McKelvey–Schofield chaos theorem

The McKelvey–Schofield chaos theorem is a result in social choice theory. It states that if preferences are defined over a multidimensional policy space, then majority rule is in general unstable: there is no Condorcet winner. Furthermore, any point in the space can be reached from any other point by a sequence of majority votes.

The theorem can be thought of as showing that Arrow's impossibility theorem holds when preferences are restricted to be concave in $\mathbb {R} ^{n}$ . The median voter theorem 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.

Definitions

The theorem considers a finite number of voters, $n$ , who vote for policies which are represented as points in Euclidean space of dimension $m$ . Each vote is between two policies using majority rule. Each voter, $i$ , has a utility function, $U i$ , which measures how much they value different policies.^{[clarification needed]}

Euclidean preferences

Richard McKelvey considered the case when preferences are "Euclidean metrics".^[1] That means every voter's utility function has the form $U_{i}(x)=\Phi _{i}\cdot d(x,x_{i})$ for all policies $x$ and some $x i$ , where $d$ is the Euclidean distance and $\Phi _{i}$ is a monotone decreasing function.

Under these conditions, there could be a collection of policies which don't have a Condorcet winner using majority rule. This means that, given a number of policies $X a$ , $X b$ , $X c$ , there could be a series of pairwise elections where:

$X a$ wins over $X b$
$X b$ wins over $X c$
$X c$ wins over $X a$

McKelvey proved that elections can be even more "chaotic" than that: If there is no equilibrium outcome^{[clarification needed]} then any two policies, e.g. $A$ and $B$ , have a sequence of policies, $X_{1},X_{2},...,X_{s}$ , where each one pairwise wins over the other in a series of elections, meaning:

$A$ wins over $X 1$
$X 1$ wins over $X 2$
...
$X s$ wins over $B$

This is true regardless of whether $A$ would beat $B$ or vice versa.

Example

An example of McKelvey's theorem

The simplest illustrating example is in two dimensions, with three voters. Each voter will then have a maximum preferred policy, and any other policy will have a corresponding circular indifference curve centered at the preferred policy. If a policy was proposed, then any policy in the intersection 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.

Generalisations

Norman Schofield extended the theorem to more general classes of utility functions, requiring only that they are differentiable. He also established conditions for the existence of a directed continuous path of policies, where each policy further along the path would win against one earlier.^[2]^[3] Some of Schofield's proofs were later found to be incorrect by Jeffrey S. Banks, who corrected his proofs.^[4]^[5]

References

^ McKelvey, Richard D. (June 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.
^ Schofield, N. (1 October 1978). "Instability of Simple Dynamic Games". The Review of Economic Studies. 45 (3): 575–594. doi:10.2307/2297259. JSTOR 2297259.
^ Cox, Gary W.; Shepsle, Kenneth A. (2007). "Majority Cycling and Agenda Manipulation: Richard McKelvey's Contributions and Legacy". In Aldrich, John Herbert; Alt, James E.; Lupia, Arthur (eds.). Positive Changes in Political Science. Analytical perspectives on politics. Ann Arbor, Michigan: University of Michigan Press. pp. 20–23. ISBN 978-0-472-06986-6.
^ Banks, Jeffrey S. (1995-01-01). "Singularity theory and core existence in the spatial model". Journal of Mathematical Economics. 24 (6): 523–536. doi:10.1016/0304-4068(94)00704-E. ISSN 0304-4068.
^ Saari, Donald G. (2008). "Deliver Us from the Plurality Vote". Disposing dictators, demystifying voting paradoxes: social choice analysis. Cambridge, New York: Cambridge University Press. ISBN 978-0-521-51605-1. OCLC 227031682.

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[1] McKelvey, Richard D. (June 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.

[2] Schofield, N. (1 October 1978). "Instability of Simple Dynamic Games". The Review of Economic Studies. 45 (3): 575–594. doi:10.2307/2297259. JSTOR 2297259.

[3] Cox, Gary W.; Shepsle, Kenneth A. (2007). "Majority Cycling and Agenda Manipulation: Richard McKelvey's Contributions and Legacy". In Aldrich, John Herbert; Alt, James E.; Lupia, Arthur (eds.). Positive Changes in Political Science. Analytical perspectives on politics. Ann Arbor, Michigan: University of Michigan Press. pp. 20–23. ISBN 978-0-472-06986-6.

[4] Banks, Jeffrey S. (1995-01-01). "Singularity theory and core existence in the spatial model". Journal of Mathematical Economics. 24 (6): 523–536. doi:10.1016/0304-4068(94)00704-E. ISSN 0304-4068.

[5] Saari, Donald G. (2008). "Deliver Us from the Plurality Vote". Disposing dictators, demystifying voting paradoxes: social choice analysis. Cambridge, New York: Cambridge University Press. ISBN 978-0-521-51605-1. OCLC 227031682.

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