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2014 | OriginalPaper | Chapter

Satisfaction Approval Voting

Authors : Steven J. Brams, D. Marc Kilgour

Published in: Voting Power and Procedures

Publisher: Springer International Publishing

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Abstract

We propose a new voting system, satisfaction approval voting (SAV), for multiwinner elections, in which voters can approve of as many candidates or as many parties as they like. However, the winners are not those who receive the most votes, as under approval voting (AV), but those who maximize the sum of the satisfaction scores of all voters, where a voter’s satisfaction score is the fraction of his or her approved candidates who are elected. SAV may give a different outcome from AV—in fact, SAV and AV outcomes may be disjoint—but SAV generally chooses candidates representing more diverse interests than does AV (this is demonstrated empirically in the case of a recent election of the Game Theory Society). A decision-theoretic analysis shows that all strategies under SAV, except approving of a least-preferred candidate, are undominated, so voters may rationally choose to approve of more than one candidate. In party-list systems, SAV apportions seats to parties according to the Jefferson/d’Hondt method with a quota constraint, which favors large parties and gives an incentive to smaller parties to coordinate their policies and forge alliances, even before an election, that reflect their supporters’ coalitional preferences.

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previous chapter Postulates and Paradoxes of Voting Power in a Noncooperative Setting

next chapter The Structure of Voters’ Preferences Induced by the Dual Culture Condition

Merrill III and Nagel (1987) were the first to distinguish between approval balloting, in which voters can approve of one or more candidates, and approval voting (AV), a method for aggregating approval ballots. SAV, as we will argue, is a method of aggregation that tends to elect candidates in multiwinner elections who are more representative of the entire electorate than those elected by AV, who are simply the most popular candidates.

Representing this diversity is not the issue when electing a single winner, such as a mayor, governor, or president. In such an election, the goal is to find a consensus choice, and we believe that AV is better suited than SAV to satisfy this goal. Scoring rules, in which voters rank candidates and scores are associated with the ranks, may also serve this end, but the optimal scoring rule for achieving particular standards of justice (utilitarianism, maximin, or maximax) is sensitive to the distribution of voter utilities (Apesteguia et al. 2011).

The latter kind of responsiveness would be reinforced if voters, in addition to being able to approve of one or more parties, could use SAV to choose a party’s nominees.

More speculatively, SAV may reduce a multiparty system to two competing coalitions of parties. The majority coalition winner would then depend, possibly, on a centrist party that can swing the balance in favor of one coalition or the other. Alternatively, a third moderate party (e.g., Kadima in Israel) might emerge that peels away supporters from the left and the right. In general, SAV is likely to make coalitions more fluid and responsive to popular sentiment.

An interesting modification of this measure was suggested by Kilgour and Marshall (2011) to apply when a voter approves of more candidates than are to be elected: Change the denominator of the satisfaction measure from |V _i| to min{|V _i|, k}. Thus, for example, if voter i approves of 3 candidates, but only k = 2 can be elected, i’s satisfaction would be 2/2 (rather than 2/3) whenever any two of his or her approved candidates are elected. This modification ensures that a voter’s influence on the election is not diluted if he or she approves of more candidates than are to be elected, but it does not preserve other properties of SAV.

We use ab to indicate the strategy of approving of the subset {a, b}, but we use {a, b} to indicate the outcome of a voting procedure. Later we drop the set-theoretic notation, but the distinction between voter strategies and election outcomes is useful for now.

Arguably, candidates c and d benefit under SAV by getting bullet votes from their supporters. While their supporters do not share their approval with other candidates, their election gives representation to a majority of voters, whereas AV does not.

Two of these systems—proportional AV and sequential proportional AV—assume that a voter’s satisfaction is marginally decreasing—the more of his or her approved candidates are elected, the less satisfaction the voter derives from having additional approved candidates elected. See http://www.nationmaster.com/encyclopedia/Proportional-approval-voting; http://www.nationmaster.com/encyclopedia/Sequential-proportional-approval-voting for a description and examples of these two systems, and Alcalde-Unzu and Vorsatz (2009) for an axiomatic treatment of systems in which the points given to a candidate are decreasing in the number of candidates of whom the voter approves, which they call “size approval voting.” More generally, see Kilgour (2010) and Kilgour and Marshall (2011) for a comparison of several different approval-ballot voting systems that have been proposed for the election of multiple winners, all of which may give different outcomes.

By contrast, under cumulative voting (CV), a voter can divide his or her votes—or, equivalently, a single vote—unequally, giving more weight to some candidates than others. However, equal and even cumulative voting (EaECV), which restricts voters to casting the same number of votes for all candidates whom they support, is equivalent to SAV, though its connection to voter satisfaction, as far as we know, has not previously been demonstrated. While CV and EaECV have been successfully used in some small cities in the United States to give representation to minorities on city councils, it seems less practicable in large elections, including those in countries with party-list systems in which voters vote for political parties (Sect. 5). See http://en.wikipedia.org/wiki/Cumulative_voting for additional information on cumulative voting.

Technically, the problem is NP hard (http://en.wikipedia.org/wiki/NP-hard), because it is equivalent to the hitting-set problem, which is a version of the vertex-covering problem (http://en.wikipedia.org/wiki/Vertex_cover) discussed in Karp (1972). Under SAV, as we showed at the beginning of this section, the satisfaction-maximizing subset of, say, k candidates can be calculated efficiently, as it must contain only candidates with satisfaction scores among the k highest. Because of this feature, the procedure is practical for multiwinner elections with many candidates.

Candidates a, b, and c receive, respectively, 10, 9, and 8 votes; the greedy algorithm first selects a (10 votes) and then b (4 votes).

AV-related systems, like proportional AV and sequential proportional AV (see note 8), seem to share AV’s vulnerability, but we do not pursue this question here.

The fact that there is exit from the council after 3 years makes the voting incentives different from a society in which (1) members, once elected, do not leave and (2) members decide who is admitted (Barberà et al. 2001).

Under SAV, whose results we present next, the satisfaction scores of voters in the GTS election are almost uncorrelated with the numbers of candidates they approved of, so the number of candidates approved of does not affect, in general, a voter’s satisfaction score—at least if he or she had voted the same as under AV (a big “if” that we investigate later).

Under the “minimax procedure” (Brams et al. 2007; Brams 2008), 4 of the 12 AV winners would not have been elected. These 4 include the 2 who would not have been elected under SAV; they would have been replaced by 2 who would have been elected under SAV. Thus, SAV partly duplicates the minimax outcome. It is remarkable that these two very different systems agree, to an extent, on which candidates to replace to make the outcome more representative.

We are grateful to Richard F. Potthoff for writing an integer program that gave the results for the GTS election that we report on next.

Notice that the numbers of votes shown in a contingency are all within 1 of each other, enabling a voter’s strategy to be decisive; these numbers need not sum to an integer, even though the total number of voters and votes sum to an integer. For example, contingency 4 can arise if there are 2 ab voters and 1 ac voter, giving satisfaction scores of 3/2, 1, and ½, respectively, to a, b, and c, which sum to 3. But this is equivalent to contingency 4 (1, ½, 0), obtained by subtracting ½ from each candidate’s score, whose values do not sum to an integer. Contingencies of the form (1, ½, ½), while feasible, are not included, because they are equivalent to contingencies of the form (½, 0, 0)—candidate a is ½ vote ahead of candidates b and c.

We have not shown contingencies in which any candidate is guaranteed a win or a loss. The 19 contingencies in Table 1 represent all states in which the strategy of a voter can make each of the three candidates a winner or a loser, rendering them 3-candidate competitive contingencies.

If there were a minimum number of votes (e.g., a simple majority) that a candidate needs in order to win, then abstention or approving of everybody could matter. But here we assume the two candidates with the most votes win, unless there is a tie, in which case we assume there is an (unspecified) tie-breaking rule.

Depending on the tie-breaking rule, the focal voter may have strict preferences over these outcomes, too. Because each allows for the possibility of any pair of winning candidates, we chose not to distinguish them. To be sure, a − b/c (second best) and c − a/b (second worst) also allow for the possibility of any pair of winning candidates, but the fact that the first involves the certain election of a, and the second the certain election of c, endows them with, respectively, a more-preferred and less-preferred status than the three outcomes among which the focal voter is indifferent.

To the degree that voters have relatively complete information on the standing of candidates (e.g., from polls), they can identify the most plausible contingencies and better formulate optimal strategies, taking into account the likely optimal strategies of voters with opposed preferences. In this situation, a game-theoretic model would be more appropriate than a decision-theoretic model for analyzing the consequences of different voting procedures. We plan to investigate such models in the future. For models of strategic behavior in proportional-representation systems—but not those that use an approval ballot—see Slinko and White (2010).

The Jefferson/d’Hondt method allocates seats sequentially, giving the next seat to the party that maximizes v/(a + 1), where v is its number of voters and a is its present apportionment. Thus, the 1st seat goes to A, because 5 > 4 > 2 when a = 0. Now a = 1 for A and remains 0 for B and C. Because 4/1 > 5/2 > 2/1, B gets the second seat. Now a = 1 for A and B and remains 0 for C. Because 5/2 > 4/2 = 2/1, A gets the third seat, giving an apportionment of (2, 1, 0) to (A, B, C). The divisor method that next-most favors large parties is the Webster/Sainte-Laguë method, under which the party that maximizes v/(a + ½) gets the next seat. After A and B get the first two seats, the third seat goes to C, because 2/(½) > 5/(3/2) > 4/(3/2), so the Webster/Sainte-Laguë method gives an apportionment of (1, 1, 1) to (A, B, C).

There are objective functions with a min/max operator that Jefferson/d’Hondt also optimizes (Balinski and Young 1982/2001, p. 105; Fernández de Córdoba and Penandés 2009), but they are more difficult to justify in the context of seat assignments.

The Jefferson/d’Hondt method with an upper-quota constraint is what Balinski and Young (1982/2001, p. 139) call Jefferson-Quota; SAV effectively provides this constraint. Balinski and Young (1982/2001, ch. 12) argue that because it is desirable that large parties be favored and coalitions encouraged in a parliament, the Jefferson/d’Hondt method should be used, but they do not impose the upper-quota constraint that is automatic under SAV. However, in earlier work (Balinski and Young 1978), they—along with Still (1979)—looked more favorably on such a constraint.

This SAV-based system could be designed for either a closed-list or an open-list system of proportional representation. In a closed-list system, parties would propose an ordering of candidates prior to the election; the results of the election would tell them how far down the list they can go in nominating their upper quotas of candidates. By contrast, in an open-list system, voters could vote for individual candidates; the candidates’ vote totals would then determine their positions on their party lists.

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Title: Satisfaction Approval Voting
Authors: Steven J. Brams
D. Marc Kilgour
Publisher: Springer International Publishing
Book: Voting Power and Procedures
Print ISBN: 978-3-319-05157-4

Electronic ISBN: 978-3-319-05158-1

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-3-319-05158-1_18