Election Inversions under Proportional Representation

It has been shown by P eter K urrild‐ K litgaard, using several empirical examples under the D anish 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, K urrild‐ K litgaard's examples result from imperfections in the D anish PR system introduced to serve goals other than proportionality. In this article, K urrild‐ K litgaard'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 I srael and the N etherlands 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. K urrild‐ K litgaard's D anish data is then re‐examined in the same fashion, as is the most recent apportionment of seats in the U nited S tates H ouse of R epresentatives, and more examples of inversions are found.