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Public Choice

Erschienen in:

05.10.2018

Multiwinner approval voting: an apportionment approach

verfasst von: Steven J. Brams, D. Marc Kilgour, Richard F. Potthoff

Erschienen in: Public Choice | Ausgabe 1-2/2019

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Abstract

To ameliorate ideological or partisan cleavages in councils and legislatures, we propose modifications of approval voting in order to elect multiple winners, who may be either individuals or candidates of a political party. We focus on two divisor methods of apportionment, first proposed by Jefferson and Webster, that fall within a continuum of apportionment methods. Our applications of them depreciate the approval votes of voters who have had one or more approved candidates elected and give approximately proportional representation to political parties. We compare a simple sequential rule for allocating approval votes with a computationally more complex simultaneous (nonsequential) rule that, nonetheless, is feasible for many elections. We find that our Webster apportionments tend to be more representative than ours based on Jefferson—by giving more voters at least one representative of whom they approve. But our Jefferson apportionments, with equally spaced vote thresholds that duplicate those of cumulative voting in two-party elections, are more even-handed. By enabling voters to express support for more than one candidate or party, these apportionment methods will tend to encourage coalitions across party or factional lines, thereby diminishing gridlock and promoting consensus in voting bodies.

Vorheriger Artikel The elimination paradox: apportionment in the Democratic Party

Nächster Artikel Titles for me but not for thee: transitional gains trap of property rights extension in Colombia

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Brill et al. (2018), both in its title (“Multiwinner Approval Rules as Apportionment Methods”) and its application of apportionment methods to approval voting, is quite close in spirit to the present paper. But there is almost no overlap in the propositions we prove and those proved in the Brill et al. article. In fact, we were unaware of their paper until a first draft of our paper was completed. Their contributions complement those in our paper, especially in their results on computational complexity (see note 4 later), details on historical background (see note 9 later), and the tie-in to the related contributions of Aziz et al. (2017) on representation (see note 9 later).

Nor have previous studies analyzed procedures for using “wasted votes”—above and beyond the number a candidate needs to win a seat—which is done in Brams and Brill (2018), based on Jefferson’s method of apportionment.

Rather than apportion sequentially, an equivalent method is to find a divisor which, when divided into the vote shares of political parties, yields the number of seats—after some kind of rounding—that each party will receive. The Jefferson method rounds down the exact entitlements (often called “quotas”) of the parties, which are not typically integers, whereas the Webster method rounds in the usual manner (rounding up the exact entitlement if its remainder is equal to or greater than 0.5, rounding down otherwise). For details, see Balinski and Young (2001).

A downside to the nonsequential versions of the apportionment methods is that they are computationally complex, not implementable in polynomial time (Brill et al. 2018). Thus, the required computational effort may become prohibitive if the number of voters or candidates is sufficiently large.

Of course, countries with presidential systems like the United States, wherein most candidates are elected from single-member districts, are highly unlikely to switch to party-list systems. In the United States, in particular, the partisan gerrymandering, on which the US Supreme Court has so far failed to offer a definitive ruling (Liptak 2018), complicates efforts to achieve proportional representation of political parties in both the House of Representatives and in state legislatures, ten of which include some multimember districts that vary in how winners are chosen in them. See https://ballotpedia.org/State_legislative_chambers_that_use_multi-member_districts.

The Webster method is used in four Scandinavian countries, whereas the Jefferson method is used in eight other countries. None of the other three divisor methods currently is used, except for Hill for the US House of Representatives. The nondivisor Hamilton method, also called “largest remainders”, is used in nine countries (Blais and Massicotte 2002; Cox 1997). For a review of apportionment methods, see Edelman (2006a), who proposed a nondivisor method that is described formally in Edelman (2006b). In Sect. 4, we will say more about why we favor the Jefferson and Webster methods for allocating seats to political parties in a parliament.

The deservingness functions of the three other divisor methods are defined similarly. The denominators of the summands are [r(b)(r(b) + 1)]^1/2 for Hill or “equal proportions,” 2r(b)[r(b) + 1]/[2r(b) + 1] for Dean or “harmonic mean,” and simply r(b) for Adams or “smallest divisors” (Balinski and Young 2001; Pukelsheim 2014).

It is interesting to compare sequential Jefferson with satisfaction approval voting (SAV) (Brams and Kilgour 2014). Voters who support only one candidate are treated the same, but voters who support more than one candidate are treated very differently. For a voter who approves of two candidates, SAV gives satisfaction scores of 1/2 to each, whereas sequential Jefferson gives the first candidate elected a score of 1 and the second a score of 1/2. Likewise, the score contributions of a voter who supports three candidates are reduced to 1/3 each under SAV, whereas under sequential Jefferson the first candidate elected receives a score of 1, the second a score of 1/2, and the third a score of 1/3. Thus, under SAV, the incentive of voters to support more than one candidate is weakened as they approve of more and more candidates.

In the parenthetic expressions, the summands begin with 1 and then decline (e.g., from 1 to 1/2 under Jefferson, from 1 to 1/3 under Webster, when a voter has two approved candidates elected). Thus, as with deservingness for the sequential versions of each method, getting a second approved candidate elected does not come close to doubling a voter’s satisfaction score relative to the first. It is worth pointing out that Jefferson and Webster never proposed these weighting sequences for apportioning seats to states in the US House of Representatives. Instead, they proposed an equivalent method in which a divisor of state populations is chosen such that, after rounding, the numbers of seats that all states receive sum to the number of seats in the House. For more on the history of weighting sequences in apportionment, see Brill et al. (2018). What we call the sequential and nonsequential versions of the Jefferson method, in particular, are referred to in the literature as sequential proportional approval voting (SPAV) and proportional approval voting (PAV). Aziz et al. (2017) show that PAV but not SPAV satisfies “justified representation” and “extended justified representation”; but it is possible for SPAV to be more “representative” than PAV, as we show in Example 5 in the “Appendix”.

We have not attempted to compute how often, on average, this would occur, but a computer simulation would shed light on this question. We hasten to add, however, that representativeness is not the be-all and end-all of apportionment; Example 6, used to prove Proposition 3 below, shows that a candidate’s level of support clearly matters.

Assume that one AC voter switches. Then the deservingness score of sequential Jefferson for c₁, after a₁ is elected, is 6, which is maximal (since the score for a₂ drops to 5 1/2) and makes a₁c₁ the outcome. Under nonsequential Jefferson, the maximal satisfaction score is 17 for two outcomes, a₁c₁ and b₁c₁, both of which are more diverse than a₁a₂.

Because f = t(s, k) for some k may be possible, a tie-breaking procedure, which we do not specify, may be required.

The thresholds under Jefferson are the same as those for cumulative voting, whereby voters can spread a fixed number of votes—usually equal to the number of seats to be filled—over one or more candidates. (Brams 2004, ch. 3). One drawback of cumulative voting is that parties must determine how many candidates to run to ensure that their supporters do not spread their votes across too many candidates, which would preclude the party from achieving proportional representation. As we noted earlier, the apportionment methods allow the parties to nominate a full slate of candidates, because they give ever-smaller weights to the election of additional approved candidates. This builds proportional representation into the apportionment method (i.e., without the necessity of strategizing about how many candidates to run), though what is considered “proportional” depends on the method (Jefferson or Webster) used.

For other justifications of Jefferson, see Brill et al. (2018) and references therein. Brill et al. (2017) analyze “lead-balancing” approaches that have similar justifications.

Toplak (2008) has argued that it is not just permissible but constitutionally imperative for apportionment of the US House of Representatives to use numbers of seats that are not integers.

Special cases to which this proposition applies are for the weights of Jefferson (h = 1), Webster (h = 1/2), and Adams (h approaches 0). These are just a few points along a continuum from h = 0 to h = 1 and beyond. Generally speaking, lower values of h afford a greater opportunity for minorities, with fewer voters, to gain representation on an elected voting body.

Emmanuel Macron, under the banner of “En Marche!”, which was later renamed “La République En Marche!”, won the French presidency in 2017 and later National Assembly elections without approval voting, but we think the ascendancy of such more or less centrist candidates and parties is likely to be facilitated by approval voting.

This example, as well as Example 5 (also in the “Appendix”), was found using an integer program.

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Titel: Multiwinner approval voting: an apportionment approach
verfasst von: Steven J. Brams
D. Marc Kilgour
Richard F. Potthoff
Publikationsdatum: 05.10.2018
Verlag: Springer US
Erschienen in: Public Choice / Ausgabe 1-2/2019
Print ISSN: 0048-5829
Elektronische ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-018-0609-2

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