McKelvey–Schofield Chaos Theorem
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The McKelvey–Schofield chaos theorem is a result in
social choice theory Social choice theory is a branch of welfare economics that extends the Decision theory, theory of rational choice to collective decision-making. Social choice studies the behavior of different mathematical procedures (social welfare function, soc ...
. It states that if preferences are defined over a multidimensional policy space, then choosing policies using
majority rule In social choice theory, the majority rule (MR) is a social choice rule which says that, when comparing two options (such as bills or candidates), the option preferred by more than half of the voters (a ''majority'') should win. In political ...
is unstable. There will in most cases be no
Condorcet winner A Condorcet winner (, ) is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condo ...
and any policy can be enacted through a sequence of votes, regardless of the original policy. This means that adding more policies and changing the order of votes ("agenda manipulation") can be used to arbitrarily pick the winner. Versions of the theorem have been proved for different types of preferences, with different classes of exceptions. A version of the theorem was first proved by Richard McKelvey in 1976, for preferences based on Euclidean distances in \mathbb^. Another version of the theorem was proved by Norman Schofield in 1978, for
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
preferences. The theorem can be thought of as showing that
Arrow's impossibility theorem Arrow's impossibility theorem is a key result in social choice theory showing that no ranked-choice procedure for group decision-making can satisfy the requirements of rational choice. Specifically, Arrow showed no such rule can satisfy the ind ...
holds when preferences are restricted to be
concave Concave or concavity may refer to: Science and technology * Concave lens * Concave mirror Mathematics * Concave function, the negative of a convex function * Concave polygon A simple polygon that is not convex is called concave, non-convex or ...
in \mathbb^. The
median voter theorem In political science and social choice theory, social choice, Black's median voter theorem says that if voters and candidates are distributed along a political spectrum, any voting method Condorcet criterion, compatible with majority-rule will elec ...
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, , who vote for policies which are represented as points in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
of dimension . Each vote is between two policies using
majority rule In social choice theory, the majority rule (MR) is a social choice rule which says that, when comparing two options (such as bills or candidates), the option preferred by more than half of the voters (a ''majority'') should win. In political ...
. Each voter, , has a
utility function In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a Normative economics, normative context, utility refers to a goal or ob ...
, , which measures how much they value different policies.


Euclidean preferences

Richard McKelvey considered the case when preferences are "Euclidean metrics". That means every voter's utility function has the form U_i(x)=\Phi_i \cdot d(x,x_i) for all policies and some , where is the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
and \Phi_i is a
monotone decreasing In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
function. Under these conditions, there could be a collection of policies which don't have a
Condorcet winner A Condorcet winner (, ) is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condo ...
using majority rule. This means that, given a number of policies , , , there could be a series of pairwise elections where: # wins over # wins over # wins over McKelvey proved that elections can be even more "chaotic" than that: If there is no equilibrium outcome then any two policies, e.g. and , 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: # wins over # wins over # ... # wins over This is true regardless of whether would beat or ''vice versa''.


Example

The simplest illustrating example is in
two dimensions A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimensional s ...
, with three voters. Each voter will then have a maximum preferred policy, and any other policy will have a corresponding circular
indifference curve In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the c ...
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 In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
. He also established conditions for the existence of a directed
continuous path Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
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.


References

{{Econ-theory-stub Voting theory Economics theorems