On the probability ratio index as a measure of electoral competition

Electoral competition, the interaction between voters and political parties, is crucial for a healthy democracy. Its effects are debated, with some research highlighting positive impacts on representation, voter turnout, economic performance, governance, and stability in new democracies, while others point to potential negative consequences or a lack of clear effects. The paper aims to address a lack of conceptual clarity by proposing a superior method for measuring electoral competitiveness. Existing measures, like the winning margin, are inadequate for multi-party systems where the number of candidates varies widely. Vote shares provide readily available data, but they don't capture all factors influencing competitiveness. The challenge lies in creating an index that meaningfully compares elections with different numbers of parties and is independent of the number of candidates. The paper specifically asks: given the vote shares of competing parties, how can we obtain a representative measure of electoral competition?

The literature on electoral competitiveness is vast. Mayhew (1974) used the winning margin, suitable only for two-party systems. Vanhanen (1997) used the largest party's vote share, which also ignores other parties. Multi-party measures include the Laakso-Taagepera index (LT-index), the fractionalization index, and the entropy index. However, the LT-index and entropy index are not directly comparable across elections with different numbers of parties due to their range limitations. The fractionalization index also suffers from limitations in cross-election comparisons. The paper highlights the need for an index that addresses the comparability issue without sacrificing other desirable properties.

The paper proposes the probability ratio index as a measure of electoral competitiveness. This index is defined as the ratio between the probabilities that two randomly selected voters (with replacement) voted for different parties under the observed vote shares and under equal vote shares across all parties. The index ranges from 0 (one party having all the votes) to 1 (all parties having equal vote shares). The authors demonstrate that the index meets several desirable properties: it's independent of the number of parties, twice differentiable (smooth), and decomposable. The decomposability feature is shown through a formula representing the index as a normalized sum of rivalry functions between parties. The index also satisfies vote share anonymity (the order of parties doesn't affect the result). The paper explores generalizations of the probability ratio index by considering different numbers of voters drawn at random (k). They show that only when k=2 (two voters drawn) does the generalized index maintain the desired properties of a competitiveness function. The authors analyze the relationships between the probability ratio index and other measures: it's inversely related to the Herfindahl-Hirschman (HH) index (and its normalized variant) and the squared coefficient of variation, and directly related to the fractionalization index. They show the probability ratio index is equivalent to a normalized version of the fractionalization index, which is based on sampling voters without replacement.

The probability ratio index offers several advantages: it's comparable across elections with different numbers of parties, it's simple to calculate and understand, and it is inversely related to the Herfindahl-Hirschman index and the squared coefficient of variation and directly related to the fractionalization index. The paper demonstrates this comparability advantage through examples, showing that the probability ratio index correctly identifies differences in competitiveness where other indices fail. For instance, comparing elections with different numbers of parties and equal vote shares among the leading parties, the probability ratio index correctly indicates different levels of competitiveness. The authors also illustrate how the probability ratio index handles scenarios where additional parties with zero vote shares are added; unlike other indices, it correctly reflects that competition has decreased due to the addition of these parties, even though the distribution of votes remains the same among the actively competing parties. The probability ratio index is shown to be closely related to the HH-index, the squared coefficient of variation, and the fractionalization index, providing a deeper understanding of its properties and relationship to existing measures. The decomposability of the index allows it to be expressed as a normalized sum of all possible rivalry functions between parties, making it a flexible and informative tool for analyzing electoral competition.

The probability ratio index provides a significant improvement in measuring electoral competition. Its ability to handle multi-party systems with varying numbers of candidates and its clear interpretation makes it a valuable tool for comparative political analysis. The close relationship between the probability ratio index and other established indices like the HH-index and fractionalization index strengthens its validity and provides a framework for interpreting the results within the broader context of political science literature. The index's sensitivity to the presence of even small parties, while initially seeming like a drawback, accurately captures the impact of such parties on overall competitiveness. This feature is seen as beneficial as it reflects changes in the dynamics of competition that may be overlooked by other indices.

The probability ratio index offers a robust and flexible method for measuring electoral competitiveness, overcoming limitations of existing indices. Its comparability across elections with different numbers of parties is a key advantage. Future research could investigate the impact of coalition formation and blocking coalitions on the index.

The study focuses solely on vote shares, acknowledging that other factors influence electoral competition. While vote share data are widely available, incorporating other factors would enrich the analysis but presents methodological challenges. The index's sensitivity to the inclusion of small parties with zero vote shares, while considered an advantage in the paper, might require further consideration depending on the research context. Future research could explore incorporating additional variables or weighting schemes to refine the model.