Advertisement

A Probabilistic Model for Detecting Gerrymandering in Partially-Contested Multiparty Elections

  • Dariusz StolickiEmail author
  • Wojciech Słomczyński
  • Jarosław Flis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11890)

Abstract

Classic methods for detecting gerrymandering fail in multiparty partially-contested elections, such as the Polish local election of 2014. A new method for detecting electoral bias, based on the assumption that voting is a stochastic process described by Pólya’s urn model, is devised to overcome these difficulties. Since the partially-contested character of the election makes it difficult to estimate parameters of the urn model, an ad-hoc procedure for estimating those parameters in a manner untainted by potential gerrymandering is proposed.

Keywords

Gerrymandering Partially-contested elections Pólya’s urn model Dirichlet distribution Seats-votes relationship 

References

  1. 1.
    Altman, M.: Modeling the effect of mandatory district compactness on partisan gerrymanders. Polit. Geogr. 17(8), 989–1012 (1998).  https://doi.org/10.1016/S0962-6298(98)00015-8CrossRefGoogle Scholar
  2. 2.
    Altman, M., Amos, B., McDonald, M.P., Smith, D.A.: Revealing preferences: why gerrymanders are hard to prove, and what to do about it. Technical report, SSRN 2583528, March 2015. https://doi.org/10.2139/ssrn.2583528
  3. 3.
    Altman, M., McDonald, M.P.: The promise and perils of computers in redistricting. Duke J. Const. Law Public Policy 5(1), 69–159 (2010)Google Scholar
  4. 4.
    Ansolabehere, S., Leblanc, W.: A spatial model of the relationship between seats and votes. Math. Comput. Model. 48(9–10), 1409–1420 (2008).  https://doi.org/10.1016/j.mcm.2008.05.028MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Apollonio, N., Becker, R.I., Lari, I., Ricca, F., Simeone, B.: The sunfish against the octopus: opposing compactness to gerrymandering. In: Simeone, B., Pukelsheim, F. (eds.) Mathematics and Democracy: Recent Advances in Voting Systems and Collective Choice, pp. 19–41. Springer, Berlin-Heidelberg (2006).  https://doi.org/10.1007/3-540-35605-3_2CrossRefGoogle Scholar
  6. 6.
    Aras, A., Costantini, M., van Erkelens, D., Nieuweboer, I.: Gerrymandering in three-party elections under various voting rules (2017)Google Scholar
  7. 7.
    Athreya, K.B.: On a characteristic property of Polya’s urn. Stud. Sci. Math. Hung. 4, 31–35 (1969)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bachrach, Y., Lev, O., Lewenberg, Y., Zick, Y.: Misrepresentation in district voting. In: Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, pp. 81–87 (2016)Google Scholar
  9. 9.
    Balinski, M.L., Demange, G.: Algorithms for proportional matrices in reals and integers. Math. Program. 45(1–3), 193–210 (1989).  https://doi.org/10.1007/BF01589103MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Balinski, M.L., Demange, G.: An axiomatic approach to proportionality between matrices. Math. Oper. Res. 14(4), 700–719 (1989).  https://doi.org/10.1287/moor.14.4.700MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Barthélémy, F., Martin, M., Piggins, A.: The architecture of the Electoral College, the house size effect, and the referendum paradox. Electoral. Stud. 34, 111–118 (2014).  https://doi.org/10.1016/j.electstud.2013.07.004CrossRefGoogle Scholar
  12. 12.
    Berg, S.: Paradox of voting under an urn model: the effect of homogeneity. Public Choice 47(2), 377–387 (1985).  https://doi.org/10.1007/BF00127533CrossRefGoogle Scholar
  13. 13.
    Blais, A.: Turnout in elections. In: Dalton, R.E., Klingemann, H.D. (eds.) Oxford Handbook of Political Behavior, pp. 621–635. Oxford University Press, Oxford (2007).  https://doi.org/10.1093/oxfordhb/9780199270125.003.0033CrossRefGoogle Scholar
  14. 14.
    Brookes, R.H.: Electoral distortion in New Zealand. Aust. J. Polit. Hist. 5(2), 218–223 (1959).  https://doi.org/10.1111/j.1467-8497.1959.tb01197.xCrossRefGoogle Scholar
  15. 15.
    Butler, D.E.: Appendix. In: Nicholas, H.G. (ed.) The British General Election of 1950, pp. 306–333. Macmillan, London (1951)Google Scholar
  16. 16.
    Cain, B.E.: Assessing the partisan effects of redistricting. Am. Polit. Sci. Rev. 79(02), 320–333 (1985).  https://doi.org/10.2307/1956652CrossRefGoogle Scholar
  17. 17.
    Campbell, C.D., Tullock, G.: A measure of the importance of cyclical majorities. Econ. J. 75(300), 853 (1965).  https://doi.org/10.2307/2229705CrossRefGoogle Scholar
  18. 18.
    Chen, J., Rodden, J.: Unintentional gerrymandering: political geography and electoral bias in legislatures. Q. J. Polit. Sci. 8(3), 239–269 (2013).  https://doi.org/10.1561/100.00012033CrossRefGoogle Scholar
  19. 19.
    Chen, J., Rodden, J.: Cutting through the thicket: redistricting simulations and the detection of partisan gerrymanders. Election Law J. Rules Politics Policy 14(4), 331–345 (2015).  https://doi.org/10.1089/elj.2015.0317CrossRefGoogle Scholar
  20. 20.
    Cirincione, C., Darling, T.A., O’Rourke, T.G.: Assessing South Carolina’s 1990s congressional districting. Polit. Geogr. 19(2), 189–211 (2000).  https://doi.org/10.1016/S0962-6298(99)00047-5CrossRefGoogle Scholar
  21. 21.
    Coleman, J.S.: Introduction to Mathematical Sociology. Free Press, New York (1964)Google Scholar
  22. 22.
    Eggenberger, F., Pólya, G.: über die Statistik verketteter Vorgänge. Zeitschrift für Angewandte Mathematik und Mechanik 3(4), 279–289 (1923).  https://doi.org/10.1002/zamm.19230030407CrossRefzbMATHGoogle Scholar
  23. 23.
    Enelow, J.M., Hinich, M.J.: The Spatial Theory of Voting: An Introduction. Cambridge University Press, Cambridge (1984)zbMATHGoogle Scholar
  24. 24.
    Engstrom, R.L., Wildgen, J.K.: Pruning thorns from the thicket: an empirical test of the existence of racial gerrymandering. Legis. Stud. Q. 2(4), 465–479 (1977).  https://doi.org/10.2307/439420CrossRefGoogle Scholar
  25. 25.
    Fifield, B., Higgins, M., Imai, K.: A new automated redistricting simulator using Markov Chain Monte Carlo. Working Paper, Princeton University, Princeton, NJ (2015)Google Scholar
  26. 26.
    Foos, F., John, P.: Parties are no civic charities: voter contact and the changing partisan composition of the electorate. Polit. Sci. Res. Methods 6(2), 283–298 (2018).  https://doi.org/10.1017/psrm.2016.48CrossRefGoogle Scholar
  27. 27.
    Friedman, J.N., Holden, R.T.: Optimal gerrymandering: sometimes pack, but never crack. Am. Econ. Rev. 98(1), 113–144 (2008).  https://doi.org/10.1257/aer.98.1.113CrossRefGoogle Scholar
  28. 28.
    Fryer, R.G., Holden, R.: Measuring the compactness of political districting plans. J. Law Econ. 54(3), 493–535 (2011).  https://doi.org/10.1086/661511CrossRefGoogle Scholar
  29. 29.
    Garand, J.C., Parent, T.W.: Representation, swing, and bias in U.S. Presidential Elections, 1872–1988. Am. J. Polit. Sci. 35(4), 1011 (1991).  https://doi.org/10.2307/2111504
  30. 30.
    Geenens, G.: Probit transformation for kernel density estimation on the unit interval. J. Am. Stat. Assoc. 109(505), 346–358 (2014).  https://doi.org/10.1080/01621459.2013.842173MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Gehrlein, W.V., Fishburn, P.C.: Condorcet’s paradox and anonymous preference profiles. Public Choice 26(1), 1–18 (1976).  https://doi.org/10.1007/BF01725789CrossRefGoogle Scholar
  32. 32.
    Gelman, A., King, G.: Estimating the electoral consequences of legislative redistricting. J. Am. Stat. Assoc. 85(410), 274–282 (1990)CrossRefGoogle Scholar
  33. 33.
    Gelman, A., King, G.: A unified method of evaluating electoral systems and redistricting plans. Am. J. Polit. Sci. 38(2), 514–554 (1994)CrossRefGoogle Scholar
  34. 34.
    Gibbs, A.L., Su, F.E.: On choosing and bounding probability metrics. Int. Stat. Rev. 70(3), 419–435 (2002).  https://doi.org/10.1111/j.1751-5823.2002.tb00178.xCrossRefzbMATHGoogle Scholar
  35. 35.
    Grofman, B.: Measures of bias and proportionality in seats-votes relationships. Polit. Methodol. 9(3), 295–327 (1983)MathSciNetGoogle Scholar
  36. 36.
    Grofman, B.: Criteria for districting: a social science perspective. UCLA Law Rev. 33(1), 77–184 (1985)Google Scholar
  37. 37.
    Grofman, B., King, G.: The future of partisan symmetry as a judicial test for partisan gerrymandering after LULAC v. Perry. Election Law J. Rules Polit. Policy 6(1), 2–35 (2007).  https://doi.org/10.1089/elj.2006.6002
  38. 38.
    Gudgin, G., Taylor, P.J.: Seats, Votes, and the Spatial Organisation of Elections. Pion, London (1979)Google Scholar
  39. 39.
    Issacharoff, S.: Gerrymandering and political cartels. Harvard Law Rev. 116(2), 593–648 (2002).  https://doi.org/10.2307/1342611CrossRefGoogle Scholar
  40. 40.
    Jamison, D., Luce, E.: Social homogeneity and the probability of intransitive majority rule. J. Econ. Theory 5(1), 79–87 (1972)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Johnson, N.L., Kotz, S.: Urn Models and Their Applications: An Approach to Modern Discrete Probability Theory. Wiley, New York (1977)zbMATHGoogle Scholar
  42. 42.
    Johnston, R.: Manipulating maps and winning elections: measuring the impact of malapportionment and gerrymandering. Polit. Geogr. 21(1), 1–31 (2002).  https://doi.org/10.1016/S0962-6298(01)00070-1CrossRefGoogle Scholar
  43. 43.
    Johnston, R.J.: Political, Electoral, and Spatial Systems: An Essay in Political Geography. Contemporary Problems in Geography. Clarendon Press; Oxford University Press, Oxford; New York (1979)Google Scholar
  44. 44.
    Karvanen, J.: Estimation of quantile mixtures via L-moments and trimmed L-moments. Comput. Stat. Data Anal. 51(2), 947–959 (2006).  https://doi.org/10.1016/j.csda.2005.09.014MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Katz, J.N., King, G.: A statistical model for multiparty electoral data. Am. Polit. Sci. Rev. 93(1), 15–32 (1999).  https://doi.org/10.2307/2585758CrossRefGoogle Scholar
  46. 46.
    Kendall, M.G., Stuart, A.: The law of the cubic proportion in election results. Br. J. Sociol. 1(3), 183–196 (1950).  https://doi.org/10.2307/588113CrossRefGoogle Scholar
  47. 47.
    King, G., Browning, R.X.: Democratic representation and partisan bias in Congressional elections. Am. Polit. Sci. Rev. 81(4), 1251 (1987).  https://doi.org/10.2307/1962588CrossRefGoogle Scholar
  48. 48.
    Kuga, K., Nagatani, H.: Voter antagonism and the paradox of voting. Econometrica 42(6), 1045–1067 (1974).  https://doi.org/10.2307/1914217MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Lepelley, D., Merlin, V., Rouet, J.L.: Three ways to compute accurately the probability of the referendum paradox. Math. Soc. Sci. 62(1), 28–33 (2011).  https://doi.org/10.1016/j.mathsocsci.2011.04.006MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Linzer, D.A.: The relationship between seats and votes in multiparty systems. Polit. Anal. 20(3), 400–416 (2012).  https://doi.org/10.1093/pan/mps017CrossRefGoogle Scholar
  51. 51.
    Mattingly, J.C., Vaughn, C.: Redistricting and the will of the people. Technical report, arXiv: 1410.8796 [physics.soc-ph], October 2014
  52. 52.
    McClurg, S.D.: The electoral relevance of political talk: examining disagreement and expertise effects in social networks on political participation. Am. J. Polit. Sci. 50(3), 737–754 (2006).  https://doi.org/10.1111/j.1540-5907.2006.00213.xCrossRefGoogle Scholar
  53. 53.
    McGhee, E.: Measuring partisan bias in single-member district electoral systems: measuring partisan bias. Legis. Stud. Q. 39(1), 55–85 (2014).  https://doi.org/10.1111/lsq.12033MathSciNetCrossRefGoogle Scholar
  54. 54.
    Mcleod, J.M., Scheufele, D.A., Moy, P.: Community, communication, and participation: the role of mass media and interpersonal discussion in local political participation. Polit. Commun. 16(3), 315–336 (1999).  https://doi.org/10.1080/105846099198659CrossRefGoogle Scholar
  55. 55.
    Merrill, S.: A comparison of efficiency of multicandidate electoral systems. Am. J. Polit. Sci. 28(1), 23–48 (1984).  https://doi.org/10.2307/2110786CrossRefGoogle Scholar
  56. 56.
    Niemi, R.G.: Relationship between votes and seats: the ultimate question in political gerrymandering. UCLA Law Rev. 33, 185–212 (1985)Google Scholar
  57. 57.
    Niemi, R.G., Deegan, J.: A theory of political districting. Am. Polit. Sci. Rev. 72(4), 1304–1323 (1978).  https://doi.org/10.2307/1954541CrossRefGoogle Scholar
  58. 58.
    Niemi, R.G., Fett, P.: The swing ratio: an explanation and an assessment. Legis. Stud. Q. 11(1), 75–90 (1986).  https://doi.org/10.2307/439910CrossRefGoogle Scholar
  59. 59.
    Nurmi, H.: Voting Paradoxes and How to Deal with Them. Springer, New York (1999).  https://doi.org/10.1007/978-3-662-03782-9CrossRefzbMATHGoogle Scholar
  60. 60.
    O’Loughlin, J.: The identification and evaluation of racial gerrymandering. Ann. Assoc. Am. Geogr. 72(2), 165–184 (1982).  https://doi.org/10.1111/j.1467-8306.1982.tb01817.xCrossRefGoogle Scholar
  61. 61.
    Penrose, L.S.: On the Objective Study of Crowd Behaviour. H.K. Lewis, London (1952)Google Scholar
  62. 62.
    Polsby, D.D., Popper, R.D.: The third criterion: compactness as a procedural safeguard against partisan gerrymandering. Yale Law Policy Rev. 9(2), 301–353 (1991)Google Scholar
  63. 63.
    Pólya, G.: Sur quelques points de la théorie des probabilités. Ann. l’inst. Henri Poincaré 1(2), 117–161 (1930)Google Scholar
  64. 64.
    Pukelsheim, F.: Biproportional scaling of matrices and the iterative proportional fitting procedure. Ann. Oper. Res. 215(1), 269–283 (2014).  https://doi.org/10.1007/s10479-013-1468-3MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Regenwetter, M., Grofman, B., Tsetlin, I., Marley, A.: Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  66. 66.
    Sano, F., Hisakado, M., Mori, S.: Mean field voter model of election to the House of Representatives in Japan. In: APEC-SSS2016, p. 011016. Journal of the Physical Society of Japan (2017).  https://doi.org/10.7566/JPSCP.16.011016
  67. 67.
    Stanton, R.G.: A result of Macmahon on electoral predictions. Ann. Discrete Math. 8, 163–167 (1980).  https://doi.org/10.1016/S0167-5060(08)70866-5MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Steerneman, T.: On the total variation and hellinger distance between signed measures; an application to product measures. Proc. Am. Math. Soc. 88(4), 684–688 (1983).  https://doi.org/10.2307/2045462MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Stephanopoulos, N.O., McGhee, E.M.: Partisan gerrymandering and the efficiency gap. Univ. Chicago Law Rev. 82(2), 831–900 (2015)Google Scholar
  70. 70.
    Szufa, S.: Optimal gerrymandering for simplified districts. Working Paper, Jagiellonian Center for Quantitative Research in Political Science, Kraków (2019)Google Scholar
  71. 71.
    Tideman, T.N., Plassmann, F.: Modeling the outcomes of vote-casting in actual elections. In: Felsenthal, D.S., Machover, M. (eds.) Electoral Systems. Paradoxes, Assumptions, and Procedures, pp. 217–251. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-20441-8_9CrossRefGoogle Scholar
  72. 72.
    Tomz, M., Tucker, J.A., Wittenberg, J.: An easy and accurate regression model for multiparty electoral data. Polit. Anal. 10(1), 66–83 (2002).  https://doi.org/10.1093/pan/10.1.66CrossRefGoogle Scholar
  73. 73.
    Tufte, E.R.: The relationship between seats and votes in two-party systems. Am. Polit. Sci. Rev. 67(2), 540–554 (1973).  https://doi.org/10.2307/1958782CrossRefGoogle Scholar
  74. 74.
    Upton, G.: Blocks of voters and the cube ‘law’. Br. J. Polit. Sci. 15(3), 388–398 (1985).  https://doi.org/10.1017/S0007123400004257CrossRefGoogle Scholar
  75. 75.
    Wildgen, J.K., Engstrom, R.L.: Spatial distribution of partisan support and the seats/votes relationship. Legis. Stud. Q. 5(3), 423–435 (1980).  https://doi.org/10.2307/439554CrossRefGoogle Scholar
  76. 76.
    Yamamoto, T.: A multinomial response model for varying choice sets, with application to partially contested multiparty elections. Working Paper, Massachusetts Institute of Technology, Cambridge, MA (2014)Google Scholar
  77. 77.
    Young, H.P.: Measuring the compactness of legislative districts. Legis. Stud. Q. 13(1), 105–115 (1988).  https://doi.org/10.2307/439947CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Dariusz Stolicki
    • 1
    • 2
    Email author
  • Wojciech Słomczyński
    • 1
    • 3
  • Jarosław Flis
    • 1
    • 4
  1. 1.Jagiellonian Center for Quantitative Research in Political ScienceJagiellonian UniversityKrakówPoland
  2. 2.Faculty of International and Political StudiesJagiellonian UniversityKrakówPoland
  3. 3.Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  4. 4.Faculty of Management and Social CommunicationJagiellonian UniversityKrakówPoland

Personalised recommendations