Arrow's impossibility theorem
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Arrow's impossibility theorem is a key result in social choice showing that no rankedchoice voting rule^{[note 1]} can produce logically coherent results with more than two candidates. Specifically, any such rule violates independence of irrelevant alternatives: the principle that a choice between and should not depend on the quality of a third, unrelated outcome .^{[1]}
The result is often cited in discussions of election science and voting theory, where is called a spoiler candidate. As a result, Arrow's theorem implies that a ranked voting system can never be completely independent of spoilers.^{[1]}
The practical consequences of the theorem are debatable, with Arrow himself noting "Most [ranked] systems are not going to work badly all of the time. All I proved is that all can work badly at times."^{[2]}^{[3]} However, the susceptibility of different systems varies greatly. Plurality and instantrunoff suffer spoiler effects more often than other methods.^{[4]}^{[5]}^{[6]} Majoritychoice methods uniquely minimize the effect of spoilers on election results, limiting them to rare^{[7]}^{[8]} situations known as cyclic ties.^{[6]}
History[edit]
Arrow's theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated it in his doctoral thesis and popularized it in his 1951 book.^{[1]}
Arrow's work is remembered as much for its pioneering methodology as its direct implications. Arrow's axiomatic approach provided a framework for proving facts about all conceivable voting mechanisms at once, contrasting with the earlier approach of investigating such rules one by one.^{[9]}
Background[edit]
Arrow's theorem falls under the branch of welfare economics known as social choice theory, which deals with aggregating preferences and information to make fair and accurate decisions for society. The goal is to create a social ordering function—a procedure for determining which outcomes are better, according to society as a whole—that satisfies the properties of rational behavior.^{[1]}
Among the most important is independence of irrelevant alternatives, which says that when deciding between and , our opinions about some irrelevant option should not affect our decision.^{[1]} Arrow's theorem shows this is not possible without relying on further information, such as rated ballots (rejected by strict behaviorists).^{[10]}
Nondegenerate systems[edit]
As background, it is typically assumed that any nondegenerate (i.e. actually useful) voting system nondictatorship:
 Nondictatorship—at least two voters can affect the outcome of the election. (The system does not just ignore every vote except one, or even ignore all of the votes and always elect the same candidate.)
Most proofs use additional assumptions to simplify deriving the result, though Robert Wilson proved these to be unnecessary.^{[11]} Older proofs have taken as axioms:
 Nonnegative response—increasing the rank of an outcome should not make them lose. In other words, a candidate should never lose as a result of winning "too many votes".^{[1]} While originally considered "obvious" for any practical system, instantrunoff fails this criterion. Arrow later gave another proof applying to systems with negative response.^{[12]}
 Pareto efficiency—if every voter agrees one candidate is better than another, the system will agree as well. (A candidate with unanimous support should win.) This assumption replaces nonnegative response in Arrow's second proof, extending it to include instantrunoff.^{[12]}
 Majority rule—if most voters prefer to , then should defeat . This form was proven by the Marquis de Condorcet with his discovery of the voting paradox.
 Universal domain—Some authors are explicit about the assumption that the social welfare function is a function over the domain of all possible preferences (not just a partial function).
 In other words, the system cannot simply "give up" and refuse to elect a candidate in some elections.
 Without this assumption, majority rule is the only system that satisfies Arrow's criteria, by May's theorem. This result is often cited as justification for Condorcet methods, which always elect a majoritypreferred candidate if possible.^{[13]}
Independence of irrelevant alternatives (IIA)[edit]
The IIA condition is an important assumption governing rational choice. The axiom says that adding irrelevant—i.e. rejected—options should not affect the outcome of a decision. From a practical point of view, the assumption prevents electoral outcomes from behaving erratically in response to the arrival and departure of candidates.^{[12]}
Arrow defines IIA slightly differently, by stating that the social preference between alternatives and should only depend on the individual preferences between and ; that is, it should not be able to go from to by changing preferences over some irrelevant alternative, e.g. whether . This is equivalent to the above statement about independence of spoiler candidates when using the standard construction of a placement function.^{[14]}
Theorem[edit]
Intuitive argument[edit]
Arrow's requirement that the social preference only depend on individual preferences is extremely restrictive. May's theorem shows that the only "fair" way to aggregate ordinal preferences for pairs of preferences is simple majority; thus, assumptions only slightly stronger than Arrow's are already enough to lock us into the class of Condorcet methods. At this point, the existence of the voting paradox is enough to show the impossibility of rational behavior for rankedchoice voting.
While the above argument is intuitive, it is not rigorous, and it requires additional assumptions not used by Arrow.
Formal statement[edit]
Let A be a set of outcomes, N a number of voters or decision criteria. We denote the set of all total orderings of A by L(A).
An ordinal (ranked) social welfare function is a function:
which aggregates voters' preferences into a single preference order on A.
An Ntuple (R_{1}, …, R_{N}) ∈ L(A)^{N} of voters' preferences is called a preference profile. We assume two conditions:
 Pareto efficiency
 If alternative a is ranked strictly higher than b for all orderings R_{1} , …, R_{N}, then a is ranked strictly higher than b by F(R_{1}, R_{2}, …, R_{N}). This axiom is not needed to prove the result,^{[11]} but is used in both proofs below.
 Nondictatorship
 There is no individual i whose strict preferences always prevail. That is, there is no i ∈ {1, …, N} such that for all (R_{1}, …, R_{N}) ∈ L(A)^{N} and all a and b, when a is ranked strictly higher than b by R_{i} then a is ranked strictly higher than b by F(R_{1}, R_{2}, …, R_{N}).
Then, this rule must violate independence of irrelevant alternatives:
 Independence of irrelevant alternatives
 For two preference profiles (R_{1}, …, R_{N}) and (S_{1}, …, S_{N}) such that for all individuals i, alternatives a and b have the same order in R_{i} as in S_{i}, alternatives a and b have the same order in F(R_{1}, …, R_{N}) as in F(S_{1}, …, S_{N}).
Formal proof[edit]
Proof by decisive coalition


Arrow's proof used the concept of decisive coalitions.^{[12]} 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^{[15]} and Ariel Rubinstein.^{[16]} 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.^{[17]} The proof given here is a simplified version based on two proofs published in Economic Theory.^{[18]}^{[19]} 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. 
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."^{[2]}^{[3]}
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.
Minimizing IIA failures: Majorityrule methods[edit]
The first set of methods economists have studied are the majorityrule methods, which limit spoilers to rare situations where majority rule is selfcontradictory, and uniquely minimize the possibility of a spoiler effect among rated methods.^{[13]} Arrow's theorem was preceded by the Marquis de Condorcet's discovery of cyclic social preferences, cases where majority votes are logically inconsistent. Condorcet believed voting rules should satisfy his majority rule principle, i.e. if most voters rank Alice ahead of Bob, Alice should defeat Bob in the election.
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. Thus Condorcet proved a weaker form of Arrow's impossibility theorem long before Arrow himself, under the stronger assumption that a voting system in the twocandidate case will agree with a simple majority vote.
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 empirically rare, suggesting they are of limited practical concern. Spatial voting models also suggest such paradoxes are likely to be infrequent^{[20]} or even nonexistent.^{[21]}
Leftright spectrum[edit]
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 compatible, and all of them will be met by any rule satisfying Condorcet's principle.^{[21]}
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 volume gets progressively too loud or too quiet, they would be increasingly dissatisfied.^{[21]}
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.^{[21]}
Unfortunately, the rule does not generalize from the political spectrum to the political compass, a result called the McKelveySchofield Chaos Theorem.^{[22]} However, a welldefined median and Condorcet winner do exist if the distribution of voters on the ideological spectrum is rotationally symmetric.^{[23]} In realistic cases, when voters' opinions follow a roughlysymmetric distribution such as a normal distribution or can be accurately summarized in one or two dimensions, Condorcet cycles tend to be rare.^{[20]}^{[24]}
Generalized stability theorems[edit]
Campbell and Kelly (2000) showed 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.^{[13]} In other words, replacing a rankedvoting 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.^{[13]}
In 1977, Ehud Kalai and Eitan Muller gave a full characterization of domain restrictions admitting a nondictatorial and rational social welfare function. These correspond to preferences for which there is a Condorcet winner.^{[25]}
Holliday and Pacuit devise a voting system that provably minimizes the potential for spoiler effects, albeit at the cost of other criteria, and find that it is a Condorcet method, albeit at the cost of occasional monotonicity failures (at a much lower rate than seen in instantrunoff voting).^{[24]}
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.^{[2]} 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).^{[21]}
While Arrow's theorem does not apply to graded systems, Gibbard's theorem still does: no electoral system can be strategyfree,^{[26]} so the informal dictum that "no voting system is perfect" still has some mathematical basis.^{[27]}
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. Taking inspiration from the strict behaviorism popular in psychology, some philosophers and economists rejected the idea of comparing internal human experiences of wellbeing.^{[28]} Such philosophers claimed it was impossible to compare preferences of different people in situations where they disagree; Sen gives as an example that it is impossible to know whether the Great Fire of Rome was good or bad, because it allowed Nero to build himself a new palace.^{[29]}
Arrow himself originally agreed with these positions and rejected cardinal utility, leading him to focus his theorem on preference rankings.^{[28]}^{[30]} However, he later reversed this opinion, admitting scoring methods can provide useful information that allows them to evade his theorem.^{[3]} Similarly, Amartya Sen first claimed interpersonal comparability is necessary for IIA, but later argued it would only require "rather limited levels of partial comparability" to hold in practice.^{[31]}
Balinski and Laraki dispute the necessity of any genuinely cardinal information for rated voting methods to pass IIA. They argue the availability of a "common language" and the use of verbal grades are sufficient to ensure IIA, because they allow voters to give consistent responses to questions about candidate quality regardless of which candidates run.^{[32]}
John Harsanyi argued his theorem^{[33]} and other utility representation theorems like the VNM theorem, which show that rational behavior implies consistent cardinal utilities. Harsanyi^{[33]} and Vickrey^{[34]} independently derived results showing such preferences could be rigorously defined using individual preferences over the lottery of birth.^{[35]}^{[36]}
These results have led to the rise of implicit utilitarian voting approaches, which model rankedchoice procedures as approximations of the utilitarian rule (i.e. score voting).^{[37]}
Nonstandard spoilers[edit]
Behavioral economists have shown human behavior can violate IIA (e.g. with decoy effects),^{[38]} suggesting human behavior might cause IIA failures even if the voting method itself does not. However, such effects tend to be small.^{[39]}
Strategic voting can also create pseudospoiler situations: if the winner with honest voting is A, but with strategy is B, then eliminating every candidate but A and B would change the strategic outcome from B to A since majority rule is strategyproof.^{[citation needed]}
Esoteric solutions[edit]
In addition to the above practical resolutions, there exist unusual (lessthanpractical) situations where Arrow's conditions can be satisfied.
Nonneutral voting rules[edit]
When equal treatment of candidates is not a necessity, Condorcet's majorityrule criterion can be modified to require a supermajority. Such situations become more practical if there is a clear default (such as doing nothing, or allowing an incumbent to complete their term in a recall election). In this situation, setting a threshold that requires a majority to select between 3 outcomes, for 4, etc. does not cause paradoxes; this result is related to the Nakamura number of voting mechanisms.^{[40]}
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).^{[41]}
Uncountable voter sets[edit]
Fishburn shows all of Arrow's conditions can be satisfied for uncountable sets of voters given the axiom of choice;^{[42]} however, Kirman and Sondermann showed this requires disenfranchising almost all members of a society (eligible voters form a set of measure 0), leading them to refer to such societies as "invisible dictatorships".^{[43]}
Fractional social choice[edit]
Maximal lotteries satisfy a probabilistic version of Arrow's criteria in fractional social choice models, where candidates can be elected by lottery or engage in powersharing agreements (e.g. where each holds office for a specified period of time).^{[44]}
Common misconceptions[edit]
Arrow's theorem does not deal with strategic voting, which does not appear in his framework. The Arrovian framework of social welfare assumes all voter preferences are known and the only issue is in aggregating them.^{[45]}
Contrary to a common misconception, Arrow's theorem deals with the limited class of rankedchoice voting systems, rather than voting systems as a whole.^{[46]}^{[47]}
See also[edit]
 Condorcet paradox
 Gibbard–Satterthwaite theorem
 Gibbard's theorem
 Holmström's theorem
 Market failure
 Voting paradox
 Comparison of electoral systems
References[edit]
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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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 ^ 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.
 ^ 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.
 ^ F. Brandl and F. Brandt. Arrovian Aggregation of Convex Preferences. Econometrica. 88(2), pages 799844, 2020.
 ^ "Arrow's Theorem". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 2019.
 ^ "Arrow's Theorem". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 2019.
 ^ 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.
 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.