Voting in legislative elections under plurality rule

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 single district elections are used to fill 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 to a more moderate policy by the median voter in a majority of districts. The mere existence of a centrist party can lead to moderate outcomes even if the party itself wins few seats. I also show that, while some voters in a district will not vote for their nationally preferred party, in many equilibria they will want the candidate for whom they vote to win that district. This is never the case in single district elections. There, some voters always want the candidate they voted for to lose.