Arrow's impossibility 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 $A$ and $B$ should not depend on the quality of a third, unrelated outcome $C$ .^[1]

The result is often cited in discussions of election science and voting theory, where $C$ is called a spoiler candidate. As a result, Arrow's theorem can be restated as saying that no ranked voting system can eliminate the spoiler effect.^[1]^[2]

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, Borda, and instant-runoff suffer spoiler effects more often than other methods,^[4] and even in situations where spoiler effects are not necessary,^[5]^[6] as they can elect candidates who would have lost in a straight majority vote. Majority-choice methods uniquely minimize the effect of spoilers on election results, limiting them to rare^[7]^[8] situations known as cyclic ties.^[5]

While originally overlooked, a large class of systems called rated methods are not affected by Arrow's theorem or IIA failures.^[2]^[9]^[10] Arrow initially asserted the information provided by these systems was meaningless and therefore could not prevent his paradox;^[11] however, he would later recognize this as a mistake,^[2] describing score voting as "probably the best" way to avoid his theorem.^[2]^[12]^[10]

Cite error: There are <ref group=note> tags on this page, but the references will not show without a {{reflist|group=note}} template (see the help page).

^ ^a ^b Cite error: The named reference :3 was invoked but never defined (see the help page).
^ ^a ^b ^c ^d ^e 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.
^ McKenna, Phil (12 April 2008). "Vote of no confidence". New Scientist. 198 (2651): 30–33. doi:10.1016/S0262-4079(08)60914-8.
^ 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
^ ^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."
^ 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.
^ 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.
^ 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.
^ Poundstone, William. (2013). Gaming the vote : why elections aren't fair (and what we can do about it). Farrar, Straus and Giroux. pp. 168, 197, 234. ISBN 9781429957649. OCLC 872601019. IRV is subject to something called the "center squeeze." A popular moderate can receive relatively few first-place votes through no fault of her own but because of vote splitting from candidates to the right and left. [...] Approval voting thus appears to solve the problem of vote splitting simply and elegantly. [...] Range voting solves the problems of spoilers and vote splitting
^ ^a ^b Ng, Y. K. (November 1971). "The Possibility of a Paretian Liberal: Impossibility Theorems and Cardinal Utility". Journal of Political Economy. 79 (6): 1397–1402. doi:10.1086/259845. ISSN 0022-3808. In the present stage of the discussion on the problem of social choice, it should be common knowledge that the General Impossibility Theorem holds because only the ordinal preferences is or can be taken into account. If the intensity of preference or cardinal utility can be known or is reflected in social choice, the paradox of social choice can be solved.
^ Cite error: The named reference :5 was invoked but never defined (see the help page).
^ 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. Dr. Arrow: Well, I'm a little inclined to think that score systems where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best.

[:3-2] Cite error: The named reference :3 was invoked but never defined (see the help page).

[:02-3] 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.

[ns122-4] McKenna, Phil (12 April 2008). "Vote of no confidence". New Scientist. 198 (2651): 30–33. doi:10.1016/S0262-4079(08)60914-8.

[Borgers22-5] 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

[Holliday2-6] 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."

[:22-7] 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.

[:532-8] 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.

[:632-9] 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.

[Poundstone,_William.-20132-10] Poundstone, William. (2013). Gaming the vote : why elections aren't fair (and what we can do about it). Farrar, Straus and Giroux. pp. 168, 197, 234. ISBN 9781429957649. OCLC 872601019. IRV is subject to something called the "center squeeze." A popular moderate can receive relatively few first-place votes through no fault of her own but because of vote splitting from candidates to the right and left. [...] Approval voting thus appears to solve the problem of vote splitting simply and elegantly. [...] Range voting solves the problems of spoilers and vote splitting

[:8-11] Ng, Y. K. (November 1971). "The Possibility of a Paretian Liberal: Impossibility Theorems and Cardinal Utility". Journal of Political Economy. 79 (6): 1397–1402. doi:10.1086/259845. ISSN 0022-3808. In the present stage of the discussion on the problem of social choice, it should be common knowledge that the General Impossibility Theorem holds because only the ordinal preferences is or can be taken into account. If the intensity of preference or cardinal utility can be known or is reflected in social choice, the paradox of social choice can be solved.

[:5-12] Cite error: The named reference :5 was invoked but never defined (see the help page).

[13] 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. Dr. Arrow: Well, I'm a little inclined to think that score systems where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best.

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