Editing Arrow's impossibility theorem (section)

==== {{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"/>