus and Giroux. pp. 168, 197, 234. ISBN 9781429957649. OCLC 872601019.
IRV is subject to something called the "center squeeze." A popular moderate can receive relatively few firstplace votes through no fault of her own but because of vote splitting from candidates to the right and left. [...] Approval voting thus appears to solve the problem of vote splitting simply and elegantly. [...] Range voting solves the problems of spoilers and vote splitting
 ^ "Modern economic theory has insisted on the ordinal concept of utility; that is, only orderings can be observed, and therefore no measurement of utility independent of these orderings has any significance. In the field of consumer's demand theory the ordinalist position turned out to create no problems; cardinal utility had no explanatory power above and beyond ordinal. Leibniz' Principle of the identity of indiscernibles demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on p. 33 by Racnchetti, Fabio (2002), "Choice without utility? Some reflections on the loose foundations of standard consumer theory", in Bianchi, Marina (ed.), The Active Consumer: Novelty and Surprise in Consumer Choice, Routledge Frontiers of Political Economy, vol. 20, Routledge, pp. 21–45
 ^ Hamlin, Aaron (20121006). "Podcast 20121006: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 20230605. 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.
 ^ ^{a} ^{b} Harsanyi, John C. (19790901). "Bayesian decision theory, rule utilitarianism, and Arrow's impossibility theorem". Theory and Decision. 11 (3): 289–317. doi:10.1007/BF00126382. ISSN 15737187. Retrieved 20200320.
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.
 ^ Hamlin, Aaron (20121006). "Podcast 20121006: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 20230605.
Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes (in spite of what I said about manipulation) is probably the best.[...] And some of these studies have been made. In France, [Michel] Balinski has done some studies of this kind which seem to give some support to these scoring methods.
 ^ Hamlin, Aaron (20121006). "Podcast 20121006: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 20230605. 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.
 ^ Gibbard, Allan (1973). "Manipulation of Voting Schemes: A General Result". Econometrica. 41 (4): 587–601. doi:10.2307/1914083. ISSN 00129682. JSTOR 1914083.
 ^ 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.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} Arrow, Kenneth Joseph Arrow (1963). Social Choice and Individual Values (PDF). Yale University Press. ISBN 9780300013641. Archived (PDF) from the original on 20221009.
 ^ ^{a} ^{b} ^{c} Wilson, Robert (December 1972). "Social choice theory without the Pareto Principle". Journal of Economic Theory. 5 (3): 478–486. doi:10.1016/00220531(72)900518. ISSN 00220531.
 ^ ^{a} ^{b} Lagerspetz, Eerik (2016), "Arrow's Theorem", Social Choice and Democratic Values, Studies in Choice and Welfare, Cham: Springer International Publishing, pp. 171–245, doi:10.1007/9783319232614_4, ISBN 9783319232614, retrieved 20240720
 ^ Quesada, Antonio (2002). "From social choice functions to dictatorial social welfare functions". Economics Bulletin. 4 (16): 1–7.
 ^ 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 20110720.
 ^ ^{a} ^{b} 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. doi:10.2307/2110496. ISSN 00925853. JSTOR 2110496.
 ^ 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 20110720.
 ^ Quesada, Antonio (2002). "From social choice functions to dictatorial social welfare functions". Economics Bulletin. 4 (16): 1–7.
 ^ ^{a} ^{b} ^{c} Pearce, David. "Individual and social welfare: a Bayesian perspective" (PDF). Frisch Lecture Delivered to the World Congress of the Econometric Society.
 ^ ^{a} ^{b} McLean, Iain (19951001). "Independence of irrelevant alternatives before Arrow". Mathematical Social Sciences. 30 (2): 107–126. doi:10.1016/01654896(95)00784J. ISSN 01654896.
 ^ ^{a} ^{b} ^{c} ^{d} 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 20110720.
 ^ ^{a} ^{b} Geanakoplos, John (2005). "Three Brief Proofs of Arrow's Impossibility Theorem" (PDF). Economic Theory. 26 (1): 211–215. CiteSeerX 10.1.1.193.6817. doi:10.1007/s0019900405567. JSTOR 25055941. S2CID 17101545. Archived (PDF) from the original on 20221009.
 ^ ^{a} ^{b} Arrow, Kenneth Joseph Arrow (1963). Social Choice and Individual Values (PDF). Yale University Press. ISBN 9780300013641. Archived (PDF) from the original on 20221009.
 ^ Sen, Amartya (20140722). "Arrow and the Impossibility Theorem". The Arrow Impossibility Theorem. Columbia University Press. pp. 29–42. doi:10.7312/mask15328003. ISBN 9780231526869.
 ^ Rubinstein, Ariel (2012). Lecture Notes in Microeconomic Theory: The Economic Agent (2nd ed.). Princeton University Press. Problem 9.5. ISBN 9781400842469. OL 29649010M.
 ^ Barberá, Salvador (January 1980). "Pivotal voters: A new proof of arrow's theorem". Economics Letters. 6 (1): 13–16. doi:10.1016/01651765(80)900506. ISSN 01651765.
 ^ Yu, Ning Neil (2012). "A oneshot proof of Arrow's theorem". Economic Theory. 50 (2): 523–525. doi:10.1007/s0019901206933. JSTOR 41486021. S2CID 121998270.
 ^ Wilson, Robert (December 1972). "Social choice theory without the Pareto Principle". Journal of Economic Theory. 5 (3): 478–486. doi:10.1016/00220531(72)900518. ISSN 00220531.
 ^ 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.
 ^ ^{a} ^{b} McKenna, Phil (12 April 2008). "Vote of no confidence". New Scientist. 198 (2651): 30–33. doi:10.1016/S02624079(08)609148.
 ^ ^{a} ^{b} ^{c} ^{d} Pearce, David. "Individual and social welfare: a Bayesian perspective" (PDF). Frisch Lecture Delivered to the World Congress of the Econometric Society.
 ^ ^{a} ^{b} ^{c} 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". Economic Theory. 15 (3): 689–700. doi:10.1007/s001990050318. JSTOR 25055296. S2CID 122290254.
 ^ ^{a} ^{b} McLean, Iain (19951001). "Independence of irrelevant alternatives before Arrow". Mathematical Social Sciences. 30 (2): 107–126. doi:10.1016/01654896(95)00784J. ISSN 01654896.
 ^ Gehrlein, William V. (19830601). "Condorcet's paradox". Theory and Decision. 15 (2): 161–197. doi:10.1007/BF00143070. ISSN 15737187.
 ^ McGann, Anthony J.; Koetzle, William; Grofman, Bernard (2002). "How an Ideologically Concentrated Minority Can Trump a Dispersed Majority: Nonmedian Voter Results for Plurality, Runoff, and Sequential Elimination Elections". American Journal of Political Science. 46 (1): 134–147. doi:10.2307/3088418. ISSN 00925853. JSTOR 3088418.
As with simple plurality elections, it is apparent the outcome will be highly sensitive to the distribution of candidates.
 ^ Van Deemen, Adrian (20140301). "On the empirical relevance of Condorcet's paradox". Public Choice. 158 (3): 311–330. doi:10.1007/s1112701301333. ISSN 15737101.
 ^ ^{a} ^{b} Wolk, Sara; Quinn, Jameson; Ogren, Marcus (20230901). "STAR Voting, equality of voice, and voter satisfaction: considerations for voting method reform". Constitutional Political Economy. 34 (3): 310–334. doi:10.1007/s10602022093893. ISSN 15729966.
 ^ Gehrlein, William V. (20020301). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 15737187.
 ^ ^{a} ^{b} Black, Duncan (1968). The theory of committees and elections. Cambridge, Eng.: University Press. ISBN 9780898381894.
 ^ 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.
 ^ McKelvey, Richard D. (1976). "Intransitivities in multidimensional voting models and some implications for agenda control". Journal of Economic Theory. 12 (3): 472–482. doi:10.1016/00220531(76)900405.
 ^ Davis, Otto A.; DeGroot, Morris H.; Hinich, Melvin J. (1972). "Social Preference Orderings and Majority Rule". Econometrica. 40 (1): 147–157. doi:10.2307/1909727. ISSN 00129682.
 ^ Dotti, V. (20160928). Multidimensional voting models: theory and applications (Doctoral thesis). UCL (University College London).
 ^ ^{a} ^{b} Holliday, Wesley H.; Pacuit, Eric (20230901). "Stable Voting". Constitutional Political Economy. 34 (3): 421–433. doi:10.1007/s10602022093839. ISSN 15729966.
 ^ Kalai, Ehud; Muller, Eitan (1977). "Characterization of Domains Admitting Nondictatorial Social Welfare Functions and Nonmanipulable Voting Procedures" (PDF). Journal of Economic Theory. 16 (2): 457–469. doi:10.1016/00220531(77)900199.
 ^ ^{a} ^{b} ^{c} Balinski, M. L.; Laraki, Rida (2010). Majority judgment: measuring, ranking, and electing. Cambridge, Mass: MIT Press. ISBN 9780262545716.
 ^ Poundstone, William (20090217). Gaming the Vote: Why Elections Are not Fair (and What We Can Do About It). Macmillan. ISBN 9780809048922.
 ^ Cockrell, Jeff (20160308). "What economists think about voting". Capital Ideas. Chicago Booth. Archived from the original on 20160326. Retrieved 20160905.
Is there such a thing as a perfect voting system? The respondents were unanimous in their insistence that there is not.
 ^ ^{a} ^{b} Lützen, Jesper (20190201). "How mathematical impossibility changed welfare economics: A history of Arrow's impossibility theorem". Historia Mathematica. 46: 56–87. doi:10.1016/j.hm.2018.11.001. ISSN 03150860.
 ^ ^{a} ^{b} "Modern economic theory has insisted on the ordinal concept of utility; that is, only orderings can be observed, and therefore no measurement of utility independent of these orderings has any significance. In the field of consumer's demand theory the ordinalist position turned out to create no problems; cardinal utility had no explanatory power above and beyond ordinal. Leibniz' Principle of the identity of indiscernibles demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on p. 33 by Racnchetti, Fabio (2002), "Choice without utility? Some reflections on the loose foundations of standard consumer theory", in Bianchi, Marina (ed.), The Active Consumer: Novelty and Surprise in Consumer Choice, Routledge Frontiers of Political Economy, vol. 20, Routledge, pp. 21–45
 ^ ^{a} ^{b} Sen, Amartya (1999). "The Possibility of Social Choice". American Economic Review. 89 (3): 349–378. doi:10.1257/aer.89.3.349.
 ^ Arrow, Kenneth Joseph Arrow (1963). "III. The Social Welfare Function". Social Choice and Individual Values (PDF). Yale University Press. pp. 31–33. ISBN 9780300013641. Archived (PDF) from the original on 20221009.
 ^ ^{a} ^{b} 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 20110720.
 ^ Hamlin, Aaron (20121006). "Podcast 20121006: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 20230605. 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.
 ^ ^{a} ^{b} Arrow, Kenneth et al. 1993. Report of the NOAA panel on Contingent Valuation.
 ^ ^{a} ^{b} Harsanyi, John C. (1955). "Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility". Journal of Political Economy. 63 (4): 309–321. doi:10.1086/257678. JSTOR 1827128. S2CID 222434288.
 ^ Neumann, John von and Morgenstern, Oskar, Theory of Games and Economic Behavior. Princeton, NJ. Princeton University Press, 1953.
 ^ Vickrey, William (1945). "Measuring Marginal Utility by Reactions to Risk". Econometrica. 13 (4): 319–333. doi:10.2307/1906925. JSTOR 1906925.
 ^ Mongin, Philippe (October 2001). "The impartial observer theorem of social ethics". Economics & Philosophy. 17 (2): 147–179. doi:10.1017/S0266267101000219 (inactive 20240802). ISSN 14740028.
{{cite journal}}
: CS1 maint: DOI inactive as of August 2024 (link)  ^ Feiwel, George, ed. (1987). Arrow and the Foundations of the Theory of Economic Policy. Springer. p. 92. ISBN 9781349073573.
...the fictitious notion of 'original position' [was] developed by Vickery (1945), Harsanyi (1955), and Rawls (1971).
 ^ Hildreth, Clifford (1953). "Alternative Conditions for Social Orderings". Econometrica. 21 (1): 81–94. doi:10.2307/1906944. ISSN 00129682.
 ^ Procaccia, Ariel D.; Rosenschein, Jeffrey S. (2006). "The Distortion of Cardinal Preferences in Voting". Cooperative Information Agents X. Lecture Notes in Computer Science. Vol. 4149. pp. 317–331. CiteSeerX 10.1.1.113.2486. doi:10.1007/11839354_23. ISBN 9783540385691.
 ^ Moore, Michael (1 July 1975). "Rating versus ranking in the Rokeach Value Survey: An Israeli comparison". European Journal of Social Psychology. 5 (3): 405–408. doi:10.1002/ejsp.2420050313. ISSN 10990992.
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.
 ^ ^{a} ^{b} 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. doi:10.1207/s15324834basp1802_4. ISSN 01973533.
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 lowdifferentiating participants. In addition, the validity of ratings was greater than rankings overall.
 ^ Conklin, E. S.; Sutherland, J. W. (1 February 1923). "A Comparison of the Scale of Values Method with the OrderofMerit Method". Journal of Experimental Psychology. 6 (1): 44–57. doi:10.1037/h0074763. ISSN 00221015.
the scaleofvalues method can be used for approximately the same purposes as the orderofmerit method, but that the scaleofvalues method is a better means of obtaining a record of judgments
 ^ ^{a} ^{b} ^{c} 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. Bibcode:2022PNAS..11910412K. doi:10.1073/pnas.2210412119. ISSN 00278424. PMC 9586273. PMID 36191179.
 ^ Huber, Joel; Payne, John W.; Puto, Christopher (1982). "Adding Asymmetrically Dominated Alternatives: Violations of Regularity and the Similarity Hypothesis". Journal of Consumer Research. 9 (1): 90–98. doi:10.1086/208899. S2CID 120998684.
 ^ ^{a} ^{b} 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. 24 (3): 549–559. doi:10.1111/0162895X.00340. ISSN 0162895X.
 ^ Huber, Joel; Payne, John W.; Puto, Christopher P. (2014). "Let's Be Honest About the Attraction Effect". Journal of Marketing Research. 51 (4): 520–525. doi:10.1509/jmr.14.0208. ISSN 00222437. S2CID 143974563.
 ^ Moulin, Hervé (19850201). "From social welfare ordering to acyclic aggregation of preferences". Mathematical Social Sciences. 9 (1): 1–17. doi:10.1016/01654896(85)900022. ISSN 01654896.
 ^ ^{a} ^{b} Caplin, Andrew; Nalebuff, Barry (1988). "On 64%Majority Rule". Econometrica. 56 (4): 787–814. doi:10.2307/1912699. ISSN 00129682. JSTOR 1912699.
 ^ Fishburn, Peter Clingerman (1970). "Arrow's impossibility theorem: concise proof and infinite voters". Journal of Economic Theory. 2 (1): 103–106. doi:10.1016/00220531(70)900153.
 ^ See Chapter 6 of Taylor, Alan D. (2005). Social choice and the mathematics of manipulation. New York: Cambridge University Press. ISBN 9780521008839 for a concise discussion of social choice for infinite societies.
 ^ ^{a} ^{b} ^{c} "Arrow's Theorem". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 2019.
 ^ Gibbard, Allan (1973). "Manipulation of Voting Schemes: A General Result". Econometrica. 41 (4): 587–601. doi:10.2307/1914083. ISSN 00129682. JSTOR 1914083.
 ^ Arrow, Kenneth Joseph Arrow (1963). Social Choice and Individual Values (PDF). Yale University Press. ISBN 9780300013641. Archived (PDF) from the original on 20221009.
 ^ Hamlin, Aaron (March 2017). "Remembering Kenneth Arrow and His Impossibility Theorem". Center for Election Science. Retrieved 5 May 2024.
Further reading
[edit] Campbell, D. E. (2002). "Impossibility theorems in the Arrovian framework". In Arrow, Kenneth J.; Sen, Amartya K.; Suzumura, Kōtarō (eds.). Handbook of social choice and welfare. Vol. 1. Amsterdam, Netherlands: Elsevier. pp. 35–94. ISBN 9780444829146. Surveys many of approaches discussed in #Alternatives based on functions of preference profiles^{[broken anchor]}.
 Dardanoni, Valentino (2001). "A pedagogical proof of Arrow's Impossibility Theorem" (PDF). Social Choice and Welfare. 18 (1): 107–112. doi:10.1007/s003550000062. JSTOR 41106398. S2CID 7589377. preprint.
 Hansen, Paul (2002). "Another Graphical Proof of Arrow's Impossibility Theorem". The Journal of Economic Education. 33 (3): 217–235. doi:10.1080/00220480209595188. S2CID 145127710.
 Hunt, Earl (2007). The Mathematics of Behavior. Cambridge University Press. ISBN 9780521850124.. The chapter "Defining Rationality: Personal and Group Decision Making" has a detailed discussion of the Arrow Theorem, with proof.
 Lewis, Harold W. (1997). Why flip a coin? : The art and science of good decisions. John Wiley. ISBN 0471296457. Gives explicit examples of preference rankings and apparently anomalous results under different electoral system. States but does not prove Arrow's theorem.
 Sen, Amartya Kumar (1979). Collective choice and social welfare. Amsterdam: NorthHolland. ISBN 9780444851277.
 Skala, Heinz J. (2012). "What Does Arrow's Impossibility Theorem Tell Us?". In Eberlein, G.; Berghel, H. A. (eds.). Theory and Decision : Essays in Honor of Werner Leinfellner. Springer. pp. 273–286. ISBN 9789400938953.
 Tang, Pingzhong; Lin, Fangzhen (2009). "Computeraided Proofs of Arrow's and Other Impossibility Theorems". Artificial Intelligence. 173 (11): 1041–1053. doi:10.1016/j.artint.2009.02.005.
External links
[edit] "Arrow's impossibility theorem" entry in the Stanford Encyclopedia of Philosophy
 A proof by Terence Tao, assuming a much stronger version of nondictatorship
 ^ in social choice, ranked rules include Firstpreference plurality and all other rules where a candidate's score can be determined from a ranking of candidates.