We built a computational model of political party competition in order to gain insight into the effect of the decrease in the number of relevant political issues (dimensions), and the change of their relative importance, on the number of surviving political parties, their strategy performance, and the degree of political party fragmentation. Particularly, we find that when there is a dimensionality reduction (i.e., a change from a two-dimensional issue space to a one-dimensional one, or, a substantial decrement in one of the issue's relative importance with respect to the other), the number of political parties declines, as does the overall degree of party fragmentation in the system. Regarding party strategies, we observe that, after the dimensionality reduction, (i) the inert parties tend to improve their performance in terms of party numbers (i.e., more inert parties survive, relatively * Corresponding author. C. García-Díaz, G. Zambrana-Cruz and A. van Witteloostuijn speaking); (ii) the population of large-size seekers declines, (iii) the few large-size seeker survivors, in general, cushion their increased mortality hazard with increased size (i.e., increased number of supporters); and, finally, (iv) the mortality hazard increases with distance to the mean voter spot.