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shows that when preferences are restricted to be single-peaked on the real line, Arrow's theorem does not hold, and the median voter's ideal point is a
Condorcet winner. The chaos theorem shows that this good news does not continue in multiple dimensions.
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of policies, where each policy further along the path would win against one earlier. Some of
Schofield's proofs were later found to be incorrect by Jeffrey S. Banks, who corrected his proofs.
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605:(2007). "Majority Cycling and Agenda Manipulation: Richard McKelvey's Contributions and Legacy". In Aldrich, John Herbert; Alt, James E.; Lupia, Arthur (eds.).
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of two voters indifference curves would beat it. Any point in the plane will almost always have a set of points that are preferred by 2 out of 3 voters.
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27:. It states that if preferences are defined over a multidimensional policy space, then majority rule is in general unstable: there is no
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McKelvey proved that elections can be even more "chaotic" than that: If there is no equilibrium outcome then any two policies, e.g.
483:, with three voters. Each voter will then have a maximum preferred policy, and any other policy will have a corresponding circular
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McKelvey, Richard D. (June 1976). "Intransitivities in
Multidimensional Voting Models and Some Implications for Agenda Control".
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considered the case when preferences are "Euclidean metrics". That means every voter's utility function has the form
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31:. Furthermore, any point in the space can be reached from any other point by a sequence of majority votes.
609:. Analytical perspectives on politics. Ann Arbor, Michigan: University of Michigan Press. pp. 20–23.
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extended the theorem to more general classes of utility functions, requiring only that they are
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Under these conditions, there could be a collection of policies which don't have a
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centered at the preferred policy. If a policy was proposed, then any policy in
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Disposing dictators, demystifying voting paradoxes: social choice analysis
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Schofield, N. (1 October 1978). "Instability of Simple
Dynamic Games".
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509:. He also established conditions for the existence of a directed
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using majority rule. This means that, given a number of policies
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196:{\displaystyle U_{i}(x)=\Phi _{i}\cdot d(x,x_{i})}
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