{"type":"ScholarlyArticle","title":"Nonparametric Detection of Gerrymandering in Multiparty Plurality Elections","authors":[{"name":"Dariusz Stolicki","slug":"dariusz-stolicki","given":"Dariusz","family":"Stolicki","affiliation":"Jagiellonian University , Center for Quantitative Political Science , Kraków"},{"name":"Wojciech Słomczyński","slug":"wojciech-s-omczynski","given":"Wojciech","family":"Słomczyński","affiliation":"Jagiellonian University , Center for Quantitative Political Science , Kraków"},{"name":"Stanisław Szufa","slug":"stanis-aw-szufa","given":"Stanisław","family":"Szufa","affiliation":"AGH University , Kraków; CNRS , LAMSADE , Université Paris Dauphine -PSL"}],"abstract":"Partisan gerrymandering, i.e., manipulation of electoral district boundaries for political advantage, is one of the major challenges to election integrity in modern day democracies. Yet most of the existing methods for detecting partisan gerrymandering are narrowly tailored toward fully contested two-party elections, and fail if there are more parties or if the number of candidates per district varies. We propose a new method, applying nonparametric statistical learning to detect anomalies in the relation between (aggregate) votes and (aggregate) seats. Unlike in most of the existing methods, we propose to learn the standard of fairness in districting from empirical data rather than assume one a priori. Finally, we test the proposed methods against experimental data as well as real-life data from 17 countries employing the plurality (FPTP) system.","doi":"10.24963/ijcai.2024/329","url":"https://ijcai.org/proceedings/2024/329","pdf":"https://ijcai.org/proceedings/2024/0329.pdf","markdown":"https://ijcai.org/proceedings/2024/329.md","bibtex":"https://ijcai.org/proceedings/2024/bibtex/329","volume":{"slug":"2024","title":"Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2024","url":"https://ijcai.org/proceedings/2024/","year":2024,"location":"Jeju, Korea","dates":"3-9 August 2024","editors":"Kate Larson","isbn":"978-1-956792-04-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2024"},"section":"Game Theory and Economic Paradigms","track":"Main Track","number":329,"pages":{"first":2967,"last":2975},"keywords":["Multidisciplinary Topics and Applications: MTA: Social sciences","Game Theory and Economic Paradigms: GTEP: Computational social choice"],"videos":[],"references":[{"position":1,"raw":"[Abramson, 1982] I. S. Abramson. On Bandwidth Variation in Kernel Estimates -A Square Root Law. Annals of Statistics, 10(4):1217-1223, December 1982.","title":"On Bandwidth Variation in Kernel Estimates -A Square Root Law","authors":"I S Abramson","year":"1982","venue":"Annals of Statistics","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Akaike, 1974] H. Akaike. A New Look at the Statistical Model Identification. IEEE Transactions on Automatic Control, 19(6):716-723, December 1974.","title":"A New Look at the Statistical Model Identification","authors":"H Akaike","year":"1974","venue":"IEEE Transactions on Automatic Control","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Autrey et al., 2019] E. Autrey, D. Carter, G. Herschlag, Z. Hunter, and J. C. Mattingly. Metropolized Forest Recombination for Monte Carlo Sampling of Graph Partitions. arXiv: 1911.01503 [cs.DS], 2019.","title":"Metropolized Forest Recombination for Monte Carlo Sampling of Graph Partitions","authors":"E Autrey, D Carter, G Herschlag, Z Hunter, J C Mattingly","year":"2019","venue":null,"doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Barbu and Zhu, 2005] A. Barbu and S.-C. Zhu. Generalizing Swendsen-Wang to Sampling Arbitrary Posterior Probabilities. IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 27(8):1239-1253, August 2005.","title":"Generalizing Swendsen-Wang to Sampling Arbitrary Posterior Probabilities","authors":"A Barbu, S.-C Zhu","year":"2005","venue":"IEEE Transactions on Pattern Analysis and Machine Intelligence","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Bernstein and Duchin, 2017] M. Bernstein and M. Duchin. A Formula Goes to Court: Partisan Gerrymandering and the Ef- ficiency Gap. Notices of the American Mathematical Society, 64(09):1020-1024, October 2017.","title":"A Formula Goes to Court: Partisan Gerrymandering and the Efficiency Gap","authors":"M Bernstein, M Duchin","year":"2017","venue":"Notices of the American Mathematical Society","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Bhattacharyya, 1943] A. K. Bhattacharyya. On a Measure of Di- vergence between Two Statistical Populations Defined by Their Probability Distributions. Bulletin of the Calcutta Mathematical Society, 35:99-109, 1943.","title":"On a Measure of Divergence between Two Statistical Populations Defined by Their Probability Distributions","authors":"A K Bhattacharyya","year":"1943","venue":"Bulletin of the Calcutta Mathematical Society","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Boratyn et al., 2022] D. Boratyn, W. Słomczyński, D. Stolicki, and S. Szufa. Spoiler Susceptibility in Multi-District Party Elec- tions, February 2022.","title":"Spoiler Susceptibility in Multi-District Party Elections","authors":"D Boratyn, W Słomczyński, D Stolicki, S Szufa","year":"2022","venue":null,"doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Boser et al., 1992] B. E. Boser, I. M. Guyon, and V. N. Vapnik. A Training Algorithm for Optimal Margin Classifiers. In Proceed- ings of the Fifth Annual Workshop on Computational Learning Theory, pages 144-152, New York, NY, USA, July 1992. ACM.","title":"A Training Algorithm for Optimal Margin Classifiers","authors":"B E Boser, I M Guyon, V N Vapnik","year":"1992","venue":"Proceedings of the Fifth Annual Workshop on Computational Learning Theory","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Breiman et al., 1977] L. Breiman, W. Meisel, and E. Purcell. Vari- able Kernel Estimates of Multivariate Densities. Technometrics, 19(2):135-144, May 1977.","title":"Variable Kernel Estimates of Multivariate Densities","authors":"L Breiman, W Meisel, E Purcell","year":"1977","venue":"Technometrics","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Browning and King, 1987] R. X. Browning and G. King. Seats, Votes, and Gerrymandering: Estimating Representation and Bias in State Legislative Redistricting. Law & Policy, 9(3):305-322, July 1987.","title":"Seats, Votes, and Gerrymandering: Estimating Representation and Bias in State Legislative Redistricting","authors":"R X Browning, G King","year":"1987","venue":"Law & Policy","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Burkill, 1984] H. Burkill. Monotonic Functions on Partially Ordered Sets. Journal of Combinatorial Theory, Series A, 37(3):248-256, November 1984.","title":"Monotonic Functions on Partially Ordered Sets","authors":"H Burkill","year":"1984","venue":"Journal of Combinatorial Theory, Series A","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Butler, 1947] D. E. Butler. Appendix III: The Relation of Seats to Votes. In R. B. McCallum and A. Readman, editors, The British General Election of 1945, pages 277-292.","title":"Appendix III: The Relation of Seats to Votes","authors":"D E Butler","year":null,"venue":"The British General Election of 1945","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"G. Cumberlege, Lon- don, 1947.","title":null,"authors":"G Cumberlege, London","year":"1947","venue":null,"doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Calvo and Rodden, 2015] E. Calvo and J. Rodden. The Achilles Heel of Plurality Systems: Geography and Representation in Multiparty Democracies. American Journal of Political Science, 59(4):789-805, October 2015.","title":"The Achilles Heel of Plurality Systems: Geography and Representation in Multiparty Democracies","authors":"Rodden Calvo, ] E Calvo, J Rodden","year":"2015","venue":"American Journal of Political Science","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Calvo, 2009] E. Calvo. The Competitive Road to Proportional Representation: Partisan Biases and Electoral Regime Change under Increasing Party Competition. World Politics, 61(2):254- 295, April 2009.","title":"The Competitive Road to Proportional Representation: Partisan Biases and Electoral Regime Change under Increasing Party Competition","authors":"E Calvo","year":"2009","venue":"World Politics","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Chen and Rodden, 2013] J. Chen and J. Rodden. Unintentional Gerrymandering: Political Geography and Electoral Bias in Leg- islatures. Quarterly Journal of Political Science, 8(3):239-269, June 2013.","title":"Unintentional Gerrymandering: Political Geography and Electoral Bias in Legislatures","authors":"J Chen, J Rodden","year":"2013","venue":"Quarterly Journal of Political Science","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Chen and Rodden, 2015] J. Chen and J. Rodden. Cutting Through the Thicket: Redistricting Simulations and the Detection of Par- tisan Gerrymanders. Election Law Journal: Rules, Politics, and Policy, 14(4):331-345, December 2015.","title":"Cutting Through the Thicket: Redistricting Simulations and the Detection of Partisan Gerrymanders","authors":"J Chen, J Rodden","year":"2015","venue":"Election Law Journal: Rules, Politics, and Policy","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Cortes and Vapnik, 1995] C. Cortes and V. Vapnik. Support- Vector Networks. Machine Learning, 20(3):273-297, September 1995.","title":"Support-Vector Networks","authors":"C Cortes, V Vapnik","year":"1995","venue":"Machine Learning","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Erikson, 1972] R. S. Erikson. Malapportionment, Gerrymander- ing, and Party Fortunes in Congressional Elections. American Political Science Review, 66(4):1234-1245, 1972.","title":"Malapportionment, Gerrymandering, and Party Fortunes in Congressional Elections","authors":"R S Erikson","year":"1972","venue":"American Political Science Review","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Fifield et al., 2015] B. Fifield, M. Higgins, and K. Imai. A New Automated Redistricting Simulator Using Markov Chain Monte Carlo. Working Paper, Princeton University, Princeton, NJ, 2015.","title":"A New Automated Redistricting Simulator Using Markov Chain Monte Carlo","authors":"B Fifield, M Higgins, K Imai","year":"2015","venue":null,"doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Fifield et al., 2020] B. Fifield, M. Higgins, K. Imai, and A. Tarr. Automated Redistricting Simulation Using Markov Chain Monte Carlo. Journal of Computational and Graphical Statistics, 29(4):715-728, October 2020.","title":"Automated Redistricting Simulation Using Markov Chain Monte Carlo","authors":"B Fifield, M Higgins, K Imai, A Tarr","year":"2020","venue":"Journal of Computational and Graphical Statistics","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Flis et al., 2023] J. Flis, W. Słomczyński, D. Stolicki, S. Szufa, and J. Sokołowski. Skrzywione szranki. Na tropie gerrymanderingu w polskich wyborach do rad gmin. Wydawnictwo UJ, Kraków, 2023.","title":"Skrzywione szranki. Na tropie gerrymanderingu w polskich wyborach do rad gmin","authors":"J Flis, W Słomczyński, D Stolicki, S Szufa, J Sokołowski","year":"2023","venue":null,"doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Gelman and King, 1990a] A. Gelman and G. King. Estimating In- cumbency Advantage without Bias. American Journal of Politi- cal Science, 34(4):1142-1164, November 1990.","title":"Estimating Incumbency Advantage without Bias","authors":"King ; Gelman, ] A Gelman, G King","year":"1990","venue":"American Journal of Political Science","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Gelman and King, 1990b] A. Gelman and G. King. Estimating the Electoral Consequences of Legislative Redistricting. Journal of the American Statistical Association, 85(410):274-282, 1990.","title":"Estimating the Electoral Consequences of Legislative Redistricting","authors":"King ; Gelman, ] A Gelman, G King","year":"1990","venue":"Journal of the American Statistical Association","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Gelman and King, 1994] A. Gelman and G. King. A Unified Method of Evaluating Electoral Systems and Redistricting Plans. American Journal of Political Science, 38(2):514-554, 1994.","title":"A Unified Method of Evaluating Electoral Systems and Redistricting Plans","authors":"A Gelman, G King","year":"1994","venue":"American Journal of Political Science","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Geyer and Thompson, 1995] C. J. Geyer and E. A. Thompson. An- nealing Markov Chain Monte Carlo with Applications to Ances- tral Inference. Journal of the American Statistical Association, 90(431):909-920, September 1995.","title":"Annealing Markov Chain Monte Carlo with Applications to Ancestral Inference","authors":"C J Geyer, E A Thompson","year":"1995","venue":"Journal of the American Statistical Association","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Gilligan and Matsusaka, 1999] T. W. Gilligan and J. G. Matsusaka. Structural Constraints on Partisan Bias Under the Efficient Ger- rymander. Public Choice, 100(1):65-84, July 1999.","title":"Structural Constraints on Partisan Bias Under the Efficient Gerrymander","authors":"T W Gilligan, J G Matsusaka","year":"1999","venue":"Public Choice","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Grofman, 1983] B. Grofman. Measures of Bias and Propor- tionality in Seats-Votes Relationships. Political Methodology, 9(3):295-327, 1983.","title":"Measures of Bias and Proportionality in Seats-Votes Relationships","authors":"B Grofman","year":"1983","venue":"Political Methodology","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Herfindahl, 1950] O. C. Herfindahl. Concentration in the U.S. Steel Industry. PhD thesis, Columbia University, New York, NY, 1950.","title":"Concentration in the U.S. Steel Industry","authors":"O C Herfindahl","year":"1950","venue":null,"doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Hirschman, 1945] A. O. Hirschman. National Power and the Structure of Foreign Trade. University of California Press, Berke- ley, CA, 1945.","title":"National Power and the Structure of Foreign Trade","authors":"A O Hirschman","year":"1945","venue":null,"doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Hurvich et al., 1998] C. M. Hurvich, J. S. Simonoff, and C.-L. Tsai. Smoothing Parameter Selection in Nonparametric Regres- sion Using an Improved Akaike Information Criterion. Journal of the Royal Statistical Society: Series B, 60(2):271-293, 1998.","title":"Smoothing Parameter Selection in Nonparametric Regression Using an Improved Akaike Information Criterion","authors":"C M Hurvich, J S Simonoff, C.-L Tsai","year":"1998","venue":"Journal of the Royal Statistical Society: Series B","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Härdle et al., 1988] W. Härdle, P. Hall, and J. S. Marron. How Far are Automatically Chosen Regression Smoothing Parameters from their Optimum? Journal of the American Statistical Asso- ciation, 83(401):86-95, March 1988.","title":"How Far are Automatically Chosen Regression Smoothing Parameters from their Optimum","authors":"W Härdle, P Hall, J S Marron","year":"1988","venue":"Journal of the American Statistical Association","doi":null,"url":null,"internal_url":null},{"position":33,"raw":"[Kendall and Stuart, 1950] M. G. Kendall and A. Stuart. The Law of the Cubic Proportion in Election Results. British Journal of Sociology, 1(3):183-196, September 1950.","title":"The Law of the Cubic Proportion in Election Results","authors":"M G Kendall, A Stuart","year":"1950","venue":"British Journal of Sociology","doi":null,"url":null,"internal_url":null},{"position":34,"raw":"[King, 1990] G. King. Electoral Responsiveness and Partisan Bias in Multiparty Democracies. Legislative Studies Quarterly, 15(2):159-181, 1990.","title":"Electoral Responsiveness and Partisan Bias in Multiparty Democracies","authors":"G King","year":"1990","venue":"Legislative Studies Quarterly","doi":null,"url":null,"internal_url":null},{"position":35,"raw":"[Kollman et al., 2023] K. Kollman, A. Hicken, D. Caramani, D. Backer, and D. Lublin. Constituency-Level Elections Archive. Dataset, Center for Political Studies, University of Michigan, Ann Arbor , MI, 2023.","title":"Constituency-Level Elections Archive","authors":"K Kollman, A Hicken, D Caramani, D Backer, D Lublin","year":"2023","venue":null,"doi":null,"url":null,"internal_url":null},{"position":36,"raw":"[Kullback and Leibler, 1951] S. Kullback and R. A. Leibler. On In- formation and Sufficiency. The Annals of Mathematical Statis- tics, 22(1):79-86, 1951.","title":"On Information and Sufficiency","authors":"S Kullback, R A Leibler","year":"1951","venue":"The Annals of Mathematical Statistics","doi":null,"url":null,"internal_url":null},{"position":37,"raw":"[Laakso and Taagepera, 1979] M. Laakso and R. Taagepera. \"Ef- fective\" Number of Parties: A Measure with Application to West Europe. Comparative Political Studies, 12(1):3-27, April 1979.","title":"Effective\" Number of Parties: A Measure with Application to West Europe","authors":"M Laakso, R Taagepera","year":"1979","venue":"Comparative Political Studies","doi":null,"url":null,"internal_url":null},{"position":38,"raw":"[Li and Racine, 2004] Q. Li and J. Racine. Cross-Validated Local Linear Nonparametric Regression. Statistica Sinica, 14:485-512, 2004.","title":"Cross-Validated Local Linear Nonparametric Regression","authors":"Q Li, J Racine","year":"2004","venue":"Statistica Sinica","doi":null,"url":null,"internal_url":null},{"position":39,"raw":"[Linzer, 2012] D. A. Linzer. The Relationship between Seats and Votes in Multiparty Systems. Political Analysis, 20(3):400-416, 2012.","title":"The Relationship between Seats and Votes in Multiparty Systems","authors":"D A Linzer","year":"2012","venue":"Political Analysis","doi":null,"url":null,"internal_url":null},{"position":40,"raw":"[Manow, 2011] P. Manow. The Cube Rule in a Mixed Elec- toral System: Disproportionality in German Bundestag Elections. West European Politics, 34(4):773-794, July 2011.","title":"The Cube Rule in a Mixed Electoral System: Disproportionality in German Bundestag Elections","authors":"P Manow","year":"2011","venue":"West European Politics","doi":null,"url":null,"internal_url":null},{"position":41,"raw":"J. G. March. Party Legislative Representation as a Function of Election Results. Public Opinion Quarterly, 21(4):521-542, 1957.","title":"Party Legislative Representation as a Function of Election Results","authors":"J G March","year":"1957","venue":"Public Opinion Quarterly","doi":null,"url":null,"internal_url":null},{"position":42,"raw":"[Marinari and Parisi, 1992] E. Marinari and G. Parisi. Simulated Tempering: A New Monte Carlo Scheme. Europhysics Letters (EPL), 19(6):451-458, July 1992.","title":"Simulated Tempering: A New Monte Carlo Scheme","authors":"E Marinari, G Parisi","year":"1992","venue":"Europhysics Letters (EPL)","doi":null,"url":null,"internal_url":null},{"position":43,"raw":"[McCartan and Imai, 2023] C. McCartan and K. Imai. Sequential Monte Carlo for Sampling Balanced and Compact Redistricting Plans. The Annals of Applied Statistics, 17(4):3300-3323, De- cember 2023.","title":"Sequential Monte Carlo for Sampling Balanced and Compact Redistricting Plans","authors":"C Mccartan, K Imai","year":"2023","venue":"The Annals of Applied Statistics","doi":null,"url":null,"internal_url":null},{"position":44,"raw":"[McDonald et al., 2011] M. D. McDonald, J. Krasno, and R. E. Best. An Objective and Simple Measure of Gerrymandering: A Demonstration from New York State. In 2011 Annual Meeting of the Midwest Political Science Association, Chicago, IL, 2011.","title":"An Objective and Simple Measure of Gerrymandering: A Demonstration from New York State","authors":"M D Mcdonald, J Krasno, R E Best","year":"2011","venue":"Annual Meeting of the Midwest Political Science Association","doi":null,"url":null,"internal_url":null},{"position":45,"raw":"[McGann et al., 2015] A. J. McGann, C. A. Smith, M. Latner, and J. A. Keena. A Discernable and Manageable Standard for Par- tisan Gerrymandering. Election Law Journal, 14(4):295-311, November 2015.","title":"A Discernable and Manageable Standard for Partisan Gerrymandering","authors":"A J Mcgann, C A Smith, M Latner, J A Keena","year":"2015","venue":"Election Law Journal","doi":null,"url":null,"internal_url":null},{"position":46,"raw":"[McGhee, 2014] E. McGhee. Measuring Partisan Bias in Single- Member District Electoral Systems. Legislative Studies Quar- terly, 39(1):55-85, February 2014.","title":"Measuring Partisan Bias in Single-Member District Electoral Systems","authors":"E Mcghee","year":"2014","venue":"Legislative Studies Quarterly","doi":null,"url":null,"internal_url":null},{"position":47,"raw":"[Nadaraya, 1965] E. A. Nadaraya. On Non-Parametric Estimates of Density Functions and Regression Curves. Theory of Probability & Its Applications, 10(1):186-190, January 1965.","title":"On Non-Parametric Estimates of Density Functions and Regression Curves","authors":"E A Nadaraya","year":"1965","venue":"Theory of Probability & Its Applications","doi":null,"url":null,"internal_url":null},{"position":48,"raw":"[Niemi and Deegan, 1978] R. G. Niemi and J. Deegan. A The- ory of Political Districting. American Political Science Review, 72(4):1304-1323, December 1978.","title":"A Theory of Political Districting","authors":"R G Niemi, J Deegan","year":"1978","venue":"American Political Science Review","doi":null,"url":null,"internal_url":null},{"position":49,"raw":"[Niemi et al., 1990] R. G. Niemi, B. Grofman, C. Carlucci, and T. Hofeller. Measuring Compactness and the Role of a Compact- ness Standard in a Test for Partisan and Racial Gerrymandering. Journal of Politics, 52(4):1155-1181, 1990.","title":"Measuring Compactness and the Role of a Compactness Standard in a Test for Partisan and Racial Gerrymandering","authors":"R G Niemi, B Grofman, C Carlucci, T Hofeller","year":"1990","venue":"Journal of Politics","doi":null,"url":null,"internal_url":null},{"position":50,"raw":"[Parzen, 1962] E. Parzen. On Estimation of a Probability Den- sity Function and Mode. Annals of Mathematical Statistics, 33(3):1065-1076, September 1962.","title":"On Estimation of a Probability Density Function and Mode","authors":"E Parzen","year":"1962","venue":"Annals of Mathematical Statistics","doi":null,"url":null,"internal_url":null},{"position":51,"raw":"[Rosenblatt, 1956] M. Rosenblatt. Remarks on Some Nonparamet- ric Estimates of a Density Function. Annals of Mathematical Statistics, 27(3):832-837, September 1956.","title":"Remarks on Some Nonparametric Estimates of a Density Function","authors":"M Rosenblatt","year":"1956","venue":"Annals of Mathematical Statistics","doi":null,"url":null,"internal_url":null},{"position":52,"raw":"Schucany, 1995","title":null,"authors":"Schucany","year":"1995","venue":null,"doi":null,"url":null,"internal_url":null},{"position":53,"raw":"W. R. Schucany. Adaptive Bandwidth Choice for Kernel Regression. Journal of the American Statistical Associa- tion, 90(430):535-540, June 1995.","title":"Adaptive Bandwidth Choice for Kernel Regression","authors":"W R Schucany","year":"1995","venue":"Journal of the American Statistical Association","doi":null,"url":null,"internal_url":null},{"position":54,"raw":"[Shannon, 1948] C. E. Shannon. A Mathematical Theory of Com- munication. The Bell System Technical Journal, 27(3):379-423, July 1948.","title":"A Mathematical Theory of Communication","authors":"C E Shannon","year":"1948","venue":"The Bell System Technical Journal","doi":null,"url":null,"internal_url":null},{"position":55,"raw":"[Silverman, 1986] B. W. Silverman. Density Estimation for Statis- tics and Data Analysis. Chapman & Hall, New York, 1986.","title":"Density Estimation for Statistics and Data Analysis","authors":"B W Silverman","year":"1986","venue":null,"doi":null,"url":null,"internal_url":null},{"position":56,"raw":"[Simpson, 1949] E. H. Simpson. Measurement of Diversity. Na- ture, 163(4148):688-688, April 1949.","title":"Measurement of Diversity","authors":"E H Simpson","year":"1949","venue":"Nature","doi":null,"url":null,"internal_url":null},{"position":57,"raw":"[Spearman, 1904] C. Spearman. The Proof and Measurement of Association between Two Things. The American Journal of Psy- chology, 15(1):72-101, 1904.","title":"The Proof and Measurement of Association between Two Things","authors":"C Spearman","year":"1904","venue":"The American Journal of Psychology","doi":null,"url":null,"internal_url":null},{"position":58,"raw":"[Stephanopoulos and McGhee, 2015] N. O. Stephanopoulos and E. M. McGhee. Partisan Gerrymandering and the Efficiency Gap. University of Chicago Law Review, 82(2):831-900, 2015.","title":"Partisan Gerrymandering and the Efficiency Gap","authors":"N O Stephanopoulos, E M Mcghee","year":"2015","venue":"University of Chicago Law Review","doi":null,"url":null,"internal_url":null},{"position":59,"raw":"[Stolicki et al., 2019] D. Stolicki, W. Słomczyński, and J. Flis. A Probabilistic Model for Detecting Gerrymandering in Partially- Contested Multiparty Elections. In Transactions on Compu- tational Collective Intelligence XXXIV, volume 11890 of Lec- ture Notes in Computer Science, pages 1-20, Berlin, Heidelberg, 2019. Springer.","title":"A Probabilistic Model for Detecting Gerrymandering in Partially-Contested Multiparty Elections","authors":"D Stolicki, W Słomczyński, J Flis","year":"2019","venue":"Transactions on Computational Collective Intelligence XXXIV","doi":null,"url":null,"internal_url":null},{"position":60,"raw":"[Stolicki, 2024] D. Stolicki. Redistricting and Gerrymandering af- ter the 2020-2021 Census. In R. Duda and M. Turek, editors, The Crossroads Elections: European Perspectives on the 2022 U.S. Midterm Elections, pages 76-95. Routledge, New York, 1 edition, March 2024.","title":"Redistricting and Gerrymandering after the 2020-2021 Census","authors":"D Stolicki","year":"2024","venue":"The Crossroads Elections: European Perspectives on the 2022 U.S. Midterm Elections","doi":null,"url":null,"internal_url":null},{"position":61,"raw":"[Swendsen and Wang, 1987] R. H. Swendsen and J.-S. Wang. Nonuniversal Critical Dynamics in Monte Carlo Simulations. Physical Review Letters, 58(2):86-88, January 1987.","title":"Nonuniversal Critical Dynamics in Monte Carlo Simulations","authors":"R H Swendsen, J.-S Wang","year":"1987","venue":"Physical Review Letters","doi":null,"url":null,"internal_url":null},{"position":62,"raw":"[Słomczyński et al., 2024] W. Słomczyński, D. Stolicki, and S. Sz- ufa. Nonparametric Detection of Gerrymandering in Multiparty Elections. arXiv: 2306.03085 [stat.AP], May 2024.","title":"Nonparametric Detection of Gerrymandering in Multiparty Elections","authors":"W Słomczyński, D Stolicki, S Szufa","year":"2024","venue":null,"doi":null,"url":null,"internal_url":null},{"position":63,"raw":"[Taagepera and Grofman, 1981] R. Taagepera and B. Grofman. Ef- fective Size and Number of Components. Sociological Methods & Research, 10(1):63-81, August 1981.","title":"Effective Size and Number of Components","authors":"R Taagepera, B Grofman","year":"1981","venue":"Sociological Methods & Research","doi":null,"url":null,"internal_url":null},{"position":64,"raw":"[Taagepera, 1973] R. Taagepera. Seats and Votes: A Generaliza- tion of the Cube Law of Elections. Social Science Research, 2(3):257-275, September 1973.","title":"Seats and Votes: A Generalization of the Cube Law of Elections","authors":"R Taagepera","year":"1973","venue":"Social Science Research","doi":null,"url":null,"internal_url":null},{"position":65,"raw":"[Taagepera, 1986] R. Taagepera. Reformulating the Cube Law for Proportional Representation Elections. American Political Sci- ence Review, 80(2):489-504, June 1986.","title":"Reformulating the Cube Law for Proportional Representation Elections","authors":"R Taagepera","year":"1986","venue":"American Political Science Review","doi":null,"url":null,"internal_url":null},{"position":66,"raw":"[Tufte, 1973] E. R. Tufte. The Relationship between Seats and Votes in Two-Party Systems. American Political Science Review, 67(2):540-554, June 1973.","title":"The Relationship between Seats and Votes in Two-Party Systems","authors":"E R Tufte","year":"1973","venue":"American Political Science Review","doi":null,"url":null,"internal_url":null},{"position":67,"raw":"[Wang, 2016] S. S.-H. Wang. Three Tests for Practical Evaluation of Partisan Gerrymandering. Stanford Law Review, 68(6):1263- 1321, 2016.","title":"Three Tests for Practical Evaluation of Partisan Gerrymandering","authors":"S S, -H Wang","year":"2016","venue":"Stanford Law Review","doi":null,"url":null,"internal_url":null},{"position":68,"raw":"[Warrington, 2018] G. S. Warrington. Quantifying Gerrymandering Using the Vote Distribution. Election Law Journal, 17(1):39-57, March 2018.","title":"Quantifying Gerrymandering Using the Vote Distribution","authors":"G S Warrington","year":"2018","venue":"Election Law Journal","doi":null,"url":null,"internal_url":null},{"position":69,"raw":"[Watson, 1964] G. S. Watson. Smooth Regression Analysis. Sankhyā: The Indian Journal of Statistics, Series A, 26(4):359- 372, 1964.","title":"Smooth Regression Analysis","authors":"G S Watson","year":"1964","venue":"Sankhyā: The Indian Journal of Statistics, Series A","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}