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Referendum Paradox for Party-List Proportional Representation

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Abstract

We consider two-tiers elections based on closed party-list proportional representation (PLPR), where party platforms involving multiple dichotomous issues are endogenously determined by their supporters’ preferences (via issue-wise simple majority voting). Assuming that voters compare platforms according to the criterion of the Hamming distance and provided a high enough number of voters and issues, we show that the outcome of PLPR may be Pareto dominated by the multiple referendum outcome defined as the issue-wise majority will in the whole electorate. We refer to this situation as the PLPR paradox. We characterize the set of party platforms for which the PLPR paradox is possible. We also investigate several restrictions upon voting situations that may be sufficient for avoiding the paradox.

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Notes

  1. “Representative democracy is at best a working model of direct democracy and is most successful when it generates decisions as close as possible to those that would be generated in a direct democracy. [...] It is direct democracy (actual or ideal) that is used as a measuring rod” (Chamberlin and Courant 1983).

  2. “The system of proportional representation ensures that virtually every constituency in the country will have a hearing in the national and provincial legislatures.” Bishop Desmond Tutu, The Rainbow People of God (1994).

  3. The argument is that the proportional rule is associated with larger constituencies than with the majority election system. This gives fewer possibilities to manipulate their borders. As several seats are assigned to parties proportionally to votes even within a constituency, the borders of a constituency become less relevant. Besides, concentrating public spending benefits in particular areas while spreading the cost among all taxpayers (pork barrelling) may become more difficult.

  4. The analysis of candidates’ strategic behavior in list proportional representation is well-documented and can be considered as part of the literature on incentives in teams. The reader may refer to Myerson (1993, 1999), Carey and Shugart (1995), Persson et al. (2003), Castanheira et al. (2010), Bowler and Farrell (2011), Zittel (2015), Crutzen et al. (2020) and references quoted there.

  5. Note that this procedure satisfies Pareto efficiency (as it chooses the platform that minimizes the total Hamming distance to supporters’ ideals), and is also strategy-proof. See Brams et al. (2007) for details.

  6. We do not specify any rounding method for the parliamentary seat distribution, as none of our results depend on which one is chosen.

  7. We define as consistent a stable partisanship map and discuss in Sect. 4.1 the existence of the paradox for consistent maps that satisfy even stronger properties.

  8. This discards the possibility of parties having no partisan.

  9. This implies assuming that each partisan votes for her party. We comment on this assumption in Sect. 4.

  10. While in this example equalizing the sizes of all parties in \(\mathcal{P} ^{\prime \prime }\) is ensured by creating \(\frac{m}{m_{p}}\) copies of each partisan in each party \(p\in \mathcal{P}^{\prime \prime }\), certain situations may require multiplying \(\frac{m}{m_{p}}\) by \(\alpha >1\) in order to get \(\overrightarrow{0}\) as PLPR outcome. For instance, if one adds 21 parties, each with \(\overrightarrow{1}\) as the platform, \(\overrightarrow{0}\) is the PLPR outcome only if \(\alpha \ge 2\).

  11. Remember we write \(X_{p}=X_{\mathcal{D}_{p}}\) as the profile of partisans’ ideals in party p. In a similar vein, we write here \(F(X_{p^{\prime }\cup \mathcal{H}})\) as the profile of ideals of voters in \(\mathcal{D}_{p^{\prime }}\cup \mathcal{H}\).

  12. “... the portrait is excellent in proportion to its being a good likeness,...the legislature ought to be the most exact transcript of the whole society... the faithful echo of the voices of the people.” (James Wilson, at the Constitutional Convention).

  13. Observe that we assume free entry to and free exit from parties, and that moves are coordinated among partisans of the same party.

  14. One may instead consider the absolute measure of homogeneity defined by \( \max \{d(x_{k},x_{n}):k,n\in \mathcal{D}_{p}\}\). Observe that the map defined in Tables 2 and 3 shows the occurence of the paradox at a map where the distance between any two partisans’ ideal in each party is equal to 2. One can show that 2 is the exact bound if any number of parties is allowed. For a 3-party map to face the paradox, the minimal distance between two ideals is at least 4. The map \((X_{\mathcal{N}}^{*},3,\mathcal{D})\) in Table 6 shows that 4 is the actual lower bound.

  15. See Laffond and Lainé (2000) for an analysis of the referendum paradox in the Arrovian setting where voters have preferences represented by linear orders over a finite set of abstract alternatives.

  16. The Ostrogorski paradox is close in spirit (but not equivalent) to the Anscombe’s paradox (Anscombe 1976; Wagner 1983, 1984; Laffond and Lainé 2013).

  17. See Benoît and Kornhauser (2010) for a generalization to the case where issues are no longer dichotomous.

  18. See Lacy and Niou (2000) and Brams et al. (1997) for the case where platforms involve complementarities between issues.

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Acknowledgements

Funding was provided by CNRS PICS exchange program (Grant No. 08001).

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Authors and Affiliations

  1. Murat Sertel Center for Advanced Economic Studies, Istanbul Bilgi University, Istanbul, Turkey

    Hayrullah Dindar, Gilbert Laffond & Jean Lainé

  2. Lirsa, Conservatoire National des Arts et Métiers, Paris, France

    Jean Lainé

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  1. Hayrullah Dindar
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  2. Gilbert Laffond
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  3. Jean Lainé
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Correspondence to Jean Lainé.

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Dindar, H., Laffond, G. & Lainé, J. Referendum Paradox for Party-List Proportional Representation. Group Decis Negot 30, 191–220 (2021). https://doi.org/10.1007/s10726-020-09713-y

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