Probably good to first consider a simpler, 1-dimensional, turn-based version of the problem here.
Suppose we have voters uniformly distributed over a 2-dimensional policy-space. Parties position themselves on this unit square, and voters vote for the party which is closest to their own position. The goal of each party is to maximise their share of the vote.
Let each of the parties be randomly assigned starting co-ordinates. We can calculate each party's share of the vote. Now every party has the chance to shift their policy position, up to, say, 0.01 units, in any direction. They choose the direction to move which will best improve their vote. At the new positions, the parties again have the chance to assess their vote share and shift their position.
What is the result of this iterative process? A static or dynamic equilibrium? Or something else. Could anyone write a program to help visualise?