Kociołek i chochelka: formuła szacowania rozkładu mandatów metodą Jeffersona-D’Hondta

Jarosław Flis, Wojciech Słomczyński, Dariusz Stolicki

Abstrakt


Przedstawiamy prostą, acz nową formułę szacowania rozkładu mandatów i odchylenia od proporcjonalności dla systemów wyborczych, w których alokacja mandatów odbywa się metodą Jeffersona-D’Hondta (JDH). Bazuje ona wyłącznie na ogólnokrajowych udziałach głosów i ustalonych parametrach danego systemu wyborczego. Zaproponowany przez nas wzór wyjaśnia odchylenie od proporcjonalności jako zjawisko zależne zarówno od liczby partii, jak i od liczby okręgów wyborczych. Pokazujemy, że zapewnia on dobre przybliżenie podziału mandatów, nawet w przypadku niewielkich naruszeń leżących u jego podstaw założeń. W tym celu przeprowadziliśmy testy empiryczne naszego wzoru na danych wyborczych z wszystkich dziewięciu krajów członkowskich Unii Europejskiej, w których podział mandatów w wyborach parlamentarnych odbywa się metodą Jeffersona-D’Hondta. Omawiamy zastosowania naszej formuły do modelowania efektów przesunięć głosów, konsolidacji i fragmentacji partii, tzw. spoiler effects, inżynierii wyborczej, progów ustawowych oraz gerrymanderingu. Ponieważ nie wymaga ona znajomości wyników wyborów na poziomie okręgów, umożliwia łatwiejsze prowadzenie symulacji wyborczych z wykorzystaniem metody JDH.

Słowa kluczowe


metoda Jeffersona-D’Hondta, zależność seats-votes, odchylenie od proporcjonalności, reprezentacja proporcjonalna

Pełny tekst / Download full text:

PDF

Bibliografia


ACE Project (2019). Electoral system (Chamber 1). ACE Electoral Knowledge Network. http://aceproject.org/epic-en/CDMap?question=ES005&f=.

Baldini, G., Pappalardo, A. (2009). Elections, electoral systems and volatile voters. Basingstoke: Palgrave Macmillan.

Balinski, M.L., Young, H.P. (1978a). The Jefferson method of apportionment. SIAM Review, 20(3), 278–284, doi: 10.1137/1020040.

Balinski, M.L., Young, H.P. (1978b). Stability, coalitions and schisms in proportional representation systems. American Political Science Review, 72(3), 848–858, doi: 10.2307/ 1955106.

Balinski, M.L., Young, H.P. (2001). Fair representation: meeting the ideal of one man, one vote. Washington, DC: Brookings Institution Press.

Barceló, J., Muraoka, T. (2018). The effect of variance in district magnitude on party system infl ation. Electoral Studies, 54, 44–55, doi: 10.1016/j.electstud.2018.04.016.

Benoit, K. (2000). Which electoral formula is the most proportional? A new look with new evidence. Political Analysis, 8(4), 381–388, doi: 10.1093/oxfordjournals.pan.a029822.

Blau, A. (2001). Partisan bias in British general elections. British Elections & Parties Review, 11(1), 46–65, doi: 10.1080/13689880108413053.

Bochsler, D. (2010). Who gains from apparentments under D’Hondt? Electoral Studies, 29(4), 617– 627, doi: 10.1016/j.electstud.2010.06.001.

Bormann, N.C., Golder, M. (2013). Democratic electoral systems around the world, 1946–2011.

Electoral Studies, 32(2), 360-369, doi: 10.1016/j.electstud.2013.01.005.

Brancati, D. (2007). Global Elections Database. http://www.globalelectionsdatabase.com.

Calvo, E., Rodden, J. (2015). The Achilles heel of plurality systems: geography and representation in multiparty democracies. American Journal of Political Science, 59(4), 789–805, doi: 10.1111/ ajps.12167.

Carey, J.M. (2017). Electoral system design in new democracies. W: Herron, E., Pekkanen, R., Shugart, M.S. (red.), Oxford handbook of electoral systems. New York: Oxford UP, 851 11, doi: 10.1093/oxfordhb/9780190258658.001.0001.

Chafee, Z. (1929). Congressional reapportionment. Harvard Law Review, 42(8), 1015–1047, doi:

2307/1331072.

Colomer, J.M. (2004). The handbook of electoral system choice. London: Palgrave Macmillan.

Deza, M.M., Deza, E. (2014) Encyclopedia of distances. Heidelberg: Springer.

Dančišin, V. (2013) Hľadanie volebného deliteľa Victorom D’Hondtom. European Electoral Studies, 10(1), 63–70.

D’Hondt, V. (1882). Système pratique et raisonné de représentation proportionnelle. Bruxelles: Libraire C. Muquardt, doi: 10.3931/e-rara-39876.

D’Hondt, V. (1883). Formule du minimum dans la représentation proportionnelle. Moyen facile de trouver le diviseur. Représentation proportionnelle. Revue mensuelle, 2, 117–128, 129–130.

D’Hondt, V. (1885). Exposé du système pratique de représentation proportionnelle. Adopté par le Comité de l‘Association Réformiste Belge. Gand: Eug. Vanderhaeghen.

Drton, M., Schwingenschlögl, U. (2005). Asymptotic seat bias formulae. Metrika, 62(1), 23–31, doi: 10.1007/s001840400352.

Equer, M. (1911). Relation entre la méthode d‘Hondt et la proportionnalité. La Grande Revue, Deuxième série, 31(10 Jan.), 130–137.

Evci, U.J., Kaminski, M.M. (2019). Shot in the foot: unintended political consequences of electoral engineering in the Turkish parliamentary elections in 2018. ECPR General Conference, Wrocław 2019. https://ecpr.eu/Filestore/PaperProposal/12fcff16-0906-4f9a-ae48-a3757b3cbc40.pdf.

Flis J., Słomczyński W. & Stolicki D. (2019) Seat allocation and seat bias under the Jefferson-D’Hondt System. arXiv: 1805.08291 [physics.soc-ph].

Gfeller, J. (1890). Du transfert des suffrages et de la répartition des sièges complémentaires. Représentation proportionnelle. Revue mensuelle, 9, 120–131.

Gudgin, G., Taylor, J.P. (1979). Seats, votes, and the spatial organisation of elections. London: Pion.

Hagenbach-Bischoff, E. (1888). Die Frage der Einführung einer Proportionalvertretung statt des absoluten Mehres. Basel: H. Georg.

Hagenbach-Bischoff, E. (1905). Die Verteilungsrechnung beim Basler Gesetz nach dem Grundsatz der Verhältniswahl. Basel: Berichthaus.

Happacher, M., Pukelsheim, F. (1996). Rounding probabilities: unbiased multipliers. Statistics & Decisions, 14(4), 373–382, doi: 10.1524/strm.1996.14.4.373.

Happacher, M., Pukelsheim, F. (2000). Rounding probabilities: maximum probability and minimum complexity multipliers. Journal of Statistical Planning and Inference, 85 (1–2), 145–158, doi:

1016/S0378-3758(99)00077-4.

Humphreys, J.H. (1911). Proportional representation: a study in methods of election. London: Methuen & Co.

Huntington, E.V. (1921). The mathematical theory of the apportionment of representatives. Proceedings of the National Academy of Sciences, 7(4), 123–127.

Huntington, E.V. (1928). The apportionment of representatives in Congress. Transactions of the American Mathematical Society, 30(1), 85–110.

Huntington, E.V. (1931). Methods of apportionment in Congress. American Political Science Review, 25(4), 961–965, doi: 10.2307/1946616.

James, E.J. (1897). The fi rst apportionment of federal representatives in the United States. Annals of the American Academy of Political and Social Science, 9(1), 1–41.

Janson S. (2014), Asymptotic bias of some election methods. Annals of Operations Research, 215(1), 89–136, doi: 10.1007/s10479-012-1141-2.

Jefferson, T. (1792). Opinion on apportionment bill. W: Oberg, B., Looney, J.J. (2008). The Papers of Thomas Jefferson, wersja cyfrowa. Charlottesville: University of Virginia Press.

Joachim, V. (1917). K otázce poměrného zastoupení. Správní obzor, 9(8), 289–298.

Kaminski, M.M. (2001). Coalitional stability of multi-party systems: evidence from Poland. American Journal of Political Science, 45(2), 294–312, doi: 10.2307/2669342.

Kaminski, M.M. (2002). Do parties benefi t from electoral manipulation? Electoral laws and heresthetics in Poland. Journal of Theoretical Politics, 14(3), 325–359, doi: 10.1177/ 095169280201400303.

Kaminski, M.M. (2018). Spoiler effects in proportional representation systems: evidence from eight Polish parliamentary elections, 1991–2015. Public Choice, 176(3–4), 441–460, doi: 10.1007/ s11127-018-0565-x.

Karpov, A. (2015). Alliance incentives under the D’Hondt method. Mathematical Social Sciences, 74(C), 1–7, doi: 10.1016/j.mathsocsci.2014.12.001.

Katz, J.N., King, G. (1999). A statistical model for multiparty electoral data. American Political Science Review, 93(1), 15–32, doi: 10.2307/2585758.

Kollman, K., Hicken, A., Caramani, D., Backer, D., Lublin, D. (2018). Constituency-Level Elections Archive. Ann Arbor: Center for Political Studies, University of Michigan. http://www.electiondataarchive.org.

Leutgäb, P., Pukelsheim, F. (2009). List apparentements in local elections – a lottery. W: Holler, M., Nurmi, H. (red.), Power, voting, and voting power: 30 years after. Berlin: Springer, 489–500, doi:10.1007/978-3-642-35929-3_7.

Li, Y., Shugart, M.S. (2016). The seat product model of the effective number of parties: a case for applied political science. Electoral Studies, 41, 23–43, doi: 10.1016/j.electstud. 2015.10.011.

Lijphart, A. (1990). The political consequences of electoral laws, 1945–85. American Political Science Review, 84(2), 481–496, doi: 10.2307/1963530.

Lijphart, A., Gibberd, R.W. (1977). Thresholds and payoffs in list systems of proportional representation. European Journal of Political Research, 5(3), 219–244, doi: 10.1111/j.1475-6765.1977. tb01289.x.

Linzer, D.A. (2012). The relationship between seats and votes in multiparty systems. Political Analysis, 20(3), 400–416, doi: 10.1093/pan/mps017.

Marshall, A.W., Olkin, I., Pukelsheim, F. (2002). A majorization comparison of apportionment methods in proportional representation. Social Choice and Welfare, 19(4), 885–900, doi: 10.1007/ s003550200164.

McGhee, E. (2014). Measuring partisan bias in single–member district electoral systems. Legislative Studies Quarterly, 39(1), 55–85, doi: 10.1111/lsq.12033.

McGhee, E. (2017). Measuring effi ciency in redistricting. Election Law Journal, 16(4), 417– 442, doi: 10.1089/elj.2017.0453.

Mora, X. (2013). La regla de Jefferson-D’Hondt i les seves alternatives. Materials matemàtics, 2013(4), 1–34.

Morse, M., Von Neumann, J., Eisenhart, L.P. (1948). Report to the President of the National Academy of Sciences.

Palomares, A., Ramírez Gonzáles, V. (2003). Thresholds of the divisor methods. Numerical Algorithms, 34(2), 405–415, doi: 10.1023/B:NUMA.0000005353.82970.ce.

Pavia, J., García-Cárceles, B. (2016). Estimating representatives from election poll proportions: the Spanish case. Statistica Applicata – Italian Journal of Applied Statistics, 25(3), 325–340.

Pólya, G. (1918a). Sur la représentation proportionnelle en matière électorale. L’Enseignement Mathématique, 20, 355–379.

Pólya, G. (1918b). Über die Verteilungssysteme der Proportionalwahl. Zeitschrift für schweizerische Statistik und Volkswirtschaft, 54, 363–387.

Pólya, G. (1919a). Proportionalwahl und Wahrscheinlichkeitsrechnung. Zeitschrift für die gesamte Staatswissenschaft, 74, 297–322.

Pólya, G. (1919b). Über die Systeme der Sitzverteilung bei Proportionalwahl. Wissen und Leben— Schweizerische Halbmonatsschrift, 12, 259–268, 307–312.

Poptcheva, E.M. (2016). Understanding the D’Hondt method. Allocation of parliamentary seats and leadership positions. European Parliamentary Research Service Briefi ng PE 580.901, http://www.europarl.europa.eu/RegData/etudes/BRIE/2016/580901/EPRS_BRI(2016)580 901_ EN.pdf.

Pukelsheim, F. (2014). Proportional representation: apportionment methods and their applications. Heidelberg: Springer.

Pukelsheim, F. (2017). Proportional representation: apportionment methods and their applications. Wyd. 2. Heidelberg: Springer.

Rae, D.W. (1967). The political consequences of electoral laws. New Haven, CT: Yale University Press.

Rae, D.W., Hanby, V.J., Loosemore, J. (1971). Thresholds of representation and thresholds of exclusion. An analytic note on electoral systems. Comparative Political Studies, 3(4), 479–488, doi: 10.1177/001041407100300406.

Rokkan, S. (1968). Elections: electoral systems. W: Sills, D.L. (red.). International encyclopaedia of the social sciences. New York: Crowell-Collier-Macmillan, 5, 6–21.

Sainte-Laguë, A. (1910). La représentation proportionnelle et la méthode des moindres carrés. Annales scientifi ques de l’École Normale Supérieure, Sér. 3(27), 529–542, doi: 10.24033/asens.627.

Shugart, M.S., Taagepera, R. (2017a). Votes from seats: logical models of electoral systems. Cambridge, UK: Cambridge University Press.

Shugart, M.S., Taagepera, R. (2017b). Electoral system effects on party systems. W: Herron, E., Pekkanen, R., Shugart, M.S. (red.). Oxford handbook of electoral systems. New York: Oxford UP, 41–68, doi: 10.1093/oxfordhb/9780190258658.001.0001.

Schuster, K., Pukelsheim, F., Drton, M., Draper, N.R. (2003). Seat biases of apportionment methods for proportional representation. Electoral Studies, 22(4), 651–676, doi: 10.1016/ S02613794(02)00027-6.

Stephanopoulos, N.O., McGhee, E.M. (2015). Partisan gerrymandering and the effi ciency gap. University of Chicago Law Review, 82(2), 831–900.

Stephanopoulos, N.O., McGhee, E.M. (2018). The measure of a metric: the debate over quantifying partisan gerrymandering. Stanford Law Review, 70(5), 1503–1568.

Szpiro, G.G. (2010). Numbers rule: the vexing mathematics of democracy, from Plato to the present. Princeton, NJ: Princeton University Press.

Taagepera, R. (1986). Reformulating the cube law for proportional representation elections. American Political Science Review, 80(2), 489–504, doi: 10.2307/1958270.

Taagepera, R. (2007). Predicting party sizes: the logic of simple electoral systems. Oxford: Oxford University Press.

Taagepera, R., Laakso, M. (1980). Proportionality profi les of West European electoral systems. European Journal of Political Research, 8(4), 423–446, doi: 10.1111/j.14756 765.1980.tb00582.x.

Taagepera, R., Shugart, M.S. (1989). Seats and votes: the effects and determinants of electoral systems. New Haven, CT: Yale University Press.

Tapp, K. (2018). Measuring political gerrymandering. American Mathematical Monthly, 126(7), 593-609, doi: 10.1080/00029890.2019.1609324.

Udina, F., Delicado, P. (2005). Estimating parliamentary composition through electoral polls. Journal of the Royal Statistical Society. Series A (Statistics in Society), 168(2), 387– 399.

Veomett, E. (2018). The effi ciency gap, voter turnout, and the effi ciency principle. Election Law Journal, 17(4), 249-263, doi: 10.1089/elj.2018.0488.




DOI: http://dx.doi.org/10.7206/DEC.1733-0092.129

Refbacks

  • There are currently no refbacks.


This journal provides immediate open access to its content under the Creative Commons BY 4.0 license: https://creativecommons.org/licenses/by/4.0/. Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license.