Editing Arrow's impossibility theorem (section)

=== {{Anchor|Minimizing}}Minimizing IIA failures: Majority-rule methods ===
{{Main|Condorcet cycle}}

[[File:Italian_food_Condorcet_cycle.png|thumb|383x383px|An example of a Condorcet cycle, where some candidate ''must'' cause a spoiler effect]]
The first set of methods studied by economists are the [[Condorcet methods|majority-rule, or ''Condorcet'', methods]]. These rules limit spoilers to situations where majority rule is self-contradictory, called [[Condorcet cycle]]s, and as a result uniquely minimize the possibility of a spoiler effect among ranked rules. (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 [[Majority rule|Condorcet method]] will adhere to Arrow's criteria.<ref name="Campbell2000"/>) Condorcet believed voting rules should satisfy both independence of irrelevant alternatives and the [[Condorcet winner criterion|majority rule principle]], i.e. if most voters rank ''Alice'' ahead of ''Bob'', ''Alice'' should defeat ''Bob'' in the election.<ref name="McLean-1995"/>

Unfortunately, as Condorcet proved, this rule can be intransitive on some preference profiles.<ref>{{Cite journal |last=Gehrlein |first=William V. |date=1983-06-01 |title=Condorcet's paradox |url=https://doi.org/10.1007/BF00143070 |journal=Theory and Decision |language=en |volume=15 |issue=2 |pages=161–197 |doi=10.1007/BF00143070 |issn=1573-7187}}</ref> Thus, Condorcet proved a weaker form of Arrow's impossibility theorem long before Arrow, under the stronger assumption that a voting system in the two-candidate case will agree with a simple majority vote.<ref name="McLean-1995" />

Unlike pluralitarian rules such as [[Instant-runoff voting|ranked-choice runoff (RCV)]] or [[first-preference plurality]],<ref name="McGann2002"/> [[Condorcet method]]s avoid the spoiler effect in non-cyclic elections, where candidates can be chosen by majority rule. Political scientists have found such cycles to be fairly rare, suggesting they may be of limited practical concern.<ref name="VanDeemen" /> [[Spatial model of voting|Spatial voting models]] also suggest such paradoxes are likely to be infrequent<ref name="Wolk-2023">{{Cite journal |last1=Wolk |first1=Sara |last2=Quinn |first2=Jameson |last3=Ogren |first3=Marcus |date=2023-09-01 |title=STAR Voting, equality of voice, and voter satisfaction: considerations for voting method reform |url=https://doi.org/10.1007/s10602-022-09389-3 |journal=Constitutional Political Economy |volume=34 |issue=3 |pages=310–334 |doi=10.1007/s10602-022-09389-3 |issn=1572-9966}}</ref><ref name="Gehrlein-2002"/> or even non-existent.<ref name="Black-1948" />

==== {{Anchor|Single peak}}Left-right spectrum ====
{{Main|Median voter theorem}}
Soon after Arrow published his theorem, [[Duncan Black]] showed his own remarkable result, the [[median voter theorem]]. The theorem proves that if voters and candidates are arranged on a [[Political spectrum|left-right spectrum]], Arrow's conditions are all fully compatible, and all will be met by any rule satisfying [[Condorcet winner criterion|Condorcet's majority-rule principle]].<ref name="Black-1948" /><ref name="Black-1968"/>

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 the volume gets progressively too loud or too quiet, they would be increasingly dissatisfied. If the domain is restricted to profiles where every individual has a 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, including being fully spoilerproof.<ref name="Black-1948" /><ref name="Black-1968"/><ref name="Campbell2000"/>

The rule does not fully generalize from the political spectrum to the political compass, a result related to the [[McKelvey-Schofield chaos theorem]].<ref name="Black-1948" /><ref>{{Cite journal |last1=McKelvey |first1=Richard D. |author-link=Richard McKelvey |year=1976 |title=Intransitivities in multidimensional voting models and some implications for agenda control |journal=Journal of Economic Theory |volume=12 |issue=3 |pages=472–482 |doi=10.1016/0022-0531(76)90040-5}}</ref> However, a well-defined Condorcet winner does exist if the [[Probability distribution|distribution]] of voters is [[Rotational symmetry|rotationally symmetric]] or otherwise has a [[Omnidirectional median|uniquely-defined median]].<ref>{{Cite journal |last1=Davis |first1=Otto A. |last2=DeGroot |first2=Morris H. |last3=Hinich |first3=Melvin J. |date=1972 |title=Social Preference Orderings and Majority Rule |url=http://www.jstor.org/stable/1909727 |journal=Econometrica |volume=40 |issue=1 |pages=147–157 |doi=10.2307/1909727 |jstor=1909727 |issn=0012-9682}}</ref><ref name="dotti2">{{Cite thesis |title=Multidimensional voting models: theory and applications |url=https://discovery.ucl.ac.uk/id/eprint/1516004/ |publisher=UCL (University College London) |date=2016-09-28 |degree=Doctoral |first=V. |last=Dotti}}</ref> In most realistic situations, where voters' opinions follow a roughly-[[normal distribution]] or can be accurately summarized by one or two dimensions, Condorcet cycles are rare (though not unheard of).<ref name="Wolk-2023" /><ref name="Holliday23222"/>

==== Generalized stability theorems ====
The Campbell-Kelly theorem shows 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.<ref name="Campbell2000" /> In other words, replacing a ranked method with its Condorcet variant (i.e. elect a Condorcet winner if they exist, and otherwise run the method) will sometimes prevent a spoiler effect, but can never create a new one.<ref name="Campbell2000" />

In 1977, [[Ehud Kalai]] and [[Eitan Muller]] gave a full characterization of domain restrictions admitting a nondictatorial and [[Strategyproofness|strategyproof]] social welfare function. These correspond to preferences for which there is a Condorcet winner.<ref>{{Cite journal |last1=Kalai |first1=Ehud |last2=Muller |first2=Eitan |year=1977 |title=Characterization of domains admitting nondictatorial social welfare functions and nonmanipulable voting procedures |url=http://www.kellogg.northwestern.edu/research/math/papers/234.pdf |journal=Journal of Economic Theory |volume=16 |issue=2 |pages=457–469 |doi=10.1016/0022-0531(77)90019-9}}</ref>

Holliday and Pacuit devised a voting system that provably minimizes the number of candidates who are capable of spoiling an election, albeit at the cost of occasionally failing [[Monotonicity criterion|vote positivity]] (though at a much lower rate than seen in [[instant-runoff voting]]).<ref name="Holliday23222"/>{{clarify|reason=Needs a quote saying what is claimed, for instance how it has fewer spoilers than other Smith methods.|date=November 2024}}