Condorcet Method, Independence of Irrelevant Alternatives, and the Size of the Bundestag

ABSTRACT

How should an excessively large parliament be effectively reduced in size without violating constitutional principles? This is a question that the German Bundestag has discussed since the introduction of the 2013 electoral reform measure. Facing a Bundestag consisting of 709 members and some public dissatisfaction, a reform measure to decrease the parliament’s size was adopted in 2020. However, the 20 Bundestag, elected in September 2021, again increased in size to 735 members. The newly adopted electoral law still uses plurality voting to select the directly elected lawmakers. It is widely known that selection using the plurality rule is flawed because it violates the principle of majority decision. Aside from this aspect, this paper demonstrates that the plurality rule itself contributes to the excessive size of the parliament. By removing the flaws caused by the plurality rule and replacing this voting scheme with a Condorcet method (simple-majority rule), the targeted reduction of the size of the Bundestag can be achieved.

Acknowledgement

The authors owe a great debt of gratitude to two anonymous reviewers. The authors are also indebted to Daniel Schunk and Nils Steiner for their helpful comments on an earlier draft. The project was born during the discussion with seminar participants of the 2021 social choice seminar at the Johannes-Gutenberg University. The authors express their appreciation to the seminar attendees.

Parts of this research were conducted using the supercomputer Mogon and/or advisory services offered by Johannes-Gutenberg University Mainz, which is a member of the AHRP (Alliance for High Performance Computing in Rhineland Palatinate) and the Gauss Alliance e.V. The authors gratefully acknowledge the computing time granted on the supercomputer Mogon at Johannes-Gutenberg University Mainz.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 According to Section 1 (1), sentence 1, Federal Electoral Act, BWahlG.

2 The number of districts required to achieve the targeted seat reduction is set to around 200, implying a reduction of roughly one third of the current number of districts.

3 This statement holds as long as there are Bundestag parties that win no or only very few constituencies relative to their (second) votes. The figure of 598 can only be reached if each party’s first- and second-vote shares are equal.

4 The ruling of July 25, 2012, on the constitutional review of the Nineteenth Act, Amending the Federal Election Law of November 25, 2011 (cf. Hesse 2013).

5 Thorough simulations by Weinmann (2013) have shown that under this election law, it would practically be impossible to meet the standard size of 598 seats, as the lowest number of seats in the entire simulation is 610, even in the case when election results are favourable to the size of the Bundestag.

6 However, the CSU and CDU are treated as different parties, which reduces the effect of this first approach.

7 Specifically, Dasgupta and Maskin (2008); Dasgutpa and Maskin (2014) consider six conditions: the well-known anonymity condition, neutrality, I, the Pareto principle, decisiveness (D), and ordinality (O). They use May (1952)’s definition of neutrality; it does not imply I as the neutrality definition used by Sen (2017) does. Furthermore, they use the Nash-IIA but claim in their follow-up paper (Dasgupta and Maskin 2020) that their results hold for Arrow’s I, too.

8 The term vote splitting is used to describe the situation in which candidate x would beat y in a one-on-one contest but loses to y when z runs too because z splits off some of the votes that otherwise would go to x (see, e.g., Maskin 2020a, 2020b; Sen 2020). The term should not be confused with split-ticket voting. The latter denotes voting for a different party for different contests in the same election (Barnes, Tchintian, and Alles 2017).

9 For the sake of exposition, we ignore possible ties.

10 The authors are indebted to an anonymous reviewer for pointing out this critical issue.

11 The authors are indebted to an anonymous reviewer for pointing out this flaw in our approach.

12 Black stands for the UNION, red for the SPD, violet for the LINKE, and green for the GRÜNE. We use a chocolate colour for the AFD.

13 An alternative/candidate being ranked last by more than 50% of the voters, which is actually the case here, is logically defeated in any head-to-head comparison; thus, this alternative is always a Condorcet loser.

14 We make use of the program Mandatsrechner.

15 Arrow’s independence of irrelevant alternatives is violated if in the collective choice involving two alternatives, anything other than the individual orderings over the tuple gets a place, including the preference intensities (see Sen 2017, Ch. 7). Indeed, the utilitarian approach itself is ruled out by condition I, as demonstrated by Sen (1970).

16 For instance, consider a voter, j, who knows which of the candidates he or she ranks first, but s/he is indifferent regarding all the other candidates. In this case, the voter can place a cross or the number 1 on the line of the ballot paper where his or her favourite candidate is listed. The other candidates, however, receive no ranking number from the voter. Thus, for voter j, nothing changes if the plurality rule is replaced with the simple-majority rule (or the Borda count).

17 A candidate with the most ‘votes’ can be the Condorcet loser. However, letting a Condorcet loser participate in a run-off would be inconsistent. Note that exactly this can happen by applying the absolute majority rule in a run-off.

18 With n candidates, there exist pairs for each anti-symmetric ordering. For each ordering, the number of candidate pairs that are ranked in the same order as in the respective ranking is counted. Each matching pair receives one point. The sum of all these points is called the Kemeny score. The winning ranking is that with the highest Kemeny score. From this ranking, the top-ranked candidate is chosen.

19 Candidates belong to the undominated set if no candidate outside the set is majority-ranked higher than any candidate inside the set. The union of minimal undominated sets is called the GOCHA set (generalised optimal choice set). It consists of those candidates that candidates outside the set did not defeat. The Smith set is similar: it consists of the candidates with the most victories.

Additional information

Notes on contributors

Salvatore Barbaro

Salvatore Barbaro is researching and teaching at the Gutenberg School of Management and Economics of the University of Mainz. His research focuses on social choice theory and public policy in federalist environments. In 2010, Salvatore Barbaro was appointed State Secretary in Rhineland-Palatinate and held this office until 2019. Salvatore Barbaro returned to academia in 2019.

Anna Specht

Anna Specht graduated from the University of Mainz with a B.Sc. in Economics. The paper is an extended version of her bachelor’s thesis. Currently, Anna Specht is working in the communications and marketing department of a global non-profit library service and research organisation.