# Arrow's impossibility theorem

(Redirected from Arrow's Theorem)

Arrow's impossibility theorem is a key result in social choice showing that no ranked-choice 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 ${\displaystyle A}$ and ${\displaystyle B}$ should not depend on the quality of a third, unrelated outcome ${\displaystyle C}$.[1]

The result is often cited in discussions of election science and voting theory, where ${\displaystyle C}$ 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 instant-runoff suffer spoiler effects more often than other methods.[4][5][6] Majority-choice methods uniquely minimize the effect of spoilers on election results, limiting them to rare[7][8] situations known as cyclic ties.[6]

## History

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

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 ${\displaystyle A}$ and ${\displaystyle B}$, our opinions about some irrelevant option ${\displaystyle C}$ 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]

### Non-degenerate systems

As background, it is typically assumed that any non-degenerate (i.e. actually useful) voting system non-dictatorship:

• Non-dictatorshipat 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:

• Non-negative 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, instant-runoff 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 non-negative response in Arrow's second proof, extending it to include instant-runoff.[12]
• Majority rule—if most voters prefer ${\displaystyle A}$ to ${\displaystyle B}$, then ${\displaystyle A}$ should defeat ${\displaystyle B}$. 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 majority-preferred candidate if possible.[13]

### Independence of irrelevant alternatives (IIA)

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 ${\displaystyle A}$ and ${\displaystyle B}$ should only depend on the individual preferences between ${\displaystyle A}$ and ${\displaystyle B}$; that is, it should not be able to go from ${\displaystyle A\succ B}$ to ${\displaystyle B\succ A}$ by changing preferences over some irrelevant alternative, e.g. whether ${\displaystyle A\succ C}$. This is equivalent to the above statement about independence of spoiler candidates when using the standard construction of a placement function.[14]

## Theorem

### Intuitive argument

Arrow's requirement that the social preference ${\displaystyle A\succ B}$ only depend on individual preferences ${\displaystyle A\succ _{i}B}$ 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 ranked-choice voting.

While the above argument is intuitive, it is not rigorous, and it requires additional assumptions not used by Arrow.

### Formal statement

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:

${\displaystyle F:\mathrm {L(A)} ^{N}\to \mathrm {L(A)} }$

which aggregates voters' preferences into a single preference order on A.

An N-tuple (R1, …, RN) ∈ 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 R1 , …, RN, then a is ranked strictly higher than b by F(R1, R2, …, RN). This axiom is not needed to prove the result,[11] but is used in both proofs below.
Non-dictatorship
There is no individual i whose strict preferences always prevail. That is, there is no i ∈ {1, …, N} such that for all (R1, …, RN) ∈ L(A)N and all a and b, when a is ranked strictly higher than b by Ri then a is ranked strictly higher than b by F(R1, R2, …, RN).

Then, this rule must violate independence of irrelevant alternatives:

Independence of irrelevant alternatives
For two preference profiles (R1, …, RN) and (S1, …, SN) such that for all individuals i, alternatives a and b have the same order in Ri as in Si, alternatives a and b have the same order in F(R1, …, RN) as in F(S1, …, SN).

### Formal proof

Proof by decisive coalition

Arrow's proof used the concept of decisive coalitions.[12]

Definition:

• A subset of voters is a coalition.
• A coalition is decisive over an ordered pair ${\displaystyle (x,y)}$ if, when everyone in the coalition ranks ${\displaystyle x\succ _{i}y}$, society overall will always rank ${\displaystyle x\succ y}$.
• A coalition is decisive if and only if it is decisive over all ordered pairs.

Our goal is to prove that the decisive coalition contains only one voter, who controls the outcome—in other words, a dictator.

The following proof is a simplification taken from Amartya Sen[15] and Ariel Rubinstein.[16] The simplified proof uses an additional concept:

• A coalition is weakly decisive over ${\displaystyle (x,y)}$ if and only if when every voter ${\displaystyle i}$ in the coalition ranks ${\displaystyle x\succ _{i}y}$, and every voter ${\displaystyle j}$ outside the coalition ranks ${\displaystyle y\succ _{j}x}$, then ${\displaystyle x\succ y}$.

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 ${\displaystyle G}$ is weakly decisive over ${\displaystyle (x,y)}$ for some ${\displaystyle x\neq y}$, then it is decisive.

Proof

Let ${\displaystyle z}$ be an outcome distinct from ${\displaystyle x,y}$.

Claim: ${\displaystyle G}$ is decisive over ${\displaystyle (x,z)}$.

Let everyone in ${\displaystyle G}$ vote ${\displaystyle x}$ over ${\displaystyle z}$. By IIA, changing the votes on ${\displaystyle y}$ does not matter for ${\displaystyle x,z}$. So change the votes such that ${\displaystyle x\succ _{i}y\succ _{i}z}$ in ${\displaystyle G}$ and ${\displaystyle y\succ _{i}x}$ and ${\displaystyle z}$ outside of ${\displaystyle G}$.

By Pareto, ${\displaystyle y\succ z}$. By coalition weak-decisiveness over ${\displaystyle (x,y)}$, ${\displaystyle x\succ y}$. Thus ${\displaystyle x\succ z}$. ${\displaystyle \square }$

Similarly, ${\displaystyle G}$ is decisive over ${\displaystyle (z,y)}$.

By iterating the above two claims (note that decisiveness implies weak-decisiveness), we find that ${\displaystyle G}$ is decisive over all ordered pairs in ${\displaystyle \{x,y,z\}}$. Then iterating that, we find that ${\displaystyle G}$ is decisive over all ordered pairs in ${\displaystyle X}$.

Group contraction lemma — If a coalition is decisive, and has size ${\displaystyle \geq 2}$, then it has a proper subset that is also decisive.

Proof

Let ${\displaystyle G}$ be a coalition with size ${\displaystyle \geq 2}$. Partition the coalition into nonempty subsets ${\displaystyle G_{1},G_{2}}$.

Fix distinct ${\displaystyle x,y,z}$. Design the following voting pattern (notice that it is the cyclic voting pattern which causes the Condorcet paradox):

{\displaystyle {\begin{aligned}{\text{voters in }}G_{1}&:x\succ _{i}y\succ _{i}z\\{\text{voters in }}G_{2}&:z\succ _{i}x\succ _{i}y\\{\text{voters outside }}G&:y\succ _{i}z\succ _{i}x\end{aligned}}}

(Items other than ${\displaystyle x,y,z}$ are not relevant.)

Since ${\displaystyle G}$ is decisive, we have ${\displaystyle x\succ y}$. So at least one is true: ${\displaystyle x\succ z}$ or ${\displaystyle z\succ y}$.

If ${\displaystyle x\succ z}$, then ${\displaystyle G_{1}}$ is weakly decisive over ${\displaystyle (x,z)}$. If ${\displaystyle z\succ y}$, then ${\displaystyle G_{2}}$ is weakly decisive over ${\displaystyle (z,y)}$. 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 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

Say there are three choices for society, call them A, B, and C. Suppose first that everyone prefers option B the least: everyone prefers A to B, and everyone prefers C to B. By unanimity, society must also prefer both A and C to B. Call this situation profile 0.

On the other hand, if everyone preferred B to everything else, then society would have to prefer B to everything else by unanimity. Now arrange all the voters in some arbitrary but fixed order, and for each i let profile i be the same as profile 0, but move B to the top of the ballots for voters 1 through i. So profile 1 has B at the top of the ballot for voter 1, but not for any of the others. Profile 2 has B at the top for voters 1 and 2, but no others, and so on.

Since B eventually moves to the top of the societal preference as the profile number increases, there must be some profile, number k, for which B first moves above A in the societal rank. We call the voter k whose ballot change causes this to happen the pivotal voter for B over A. Note that the pivotal voter for B over A is not, a priori, the same as the pivotal voter for A over B. In part three of the proof we will show that these do turn out to be the same.

Also note that by IIA the same argument applies if profile 0 is any profile in which A is ranked above B by every voter, and the pivotal voter for B over A will still be voter k. We will use this observation below.

### Part two: The pivotal voter for B over A is a dictator for B over C

In this part of the argument we refer to voter k, the pivotal voter for B over A, as the pivotal voter for simplicity. We will show that the pivotal voter dictates society's decision for B over C. That is, we show that no matter how the rest of society votes, if pivotal voter ranks B over C, then that is the societal outcome. Note again that the dictator for B over C is not a priori the same as that for C over B. In part three of the proof we will see that these turn out to be the same too.

In the following, we call voters 1 through k − 1, segment one, and voters k + 1 through N, segment two. To begin, suppose that the ballots are as follows:

• Every voter in segment one ranks B above C and C above A.
• Pivotal voter ranks A above B and B above C.
• Every voter in segment two ranks A above B and B above C.

Then by the argument in part one (and the last observation in that part), the societal outcome must rank A above B. This is because, except for a repositioning of C, this profile is the same as profile k − 1 from part one. Furthermore, by unanimity the societal outcome must rank B above C. Therefore, we know the outcome in this case completely.

Now suppose that pivotal voter moves B above A, but keeps C in the same position and imagine that any number (even all!) of the other voters change their ballots to move B below C, without changing the position of A. Then aside from a repositioning of C this is the same as profile k from part one and hence the societal outcome ranks B above A. Furthermore, by IIA the societal outcome must rank A above C, as in the previous case. In particular, the societal outcome ranks B above C, even though Pivotal Voter may have been the only voter to rank B above C. By IIA, this conclusion holds independently of how A is positioned on the ballots, so pivotal voter is a dictator for B over C.

### Part three: There exists a dictator

In this part of the argument we refer back to the original ordering of voters, and compare the positions of the different pivotal voters (identified by applying parts one and two to the other pairs of candidates). First, the pivotal voter for B over C must appear earlier (or at the same position) in the line than the dictator for B over C: As we consider the argument of part one applied to B and C, successively moving B to the top of voters' ballots, the pivot point where society ranks B above C must come at or before we reach the dictator for B over C. Likewise, reversing the roles of B and C, the pivotal voter for C over B must be at or later in line than the dictator for B over C. In short, if kX/Y denotes the position of the pivotal voter for X over Y (for any two candidates X and Y), then we have shown

kB/C ≤ kB/AkC/B.

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

kC/BkB/C.

Therefore, we have

kB/C = kB/A = kC/B

and the same argument for other pairs shows that all the pivotal voters (and hence all the dictators) occur at the same position in the list of voters. This voter is the dictator for the whole election.

## Interpretation and practical solutions

Arrow's theorem establishes that no ranked voting rule can always satisfy independence of irrelevant alternatives, but it says nothing about the frequency of spoilers. This led Arrow to remark that "Most systems are not going to work badly all of the time. All I proved is that all can work badly at times."[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 spoiler-prone methods, or dropping his assumption of ranked voting to focus on studying rated voting rules.

### Minimizing IIA failures: Majority-rule methods

The first set of methods economists have studied are the majority-rule methods, which limit spoilers to rare situations where majority rule is self-contradictory, 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 self-contradictory (intransitive), because there can be a rock-paper-scissors 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 two-candidate case will agree with a simple majority vote.

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 empirically rare, suggesting they are of limited practical concern. Spatial voting models also suggest such paradoxes are likely to be infrequent[20] or even non-existent.[21]

#### Left-right spectrum

Duncan Black showed his own remarkable result, the median voter theorem. The theorem proves that if voters and candidates are arranged on a left-right spectrum, Arrow's conditions are compatible, and all of them will be met by any rule satisfying Condorcet's principle.[21]

More formally, Black's theorem assumes preferences are single-peaked: a voter's happiness with a candidate goes up and then down as the candidate moves along some spectrum. For example, in a group of friends choosing a volume setting for music, each friend would likely have their own ideal volume; as 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 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.[21]

Unfortunately, the rule does not generalize from the political spectrum to the political compass, a result called the McKelvey-Schofield Chaos Theorem.[22] However, a well-defined 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 roughly-symmetric 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

Campbell and Kelly (2000) showed 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.[13] In other words, replacing a ranked-voting 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 instant-runoff voting).[24]

### Eliminating IIA failures: Rated voting

As shown above, the proof of Arrow's theorem relies crucially on the assumption of ranked voting, and is not applicable to rated voting systems. As a result, systems like score voting and graduated majority judgment pass independence of irrelevant alternatives.[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 strategy-free,[26] so the informal dictum that "no voting system is perfect" still has some mathematical basis.[27]

#### Meaningfulness of cardinal information

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 well-being.[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 ranked-choice procedures as approximations of the utilitarian rule (i.e. score voting).[37]

#### Nonstandard spoilers

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 pseudo-spoiler 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 strategy-proof.[citation needed]

### Esoteric solutions

In addition to the above practical resolutions, there exist unusual (less-than-practical) situations where Arrow's conditions can be satisfied.

#### Non-neutral voting rules

When equal treatment of candidates is not a necessity, Condorcet's majority-rule 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 ${\displaystyle 2/3}$ majority to select between 3 outcomes, ${\displaystyle 3/4}$ for 4, etc. does not cause paradoxes; this result is related to the Nakamura number of voting mechanisms.[40]

In spatial (n-dimensional ideology) models of voting, this can be relaxed to require only ${\displaystyle 1-e^{-1}}$ (roughly 64%) of the vote to prevent cycles, so long as the distribution of voters is well-behaved (quasiconcave).[41]

#### Uncountable voter sets

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

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 power-sharing agreements (e.g. where each holds office for a specified period of time).[44]

## Common misconceptions

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 ranked-choice voting systems, rather than voting systems as a whole.[46][47]

## References

1. Arrow, Kenneth J. (1950). "A Difficulty in the Concept of Social Welfare" (PDF). Journal of Political Economy. 58 (4): 328–346. doi:10.1086/256963. JSTOR 1828886. S2CID 13923619. Archived from the original (PDF) on 2011-07-20.
2. ^ a b c McKenna, Phil (12 April 2008). "Vote of no confidence". New Scientist. 198 (2651): 30–33. doi:10.1016/S0262-4079(08)60914-8.
3. ^ a b c Aaron, Hamlin (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. 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.
4. ^ "The Spoiler Effect". The Center for Election Science. Retrieved 2024-03-03.
5. ^ Borgers, Christoph (2010-01-01). Mathematics of Social Choice: Voting, Compensation, and Division. SIAM. ISBN 9780898716955. Candidates C and D spoiled the election for B ... With them in the running, A won, whereas without them in the running, B would have won. ... Instant runoff voting ... does not do away with the spoiler problem entirely, although it ... makes it less likely
6. ^ a b Holliday, Wesley H.; Pacuit, Eric (2023-02-11), Stable Voting, arXiv:2108.00542, retrieved 2024-03-11. "This is a kind of stability property of Condorcet winners: you cannot dislodge a Condorcet winner A by adding a new candidate B to the election if A beats B in a head-to-head majority vote. For example, although the 2000 U.S. Presidential Election in Florida did not use ranked ballots, it is plausible (see Magee 2003) that Al Gore (A) would have won without Ralph Nader (B) in the election, and Gore would have beaten Nader head-to-head. Thus, Gore should still have won with Nader included in the election."
7. ^ Gehrlein, William V. (2002-03-01). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 1573-7187.
8. ^ Van Deemen, Adrian (2014-03-01). "On the empirical relevance of Condorcet's paradox". Public Choice. 158 (3): 311–330. doi:10.1007/s11127-013-0133-3. ISSN 1573-7101.
9. ^ Suzumura, Kōtarō (2002). "Introduction". In Arrow, Kenneth J.; Sen, Amartya K.; Suzumura, Kōtarō (eds.). Handbook of social choice and welfare. Vol. 1. Amsterdam, Netherlands: Elsevier. p. 10. ISBN 978-0-444-82914-6.
10. ^ Pearce, David. "Individual and social welfare: a Bayesian perspective" (PDF). Frisch Lecture delivered to the World Congress of the Econometric Society.
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12. ^ a b c d Arrow, Kenneth Joseph Arrow (1963). Social Choice and Individual Values (PDF). Yale University Press. ISBN 978-0300013641. Archived (PDF) from the original on 2022-10-09.
13. ^ a b c d 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.
14. ^ Quesada, Antonio (2002). "From social choice functions to dictatorial social welfare functions". Economics Bulletin. 4 (16): 1–7.
15. ^ Sen, Amartya (2014-07-22). "Arrow and the Impossibility Theorem". The Arrow Impossibility Theorem. Columbia University Press. pp. 29–42. doi:10.7312/mask15328-003. ISBN 978-0-231-52686-9.
16. ^ Rubinstein, Ariel (2012). Lecture Notes in Microeconomic Theory: The Economic Agent (2nd ed.). Princeton University Press. Problem 9.5. ISBN 978-1-4008-4246-9. OL 29649010M.
17. ^ Barberá, Salvador (January 1980). "Pivotal voters: A new proof of arrow's theorem". Economics Letters. 6 (1): 13–16. doi:10.1016/0165-1765(80)90050-6. ISSN 0165-1765.
18. ^ 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/s00199-004-0556-7. JSTOR 25055941. S2CID 17101545. Archived (PDF) from the original on 2022-10-09.
19. ^ Yu, Ning Neil (2012). "A one-shot proof of Arrow's theorem". Economic Theory. 50 (2): 523–525. doi:10.1007/s00199-012-0693-3. JSTOR 41486021. S2CID 121998270.
20. ^ a b Wolk, Sara; Quinn, Jameson; Ogren, Marcus (2023-09-01). "STAR Voting, equality of voice, and voter satisfaction: considerations for voting method reform". Constitutional Political Economy. 34 (3): 310–334. doi:10.1007/s10602-022-09389-3. ISSN 1572-9966.
21. Black, Duncan (1968). The theory of committees and elections. Cambridge, Eng.: University Press. ISBN 978-0-89838-189-4.
22. ^ McKelvey, Richard D. (1976). "Intransitivities in multidimensional voting models and some implications for agenda control". Journal of Economic Theory. 12 (3): 472–482. doi:10.1016/0022-0531(76)90040-5.
23. ^ See Valerio Dotti's thesis "Multidimensional Voting Models" (2016).
24. ^ a b Holliday, Wesley H.; Pacuit, Eric (2023-09-01). "Stable Voting". Constitutional Political Economy. 34 (3): 421–433. doi:10.1007/s10602-022-09383-9. ISSN 1572-9966.
25. ^ 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:10.1016/0022-0531(77)90019-9.
26. ^ Poundstone, William (2009-02-17). Gaming the Vote: Why Elections Are not Fair (and What We Can Do About It). Macmillan. ISBN 9780809048922.
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28. ^ 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
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