Election inversions, coalitions and proportional representation: Examples from Danish elections

When collective choices are made in more than one round and with dif¬ferent groups of decision-makers, so-called election inversions may take place, where each group have different majority outcomes. We identify two ver¬sions of such compound majority paradoxes specifically, but not ex¬clu¬si¬ve¬ly, relevant for systems of proportional representation with governing coalitions: The “Threshold Paradox” and the “Federal Paradox”. The empirical relevance of the two paradoxes is illustrated with examples from three Danish elections (1971, 1990, 2011), where a majority of the voters voted for one bloc of parties but where a majority of the seats fell to another.