Voting in legislative elections under plurality rule

Conventional models of single district plurality elections show that with three parties anything can happen extreme policies can win regardless of voter preferences. I show that when there are multiple district elections for a legislature we get back to a world where the median voter matters: an extreme policy will generally only come about if it is preferred by the median voter in a majority of districts, while the existence of a centrist party can lead to moderate outcomes even if the party itself wins few seats. Furthermore, I show that while single district elections always have misaligned voting i.e. some voters do not vote for their preferred choice, equilibria of the legislative election exist with no misaligned voting in any district. Finally, I show that when parties are impatient, a fixed rule on how legislative bargaining occurs will lead to more coalition governments, while uncertainty will favour single party governments.