On the probability ratio index as a measure of electoral competition

Abstract

This paper deals with the fundamental issue of quantifying the degree of competitiveness underlying an election in a society. That is, given the vote shares of competing political parties in the election one needs to arrive at a representative metric of electoral competition. The vote share of a party represents its influence on the outcome of the election. A summary measure of differences of such influences across parties, aggregated in an unambiguous way, is an index of electoral competitiveness. The literature contains several such indices. Implicit under the choice of an index is a set of properties that the index satisfies. A highly attractive feature of the probability ratio index we suggest in this article, as a summary measure of electoral competition, is its suitability for making comparisons of competitiveness across elections with different number of parties. The probability ratio index is simply the ratio between the probabilities that the two voters drawn at random with (without) replacement have voted for different parties under actual vote shares across the competing parties and under equal vote shares across them. The comparability feature of the probability ratio index can be maintained without sacrificing any nice characteristic expected from an electoral competition measure. The probability ratio index is inversely related to the well-known Herfindahl–Hirschman concentration index that indicates dominance of large parties, directly related to the fractionalization index, which indicates how vote shares are split among the parties, and also inversely related to the squared coefficient of variation, a highly popular measure of inequality. It is argued explicitly how the new index can be applied to measure competitiveness for true data. Simple examples are used to demonstrate why probability ratio index has a comparative advantage over some other well-known indices, particularly, in terms of cross-election comparability.