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⁴⁴ Singer and Stephenson, ‘The Political Context and Duverger's Theory’.

⁴⁵ Reed, ‘Duverger's Law is Working in Italy’.

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⁴⁷ Cox and Schoppa, ‘Interaction Effects in Mixed-Member Electoral Systems’.

⁴⁸ Singer and Stephenson, ‘The Political Context and Duverger's Theory’.

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⁵⁶ For example, a distribution of votes across three parties in which the winner receives 39 per cent of the vote, the second-place party 38 per cent and the third-place party 13 per cent yields an effective number of parties of 2.5. A district where the top three parties receive 57, 23 and 20 per cent of the vote, respectively, also qualifies as a two-party competition under this standard. Yet both of these scenarios imply that there is a substantial third candidate in the district (even if in the second case, strategic coordination by the two losing parties could not have changed the outcome).

⁵⁷ While the empirical tests focus on the support for parties finishing third or worse, I supplement the discussion by looking at the distribution of votes for parties finishing first and second as well. This approach follows the advice of Taagepera, Predicting Party Sizes, p. 106 to differentiate whether FPTP yields distributions of the vote of ‘52-48 or 50-40-10’. I do not have a specific hypothesis about support for first- and second-place parties because Duverger's Law does not have any implications for cases in which there is only demand for one party in a district (Clark and Golder, ‘Rehabilitating Duverger's Theory’). In other words, Duverger's Law can be satisfied when support for the second-place party ranges from 0–50 per cent as long as the winning party gets the rest of the votes in the district.

⁵⁸ Gaines, ‘Duverger's Law and the Meaning of Canadian Exceptionalism’. The distributions of these various dependent variables are graphed in the online appendix.

⁵⁹ Cox, Making Votes Count.

⁶⁰ Cox, Making Votes Count considers cases with an SF ratio of 1 to be potential exemplars of a non-Duvergerian equilibrium, in which voters do not know which party is in third place and thus should be abandoned. However, the key point for this analysis is that such a scenario does not correspond to a two-party competition.

⁶¹ Several contributors to Grofman, Blais and Bowler, Duverger's Law of Plurality Voting argue that another indicator of strategic coordination failure by voters and elites is if support for third place or worse parties is greater than the margin between the first- and second-place parties. In districts where small parties gained more support than the victor's margins, strategic coordination by all small party supporters could potentially have changed the race's outcome. This is equivalent to modelling whether the winning party received 50 percent of the vote or not. This kind of outcome should be relatively rare under the strategic outcome Duverger envisions. I have thus also analysed this question and present the results in the online appendix.

⁶² The 2010 British election was more fragmented than was the 1997 election used in this dataset, but the substantive conclusions do not change if we use the 2010 elections instead (see the online appendix).

⁶³ I exclude Papua New Guinea from the analysis because it is such a large outlier from the dominant pattern in other countries using the same system, even after controlling for other factors (Reilly, ‘Party Politics in Papua New Guinea’; Singer and Stephenson, ‘The Political Context and Duverger's Theory’).

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⁷⁶ Since none of the countries in this sample combine plurality electoral rules with majoritarian presidential rules, there is no control for majoritarian presidential elections in this model.

⁷⁷ E.g., Gaines, ‘Duverger's Law and the Meaning of Canadian Exceptionalism’; Diwaker, ‘Duverger's Law and the Size of the Indian Party System’; Grofman, Blais and Bowler, Duverger's Law of Plurality Voting.

⁷⁸ Moreover, the model provides no evidence in the initial specification that size or federalism raises fragmentation generally in FPTP countries, so we cannot attribute the different outcomes in Canada, India or the UK to these characteristics.

⁷⁹ Grofman, Blais and Bowler, Duverger's Law of Plurality Voting uses the standard of whether votes for parties outside the top two parties were greater than the margin of victory to identify districts in which strategic voting potentially could have swung the election (which is equivalent to asking if the winner got a majority of the vote) and thus to diagnose coordination failures. However, this only defines an upper limit on the number of cases in which strategic behaviour could change the outcome. In some of these cases, small party voters may have been indifferent between the top two or preferred the winner, meaning that strategic behaviour would have left the outcome unchanged.

⁸⁰ The third-place party was only bigger than the margin between the top two parties in 1.4 per cent of American districts, but could have swung the election in 30 per cent of the districts in Canada, 36 per cent of the districts in the UK, 43 per cent of the districts in India or Zambia, and 75 per cent of the districts in Nepal.

⁸¹ Clark and Golder, ‘Rehabilitating Duverger's Theory’.

⁸² Singer and Stephenson, ‘The Political Context and Duverger's Theory’.

⁸³ The TF coefficient is significant at 0.05 in the results presented in online appendix 9; it is significant at the 0.10 level here.

⁸⁴ Monroe and Rose, ‘Electoral Systems and Unimagined Consequences’; Singer and Stephenson, ‘The Political Context and Duverger's Theory’.

⁸⁵ See online appendix 9.

⁸⁶ Moser, Unexpected Outcomes; Moser and Scheiner, ‘Mixed Electoral Systems and Electoral System Effects’.

⁸⁷ Reed, ‘Duverger's Law is Working in Italy’.

⁸⁸ Chhibber and Kollman, The Formation of National Party Systems; Singer and Stephenson, ‘The Political Context and Duverger's Theory’.

⁸⁹ Singer and Stephenson, ‘The Political Context and Duverger's Theory’.

⁹⁰ Monroe and Rose, ‘Electoral Systems and Unimagined Consequences’.

⁹¹ Chhibber and Kollman, The Formation of National Party Systems.

⁹² A cross-sectional correlation between the number of parties winning votes/seats nationally versus locally may reflect a feedback loop, or may be the result of extrapolating district-level electoral fragmentation onto the national level.

⁹³ Support for the first, second, third and remaining parties averages 46-30-11-12 in MMP districts, 46-30-12-11 in PR districts and 49-30-13-7 in Canada, India and the UK.

⁹⁴ The top two parties receive 90 per cent or more of the vote 72 per cent of the time in plurality elections outside of Canada, India and the UK, compared to 32 per cent of the time in mixed systems, 18 per cent of the time in Canada, India and the UK, and less than 1 per cent of districts under majority rule.