The Dynamics of Adaptive Parties under Spatial Voting

We explore the dynamics of a model of two-party competition under spatial voting. the parties are allowed to {\it incrementally} adapt their platforms by following the voting gradient imposed by the preferences of the electorate and platform of the opposition. The emphasis in this model is on the dynamic system formed by these conditions, in particular, we examine the characteristics of the transient paths and the convergence points of the evolving platforms. We find that in a simple spatial model with probabilistic voting, regardless of the initial platsforms of each party, platforms eventually converge to a unique, globally stable equilibrium matching the strength-weighted mean of the voters' preferred positions. This result holds even if we allow simple cross-issue weightings, however, if we allow nonlinear weighting functions many dynamic possibilities occur, including multiple equilibria and, perhaps, limit cycles.