TheMcKelvey–Schofield chaos theorem is a result insocial choice theory. It states that if preferences are defined over a multidimensional policy space, then choosing policies usingmajority rule is unstable. There will in most cases be noCondorcet winner 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.^[1]

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 byRichard McKelvey in 1976, for preferences based onEuclidean distances in${\displaystyle {\mathbb{R}}^n}$. Another version of the theorem was proved byNorman Schofield in 1978, fordifferentiable preferences.

The theorem can be thought of as showing thatArrow's impossibility theorem holds when preferences are restricted to beconcave in${\displaystyle {\mathbb{R}}^n}$. Themedian 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.

The theorem considers a finite number of voters,n, who vote for policies which are represented as points inEuclidean space of dimensionm. Each vote is between two policies usingmajority rule. Each voter,i, has autility function,U_i, which measures how much they value different policies.^{[clarification needed]}

Richard McKelvey considered the case when preferences are "Euclidean metrics".^[2] That means every voter's utility function has the form$${\displaystyle U_i(x)={\mathrm{\Phi }}_i\cdot d(x,x_i)}$$for all policiesx and somex_i, whered is theEuclidean distance and${\displaystyle {\mathrm{\Phi }}_i}$ is amonotone decreasing function.

Under these conditions, there could be a collection of policies which don't have aCondorcet winner using majority rule. This means that, given a number of policiesX_a,X_b,X_c, there could be a series of pairwise elections where:

1. X_a wins overX_b
2. X_b wins overX_c
3. X_c wins overX_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 andB, have a sequence of policies,${\displaystyle X_1,X_2,...,X_s}$, where each one pairwise wins over the other in a series of elections, meaning:

1. A wins overX₁
2. X₁ wins overX₂
3. ...
4. X_s wins overB

This is true regardless of whetherA would beatB orvice versa.

The simplest illustrating example is intwo dimensions, with three voters. Each voter will then have a maximum preferred policy, and any other policy will have a corresponding circularindifference curve centered at the preferred policy. If a policy was proposed, then any policy inthe 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.

Norman Schofield extended the theorem to more general classes of utility functions, requiring only that they aredifferentiable. He also established conditions for the existence of a directedcontinuous path of policies, where each policy further along the path would win against one earlier.^[3][1] Some of Schofield's proofs were later found to be incorrect by Jeffrey S. Banks, who corrected his proofs.^[4][5]

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