What are the effects of entry of new extremist parties on the policy platforms of mainstream parties?

We examine the consequences to policy-seeking, center-left and center-right parties under proportional representation following entry by extremist parties either at one or both ends of a unidimensional political spectrum. Assuming a symmetric, unimodal voter distribution, we show that the entry of a single extremist party on either the left or right drives both mainstream parties in the direction opposite to the extremist party. We argue that this three-party scenario is the most empirically relevant case in contemporary European politics. We also extend results of Casamatta and De Donder that project moderation of mainstream parties at equilibrium for PR elections with two extremist parties – one on each end – to a symmetric, unimodal voter distribution. In a setting not considered by these authors, we show that this moderating effect on the mainstream parties for a symmetric voter distribution is reversed if the voter distribution is sufficiently bimodal.