Election Inversions under Proportional Representation

It has been shown by Peter Kurrild‐Klitgaard, using several empirical examples under the Danish electoral system, that proportional representation (PR) can produce ‘election inversions’ such that a coalition of parties collectively supported by a majority of voters fails to win a majority of parliamentary seats. However, Kurrild‐Klitgaard's examples result from imperfections in the Danish PR system introduced to serve goals other than proportionality. In this article, Kurrild‐Klitgaard's analysis is carried a step further by showing that election inversions can occur even under the purest type of PR – namely, one with (i) a single national constituency, (ii) no explicit seat threshold, and (iii) a highly proportional electoral formula. Inversions result from the unavoidable ‘whole number problem’. Recent election data from Israel and the Netherlands is examined and examples of inversions under their relatively pure PR systems are found. Inversions are also found after recalculating seat allocations without a threshold, and on the basis of the most proportional electoral formulas and when the analysis is restricted to seat‐winning parties. Kurrild‐Klitgaard's Danish data is then re‐examined in the same fashion, as is the most recent apportionment of seats in the United States House of Representatives, and more examples of inversions are found.