McKelvey–Schofield chaos theorem

Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results The McKelvey–Schofield chaos theorem is a result in social choice theory.

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 by Richard McKelvey in 1976, for preferences based on Euclidean distances in

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.

[clarification needed] Richard McKelvey considered the case when preferences are "Euclidean metrics".

Under these conditions, there could be a collection of policies which don't have a Condorcet winner using majority rule.

, where each one pairwise wins over the other in a series of elections, meaning: This is true regardless of whether A would beat B or vice versa.

Norman Schofield extended the theorem to more general classes of utility functions, requiring only that they are differentiable.