Many countries have in recent years experienced political polarization and difficulties in government formation. Political candidates and parties differ not only in the traditional left-right dimension but also concerning environmental issues, gender roles, cultural values, migration, international trade, defense etc. To label certain new parties “extreme right wing”, as many journalists do, is an over-simplification due to the presumed one-dimensionality of the political space. Moreover, an often neglected but arguably important phenomenon is abstention from voting because of alienation. In this study we outline a mathematical framework for the analysis of direct and representative democracy in a multidimensional political space with endogenous voter abstention due to alienation. Our approach combines methods from non-cooperative and cooperative game theory, geometry, and random-utility theory. We illustrate the framework by an application to the current Swedish parliament.