Adaptive platform dynamics in multi-party spatial voting

The dynamics of multiple competing political parties under spatial voting is explored. Parties are allowed to modify their positions adaptively in order to gain more votes. The parties in this model are opportunistic, in the sense that they try to maximize their share of votes regardless of any ideological position. Each party makes small corrections to its current platform in order to increase its own utility by means of the steepest ascent in the variables under its own control, i.e., by locally optimizing its own platform.We show that in models with more than two parties bifurcations at the trivial equilibrium occur if only the voters are critical enough, that is, if they respond strongly to small changes in relative utilities. A numerical survey in a three-party model yields multiple bifurcations, multi-stability, and stable periodic at tractors that arise through Hopf bifurcations. Models with more than two parties can thus differ substantially from the two-party case, where it has been shown that under the assumptions of quadratic voter utilities and complete voter participation there is always a globally stable equilibrium that coincides with the mean voter position.