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 choosing policies using majority rule is unstable. There will in most cases be no Condorcet 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.

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 ${\displaystyle \mathbb {R} ^{n}}$. Another version of the theorem was proved by Norman Schofield in 1978, for differentiable preferences.

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