Rein Taagepera and Bernard Grofman, "Mapping the Indices of Seats-Votes Disproportionality and Inter-Election Volatility,"Party Politics, 9 (November 2003), 659-677.

First Paragraph:
Deviation from proportional representation (PR) means the difference in party vote shares and party seat shares in some given election (vit versus si). Volatility of votes means the difference in party vote shares (vi) from one election to another (vit versus vit'); similarly, volatility of seats means the difference in party seat shares (si) from one election to another (s sit versus sit' How do we best measure the proportionality/disproporrionality of the translation of party vote shares into party seat shares in a given election? How do we best measure the stability/volatility of party vote shares (or party seat shares) across elections?

Figures and Tables:
Table 1: Evaluation of indices of disproportionality/volatility based on absolute values of difference
Table 2: Evaluation of indices of disproportionality/volatility based on suqare of difference
Table 3: Evaluation of indices of disproportionality/volatility with bases other than absolute values of difference or square of difference

Last Paragraph:
What have we added to Monroe's (1994) and Pennisi's (1998) earlier studies of disproportionality and malapportionment? First, we have updated and complemented the zoo of indices proposed and used by various researchers, pointing out parallel developments in measuring volatility and possible extension toward systematic measurement of vote splitting. Second, we have composed two sets of desirable criteria (theoretical and practical), rating each of the 19 indices explicitly on those 12 criteria. The most widely used indices, Gallager's Gh and Loosemore-Hanby's D, satisfy more criteria than any other, including Dp, proposed by Monroe (1994).

Third, we have tried to specify a constellation that would he intuitively halfway from perfect concordance to utmost discordance; once more, Gh and D are among the few indices that yield 50 percent for such a constellation. We have located a dilemma when Dalton's principle of transfers is applied to parties rather than individuals: which party is the richer one when one has more extra seats while another is overpaid by a larger ratio? If the number of excess seats is taken as the preferred criterion, Gh becomes slightly preferable to D, in view of the latter's ambiguity when data sources lump several small parties. Finally, we tried to devise an even better index than Gh or D but came up short. [First two paragraphs of the Conclusion.]