Single vote multiple seats elections. Didactics of district versus proportional representation, using the examples of the United Kingdom and The Netherlands

Colignatus, Thomas (2010): Single vote multiple seats elections. Didactics of district versus proportional representation, using the examples of the United Kingdom and The Netherlands.

Abstract

No new issues are discussed but we try to improve on the didactics of some well-known elementary features of multiple seats elections that rely on a single vote such as common elections for Parliament or the U.S. Congress. The didactics concentrate on proportionality versus districts. Since some people in the UK want more proportionality and some people in Holland want more districts, the examples of the UK 2010 and Dutch 2006 general elections are developed in some detail. Subordinate issues are (1) majority versus plurality, and (2) threshold methods versus the mechanisms of highest average, greatest remainder and the principle of Sainte-Laguë & Webster. The latter can be optimal for apportionment of states or districts that will get at least one seat. That kind of optimality can be dubious for political parties. Firstly because a party with a majority in the turnout may miss out on majority in Parliament and secondly since voters for some party A may not want that their vote, if wasted, goes to some party B. A proportional representation of the wasted vote w in total n is also possible by leaving seats empty or by filling the seats and taking a qualified majority f = 1/2 * n / (n - w). We thus should distinguish the mirroring of the proportions in the vote and the mirroring of a majority (and it is not quite true that the first takes care of the latter). For a coalition formed after the elections there is the more complex threshold of a "coalition qualified majority" since the coalition may not always be a solid block. A compromise of proportionality and districts is to allow free (non-district) seats for the overflow. E.g. if half of the seats in Parliament are for single seat districts then the district size can be twice the electoral quota and a district candidate is (ideally) elected when gaining a majority of at least one quota. An algorithm is given that includes such rules and some simulations are shown. A multiple seats election is not quite the same as a series of single seat elections. Direct single seat elections such as for the chief executive (President) are riddled with voting paradoxes. Superior to a single vote are some methods with preference orderings like the Borda Fixed Point but these are somewhat complex. Optimal seems the indirect method where the electorate chooses Parliament in a single vote multiple seats election and that Parliament then applies the complexer preference methods for the single seat election of the Premier. For example, though voters only gave a single vote, David Cameron would be the Borda Fixed Point winner, second to Nick Clegg in a Borda count but still winning in a pairwise vote. It is also explained how to use some new routines in Mathematica. (Adapted with data from The Guardian on the votes per MP.)

• Item Type: MPRA Paper
• Institution: Thomas Cool Consultancy & Econometrics
• Language: English
• Keywords: Political economy; political science; public choice; optimal representation; electoral systems; elections; voting; district; proportional representation; electoral quota; majority; pure threshold; qualified majority; greatest remainder; highest average; Webster; Sainte-Laguë; apportionment; wasted vote; multiple seats; single seat; empty seat; free seat; additional-members system; mixed proportionality; political party; party list; coalition
• Subjects: D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations; C - Mathematical and Quantitative Methods > C8 - Data Collection and Data Estimation Methodology ; Computer Programs > C88 - Other Computer Software; A - General Economics and Teaching > A2 - Economic Education and Teaching of Economics

References

Thomas Colignatus is the preferred name of Thomas Cool in science.

Arrow, K. (1951, 1963), “Social choice and individual values”, J. Wiley

BBC (2010), “Election 2010 website; Results”, retrieved on May 9 2010, http://news.bbc.co.uk/2/shared/election2010/results/

Bielasiak, J. (2010), “Electoral Systems and Political Parties”, retrieved April 28, http://apcentral.collegeboard.com/apc/members/courses/teachers_corner/50299.html

Colignatus (2005), “Definition and Reality in the General Theory of Political Economy”, 2nd edition, Dutch University Press, http://www.dataweb.nl/~cool/Papers/Drgtpe/Index.html

Colignatus (2005a), ““Approval Voting” lacks a sound moral base for the individual voter's choice of approval versus non-approval, especially when the Status Quo is neglected”, ewp-get/0503014, March 26 2005, http://www.dataweb.nl/~cool/Papers/SocialWelfare/ApprovalVoting.pdf

Colignatus (2007), “Voting theory for democracy. Using The Economics Pack Applications of Mathematica for Direct Single Seat Elections”, 2nd edition, Thomas Cool Econometrics & Consultancy, http://www.dataweb.nl/~cool/Papers/VTFD/Index.html

Colignatus (2007a), “In a democracy, Bayrou would have won. Application of the Borda Fixed Point method to the 2007 French presidential elections”, June 27 2007, MPRA 3726, http://mpra.ub.uni-muenchen.de/3726/

Colignatus (2007b), “Why one would accept Voting Theory for Democracy and reject the Penrose Square Root Weights”, July 6 2007, MPRA 3885, http://mpra.ub.uni-muenchen.de/3885/

Colignatus (2008), “Review of Howard DeLong (1991), “A refutation of Arrow’s theorem”, with a reaction, also on its relevance in 2008 for the European Union”, July 22 2008, MPRA 9661, http://mpra.ub.uni-muenchen.de/9661/

Cool (1999, 2001), “The Economics Pack, Applications for Mathematica”, Scheveningen, JEL-99-0820, ISBN 90-804774-1-9 website update 2009, http://www.dataweb.nl/~cool/TheEconomicsPack/index.html

Curtice, J. (2009), “Recent History of Second Preferences”, retreived May 8 2010, http://news.bbc.co.uk/nol/shared/spl/hi/uk_politics/10/alternative_vote/alternative_vote_june_09_notes.pdf

Dahl & Lindblom (1976), “Politics, economics and welfare”, Chicago

DeLong, H. (1991), “A refutation of Arrow’s theorem, University Press of America

Grofman, B. (1985), “A review of macro election systems”, Sozialwissenschaftliches Jahrbuch fur Politiek, Band 4, p303-352, http://www.socsci.uci.edu/~bgrofman/7-Grofman.%20A%20Review%20of%20Macro%20Election%20Systems..pdf

Guardian (2010), “UK election results: data for every candidate in every seat”, retrieved May 18, http://www.guardian.co.uk/news/datablog/2010/may/07/uk-election-results-data-candidates-seats and http://spreadsheets.google.com/ccc?key=tdLut_gO0qo_C0JevIxnZ2g

Kestelman, P. (2005), “Apportionment and Proportionality: A Measured View”, Voting Matters 20, June, http://www.votingmatters.org.uk/MAIN.HTM

Kiesraad (Dutch Electoral Office), http://www.kiesraad.nl, http://www.kiesraad.nl/nl/Onderwerpen/Uitslagen/Uitslagberekening.html, http://www.kiesraad.nl/nl/Onderwerpen/Uitslagen/Toewijzing_zetels.html, http://www.kiesraad.nl/nl/Verkiezingen/(2128)-Verkiezingen-Software_verkiezingen_Tweede_Kamer.html

Malkevitch, J. (2002a), “Apportionment I”, AMS, Feature Column May, http://www.ams.org/samplings/feature-column/fcarc-index

Malkevitch, J. (2002b), “Apportionment II”, AMS, Feature Column June, http://www.ams.org/samplings/feature-column/fcarc-index

Mas-Colell, A., M. Whinston and J. Green (1995), “Microeconomic theory”, Oxford

Mellows-Facer, A. (2006), “General Election 2005”, RESEARCH PAPER 05/33, [Final edition – 10 March 2006], HOUSE OF COMMONS LIBRARY

Moulin, H. (1988), “Axioms of cooperative decision making”, Econometric Society monographs ; 15, CUP

Mueller (1989), “Public Choice II”, Cambridge

Saari, D, (internet), “The symmetry and complexity of elections” (S&C)

Saari, D, (internet), “Explaining all three alternative voting outcomes” (E3)

Saari, D, (internet), “Connecting and resolving Sen's and Arrow's theorems” (C&R)

Saari, D. (2001a), “Chaotic elections”, AMS, 2001, www.ams.org

Saari, D. (2001b), “Decisions and Elections. Explaining the unexpected”, CUP

Sen, A. (1970), “Collective choice and social welfare”, North Holland

Sen, A. (1986), “Social choice theory,” p1073-1181 in Arrow & Intrilligator eds. (1986), “Handbook of mathematical economics, Volume III”, North Holland

Wolfram, S. (1996), “Mathematica 3.0”, Cambridge, www.wolfram.com

Young, H.P. (2004), “Fairness in Apportionment”, Mimeo, Prepared for the U. S. Census Bureau Symposium, http://www.census.gov/history/pdf/Fairness_in_Apportionment_Young.pdf