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.

Richard McKelvey initially proved the theorem for Euclidean preferences.^[1] Norman Schofield extended the theorem to the more general class of concave preferences.^[2]

References

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.