## Abstract

Electoral systems promote strategic voting and affect party systems. Duverger (Les partis politiques, 1951) proposed that plurality rule leads to bi-partyism and proportional representation leads to multi-partyism. We show that in a dynamic setting, these static effects also lead to a higher option value for existing minor parties under plurality rule, so their incentive to exit the party system is mitigated by their future benefits from continued participation. The predictions of our model are consistent with multiple cross-sectional predictions on the comparative number of parties under plurality rule and proportional representation. In particular, there could be more parties under plurality rule than under proportional representation at any point in time. However, our model makes a unique time-series prediction: the number of parties under plurality rule should be less variable than under proportional representation. We provide extensive empirical evidence in support of these results.

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## Notes

- 1.
Our data come from the Constituency-Level Elections (CLE) Dataset (Brancati 2013).

- 2.
- 3.
See references therein for related finding in experimental settings.

- 4.
An alternative would be to assume that, say, a party of type 1 is present in all elections, and that two left of center interest groups, of types -1 and 0, decide whether or not to support parties in each election. Such a model is almost equivalent to our specification and would yield closely related results.

- 5.
We could allow for persistence in electoral states, although this would add computational complexity without affecting our central conclusions. Likewise, the simplifying assumption that non-centrist preference states \(s_{1}\) and \(s_{-1}\) occur with equal probability allows us to exploit symmetry, but it is not essential.

- 6.
- 7.
That the centrist party never benefits from the coordination costs imposed on minor parties is assumed for convenience and is not important for our results.

- 8.
- 9.
Interest group

*j*’s actions are not yet specified if the preference state is \(s_{-j}\) and no interest groups supported parties in the previous elections (i.e., \(\phi ^{t}=\emptyset\)). These histories only occur off the equilibrium path, and the details are in the proof of proposition 2 in the online Supplementary appendix. - 10.
Condition (2) does not play a role in the proof of proposition 2, but is included in order to establish that the equilibria \(\underline{\sigma }^{{\text{PL}}}\) and \(\overline{\sigma }^{{\text{PL}}}\) can exist under plurality under parametric restrictions that ensure that \(\sigma ^{{\text{PR}}}\) is the unique equilibrium under proportional representation. That the conditions of proposition 2 can be met for some parameter values can be shown by example: the neighborhood of the point with \(\delta \approx 1\), \(\underline{c}=\overline{c}=\frac{1}{4}\), \(\underline{p}=p=\frac{1}{4}\), \(\beta =\frac{1}{9}\), \(\overline{u}-u=1\) and \(u-\underline{u}=\frac{3}{2}\) contains an open set of parameters for which the conditions of proposition 2 are met and for which \(\underline{\alpha }<\underline{p}-\beta\) and \(\overline{\alpha }>0\), so that both \(\overline{\sigma }^{{\text{PL}}}\) and \(\underline{\sigma }^{{\text{PL}}}\) can be equilibria at those parameters, depending on the value of \(\alpha\).

- 11.
In all but one country in our sample, this binary variable does not change over time.

- 12.
Effective district magnitude can differ from average district magnitude, which is defined as the total number of legislative seats divided by the number of electoral districts. Taagepera and Shugart (1989) argue that effective district magnitude is the superior measure of the proportionality of an electoral system. To the extent that a legislature does not feature at-large seats, these measures are identical.

- 13.
- 14.
We report the countries and elections covered by our data in Table B1 of the supplementary appendix. The CLE unfortunately does not contain data on all democratic elections since 1945. Indeed, no single source does. We use only those elections contained in the CLE for our analysis and do not supplement our data set with data from other sources in order to maintain consistent reporting. We replicated our analysis using a similar (thought not identical) sample of elections from the CLEA data set and obtained similar results. We report results using only the CLE because this is the data set that has been primarily used to construct disproportionality indices (Gallagher and Mitchell 2005).

- 15.
Our decade dummies are defined for the periods 1940–1949, 1950–1959,…, 2000–2009. We replicated our analysis defining decade dummies for all possible periods (e.g., 1948–1957,…) and obtained results that were statistically indistinguishable from those presented.

- 16.
In the results presented, we include polynomials of all orders up to 6 in \(D_{ct}\) and \(\log D_{ct}\) (i.e., 12 additional covariates). As a robustness check, we replicated our analysis with polynomials of all orders up to 10 and obtained qualitatively similar and precise estimates of our coefficients of interest.

- 17.
We specify flexible controls for the number of parties in an analogous manner to the number of districts (i.e., 12 additional covariates).

- 18.
The total numbers of entries, exits, movements, districts and parties are all considered continuous variables. By specifying these variables in logarithms, we mitigate measurement error by ensuring that electoral systems with many parties (which tend to be more proportional, per the static results) do not simply exhibit a large amount of partisan churn by construction. Rather, any such relationship between proportionality and partisan churn should be interpreted as independent of the total number of parties. We provide further support for this interpretation by flexibly controlling for the number of parties in some specifications. Because elections may feature zero entries, exits or net movements, we transform these variables as \(\log \left( 1+x\right)\) in order to conserve data.

- 19.
We present results with and without flexible controls for the number of parties because the inclusion of these controls may adversely affect the interpretation of the relationship of interest when using the disproportionality index.

- 20.
In Tables B2 and B3 of the supplementary appendix, we reproduce these results using total party entries and total party exits as the dependent variables respectively. We find broadly consistent and robust results.

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## Acknowledgments

We thank Catherine Bobtcheff, Daniel Diermeier, Carlo Prato, seminar audiences at Carleton, UQAM and Ryerson, conference participants at CPEG 2013, Elections and Electoral Institutions IAST 2014, MPSA 2014, Formal Theory and Comparative Politics 2015, and Scott Legree for excellent research assistance. Finally, the editors and two referees provided excellent comments and suggestions.

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Forand, J.G., Maheshri, V. A dynamic Duverger’s law.
*Public Choice* **165, **285–306 (2015). https://doi.org/10.1007/s11127-016-0309-8

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### Keywords

- Duverger’s law
- Electoral rules
- Party entry and exit

### JEL Classification

- D72
- C73