Arrow's impossibility theorem

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Arrow's impossibility theorem is a celebrated result and key impossibility theorem in social choice theory. It shows that no ranked-choice voting rule[note 1] can produce a logically coherent ranking of candidates. Specifically, no such procedure can satisfy a key criterion of decision theory, called independence of irrelevant alternatives or independence of spoilers: that the choice between and should not depend on the quality of a third, unrelated outcome .

The theorem is often cited in discussions of election science and voting theory. In an electoral context, is called a spoiler candidate; thus, Arrow's theorem shows that ranked voting systems can never be completely independent of spoilers.

The practical consequences of the theorem are debatable. Arrow noted that "Most systems are not going to work badly all of the time. All I proved is that all can work badly at times."[1][2] Spoiler effects are common in some ranked systems (like instant-runoff and plurality), but rare in Condorcet methods (see below). A large class of systems using rated voting (such as score voting or expanding approvals) are not affected by Arrow's theorem or spoilers at all.[3]

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.[4]

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.[5]

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.

Among the most important of these is independence of irrelevant alternatives, which says that when deciding between A and B, our opinions about C should not affect our decision. Arrow's theorem shows this is not possible without relying on further information, such as rated ballots (which are rejected by strict behaviorists and many philosophers).

Non-degenerate systems[edit]

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

Non-dictatorship—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.[6]

  • Arrow's first proof of the theorem assumed monotonicity (increasing the support for a candidate should not cause them to lose) and non-imposition (every candidate could be elected, in theory, under some combination of ballots; i.e. the social choice function is surjective).
  • Arrow's second proof assumed the weaker Pareto efficiency concept: if every voter agrees one candidate is better than another, the system will agree as well. (A candidate with unanimous support should win.)
  • Many proofs assume anonymity (voter equality), i.e. that every voter has an equal say in the election.
  • The Marquis de Condorcet gave an earlier proof for the case of his majority-rule principle: if most voters prefer to , then should defeat (unless this causes a voting paradox).
  • Some include "universal domain"—i.e. that the social welfare function is a total function over preferences.
    • In other words, the system cannot simply "give up" and refuse to elect a candidate.

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.

In practical situations, the assumption prevents electoral outcomes from behaving erratically in response to the arrival and departure of new candidates.[7]

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.

Theorem[edit]

Intuitive argument[edit]

Arrow's requirement that the social preference only depend on individual preferences is extremely restrictive—it effectively locks us into the class of methods based on weighted paired comparisons. (Already it shows us that the only methods with any hope of behaving coherently are the class of tournament solutions, e.g. ranked pairs, and rules out most systems like first-preference votes or instant-runoff voting.)

At this point, the voting paradox is enough to show the impossibility of rational behavior for a group where multiple voters have non-zero weight. In such cases we can simply create a new paradox by modifying the numbers given in Condorcet's example.

While the above argument conveys the intuition behind Arrow's theorem, it is not a rigorous proof.

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 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,[6] 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[edit]

Proof by decisive coalition

Arrow's proof used the concept of decisive coalitions.[7]

Definition:

  • A subset of voters is a coalition.
  • A coalition is decisive over an ordered pair if, when everyone in the coalition ranks , society overall will always rank .
  • A coalition is decisive if and only if it is decisive over all ordered pairs.

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

The following proof is a simplification taken from Amartya Sen[8] and Ariel Rubinstein.[9] The simplified proof uses an addition concept:

  • A coalition is weakly decisive over if and only if when every voter in the coalition ranks , and every voter outside the coalition ranks , then .

Thenceforth assume that the social choice system satisfies unrestricted domain, Pareto efficiency, and IIA. Also assume that there are at least 3 distinct outcomes.

Field expansion lemma — if a coalition is weakly decisive over for some , then it is decisive.

Proof

Let be an outcome distinct from .

Claim: is decisive over .

Let everyone in vote over . By IIA, changing the votes on does not matter for . So change the votes such that in and and outside of .

By Pareto, . By coalition weak-decisiveness over , . Thus .

Similarly, is decisive over .

By iterating the above two claims (note that decisiveness implies weak-decisiveness), we find that is decisive over all ordered pairs in . Then iterating that, we find that is decisive over all ordered pairs in .

Group contraction lemma — If a coalition is decisive, and has size , then it has a proper subset that is also decisive.

Proof

Let be a coalition with size . Partition the coalition into nonempty subsets .

Fix distinct . Design the following voting pattern (notice that it is the cyclic voting pattern which causes the Condorcet paradox):

(Items other than are not relevant.)

Since is decisive, we have . So at least one is true: or .

If , then is weakly decisive over . If , then is weakly decisive over . Now apply the field expansion lemma.

By Pareto, the entire set of voters is decisive. Thus by the group contraction lemma, there is a size-one decisive coalition—a dictator.

Proof by pivotal voter

Proofs using the concept of the pivotal voter originated from Salvador Barberá in 1980.[10] The proof given here is a simplified version based on two proofs published in Economic Theory.[11][12]

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]

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

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.

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

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

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

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

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

Part three: There exists a dictator[edit]

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

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

kB/C ≤ kB/AkC/B.

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

kC/BkB/C.

Therefore, we have

kB/C = kB/A = kC/B

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

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."[1][2]

Attempts at dealing with the effects of Arrow's theorem generally 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 the study of rated voting rules.

Minimizing spoilers: Condorcet methods[edit]

An example of a Condorcet cycle, where some candidate must be a spoiler.

The first set of methods economists have studied are the Condorcet 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 should always be consistent with majority rule.

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 the paradox is infrequent[14] or even non-existent.[15]

Left-right 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 left-right spectrum, Arrow's conditions are compatible, and all of them will be met by any rule satisfying Condorcet's principle.

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.

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.[15]

Unfortunately, the rule does not generalize from the political spectrum to the political compass, a result called the McKelvey-Schofield Chaos Theorem.[16] However, a well-defined median and Condorcet winner do exist if the distribution of voters on the ideological spectrum is rotationally symmetric.[17] In realistic cases, when voters' opinions are roughly bell-shaped distribution or can be accurately summarized by one or two dimensions, Condorcet cycles are rare.[14][18]

Generalized stability theorems[edit]

Campbell and Kelly (2000) proved 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. In other words, replacing a ranked-voting method with its Condorcet variant (i.e. eliminate all candidates outside the Smith set, then run the method) will sometimes eliminate spoiler effects, but will never cause a new spoiler effect.[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.[19]

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 occassional monotonicity failures (at a much lower rate than seen in instant-runoff voting).[18]

Eliminating spoilers: 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 do not have a spoiler effect.[1] 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).

Interpersonal comparisons of utility[edit]

Arrow himself originally rejected cardinal utility as a meaningful tool for expressing social welfare, leading him focus his theorem on preference rankings.[20] However, Arrow later reversed his opinion on the issue, coming to believe that a "score system[...] is probably the best".[2] Further support for this position comes from results like Harsanyi's utilitarian theorem[21] and other utility representation theorems like the VNM theorem, which typically establish that consistent cardinal utilities are a requirement for coherent behavior. This has led to the rise of implicit utilitarian voting approaches, which model ordinal decision procedures as approximations of cardinal utility.

Arrow's framework assumed individual and social preferences are "orderings" (i.e. satisfy completeness and transitivity) on the set of alternatives. Much like the strict behaviorists of the time, some philosophers and economists rejected internal cardinal utilities as unfalsifiable and therefore unscientific. This assumption of ordinal preferences, which precludes interpersonal comparisons of utility, is the key assumption of Arrow's theorem. Critics of rated voting methods claim it is similarly impossible to aggregate self-reported measures of utility, as one person's 8/10 rating may be another person's 3/10.

Note that while Arrow's theorem does not apply to rated voting systems, Gibbard's theorem still does: no electoral system is fully strategy-free, so the informal dictum that "no voting system is perfect" still has some mathematical basis.[22][23]

Non-Arrovian spoilers[edit]

Behavioral economists and cognitive scientists have shown individual decisions sometimes violate independence of irrelevant alternatives. A well-known example of this is the decoy effect, where including a useless option can slightly increase ratings of another product.[24] As a result, human psychology can create spoiler effects in graded voting systems, even when the voting system itself does not cause a spoiler effect.

Strategic voting can also create pseudo-spoiler situations, as seen in the Burr dilemma. In the Myerson-Weber model of strategic score (or approval) voting, a new candidate entering the race leads voters to reassess which candidates they should support and change their ballots, which can affect the result.[25] It is worth noting that under this model, voters will select a Condorcet winner, suggesting the incidence of strategic spoilers would be no greater than the incidence of honest spoilers under Condorcet's method.[25]

Esoteric solutions[edit]

Supermajorities[edit]

Replacing Condorcet's majority-rule criterion with a supermajority-rule can prevent cycles when choosing between alternatives if the threshold for action is set at:[26]

(e.g. 23 for 3 outcomes, 34 for 4, etc.); this result is related to the Nakamura number of a game. However, such rules violate the universal domain principle (some candidates may not have enough pairwise-defeats against another).

Infinitely-many voters[edit]

If we take the axiom of choice, Fishburn shows all of Arrow's conditions can be satisfied for uncountable sets of voters;[27] however, Kirman and Sondermann show this requires disenfranchising almost all members of a society (eligible voters form a set of measure 0).[28]

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 can engage in power-sharing agreements (e.g. where each holds office for a specified period of time).[29]

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 (and most of the field of mathematical voting theory) assumes all voter preferences are known and the only issue is in aggregating them. The study of strategic voting generally falls under game theory, which has produced results like Gibbard's theorem and the semi-honesty of cardinal votes in Poisson games.

Contrary to common misconception, Arrow's theorem deals with the limited class of ranked-choice voting systems, rather than voting systems as a whole. Arrow's theorem is also not limited to methods of paired comparison; as noted above, every rule not based on paired comparisons trivially fails IIA, as any such rule directly relies on preferences other than those involving both candidates.

See also[edit]

References[edit]

  1. ^ 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.
  2. ^ a b c Aaron, Hamlin (25 May 2015). "CES Podcast with Dr Arrow". Center for Election Science. CES. Retrieved 9 March 2023.
  3. ^ However, a modified version of Arrow's theorem may still apply to such methods (e.g., Brams; Fishburn (2002). "Chapter 4". In Arrow, Kenneth J.; Sen, Amartya K.; Suzumura, Kōtarō (eds.). Handbook of social choice and welfare. Vol. 1. Amsterdam, Netherlands: Elsevier. Theorem 4.2 framework. ISBN 978-0-444-82914-6.
  4. ^ 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.
  5. ^ 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.
  6. ^ a b Wilson, Robert (December 1972). "Social choice theory without the Pareto Principle". Journal of Economic Theory. 5 (3): 478–486. doi:10.1016/0022-0531(72)90051-8. ISSN 0022-0531.
  7. ^ a b 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.
  8. ^ 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.
  9. ^ 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.
  10. ^ 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.
  11. ^ 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.
  12. ^ 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.
  13. ^ a b Indeed, many different social welfare functions can meet Arrow's conditions under such restrictions of the domain. It has been proved, however, that under any such restriction, if there exists any social welfare function that adheres to Arrow's criteria, then the majority rule 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. ^ a b "Voter Satisfaction Efficiency (VSE) FAQ". Voter Satisfaction Efficiency Simulator. Retrieved 2024-03-24.
  15. ^ a b Black, Duncan (1968). The theory of committees and elections. Cambridge, Eng.: University Press. ISBN 978-0-89838-189-4.
  16. ^ 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.
  17. ^ See Valerio Dotti's thesis "Multidimensional Voting Models" (2016).
  18. ^ 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.
  19. ^ Kalai, Ehud; Muller, Eitan (1977). "Characterization of Domains Admitting Nondictatorial Social Welfare Functions and Nonmanipulable Voting Procedures". Journal of Economic Theory. 16 (2): 457–469.
  20. ^ "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
  21. ^ 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.
  22. ^ Poundstone, William (2009-02-17). Gaming the Vote: Why Elections Are not Fair (and What We Can Do About It). Macmillan. ISBN 9780809048922.
  23. ^ Cockrell, Jeff (2016-03-08). "What economists think about voting". Capital Ideas. Chicago Booth. Archived from the original on 2016-03-26. Retrieved 2016-09-05. Is there such a thing as a perfect voting system? The respondents were unanimous in their insistence that there is not.
  24. ^ 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 0022-2437. S2CID 143974563.
  25. ^ a b Myerson, Roger B.; Weber, Robert J. (March 1993). "A Theory of Voting Equilibria". American Political Science Review. 87 (1): 102–114. doi:10.2307/2938959. ISSN 0003-0554.
  26. ^ Moulin, Hervé (1985-02-01). "From social welfare ordering to acyclic aggregation of preferences". Mathematical Social Sciences. 9 (1): 1–17. doi:10.1016/0165-4896(85)90002-2. ISSN 0165-4896.
  27. ^ Fishburn, Peter Clingerman (1970). "Arrow's impossibility theorem: concise proof and infinite voters". Journal of Economic Theory. 2 (1): 103–106. doi:10.1016/0022-0531(70)90015-3.
  28. ^ See Chapter 6 of Taylor, Alan D. (2005). Social choice and the mathematics of manipulation. New York: Cambridge University Press. ISBN 978-0-521-00883-9 for a concise discussion of social choice for infinite societies.
  29. ^ F. Brandl and F. Brandt. Arrovian Aggregation of Convex Preferences. Econometrica. 88(2), pages 799-844, 2020.

Further reading[edit]

External links[edit]

  1. ^ in social choice, ranked-choice rules include plurality and all other non-rated rules.