This paper addresses two problems: how can we identify a verisimilar policy space and how can we detect Nash equilibria in this space for parties’ policy positions? We argue that the ideological party positions that voters perceive are fixed during the time span of one electoral campaign and that they constrain the policies parties offer the electorate in search of optimal vote shares. We apply the valence model developed by Schofield to party competition during the German federal election campaign 2009. First three issue scales are combined with a left–right scale to form one homogeneous space in which equilibrium locations of parties are sought. The results show that local Nash equilibria in this combined space depend heavily on the start values and are implausible. Fixing the ideological dimension leads to a stable and meaningful equilibrium configuration in which large parties move to more central positions and smaller parties move to more peripheral positions in the policy space.