Ideology as a Higher Dimensional Space and Electoral Politics

Abstract

Most classic models of elections that we are familiar with imagine ideology as a straight line (left-right) and show that candidates win by moving toward the middle voter. This result, known as the median voter theorem, works well when political ideologies can be reduced to a single issue space or a single dimension. However, once politics involves many issues at the same time- that is, when we can say that the data is directional, this logic does not necessarily hold. In higher-dimensional settings, there may not be a single middle position that beats all others, and candidates may fail to converge at all. This failure of the median voter theorem in richer policy spaces is the central motivation for this thesis. Modern political conflicts are rarely one-dimensional. Voters care about bundles of issues, and political divisions often cut across the traditional left–right spectrum. This thesis studies electoral competition in policy spaces that better reflect these realities. It examines two settings: one where policies are com binations of yes–no positions on many issues (on a Hypercube), and another where ideology is better understood as a direction on a spherical surface. A key intuition behind the second setting comes from the idea that political ideology can have a “horseshoe” shape. Groups at opposite ends of the left–right spectrum may appear far apart, yet share similar attitudes toward candidates. A curved representation of ideology naturally captures this feature by allowing political extremes to lie close to one another, rather than forcing them to be opposites on a straight line. This work lays the groundwork for future research that can address questions such as, what can we say about the existence and the uniqueness of an equilibrium in n-dimensional spaces? And under what conditions does competition on a multi-issue space deliver an equilibrium platform that coincides with a robust median, or any ideological ‘centre’ of voter preferences.