Statistical Science

Why does voting get so complicated? A review of theories for analyzing democratic participation

Jason Gainous and Jeff Gill

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The purpose of this article is to present a sample from the panoply of formal theories on voting and elections to Statistical Science readers who have had limited exposure to such work. These abstract ideas provide a framework for understanding the context of the empirical articles that follow in this volume. The primary focus of this theoretical literature is on the use of mathematical formalism to describe electoral systems and outcomes by modeling both voting rules and human behavior. As with empirical models, these constructs are never perfect descriptors of reality, but instead form the basis for understanding fundamental characteristics of the studied system. Our focus is on providing a general, but not overly simplified, review of these theories with practical examples. We end the article with a thought experiment that applies different vote aggregation schemes to the 2000 presidential election count in Florida, and we find that alternative methods provide different results.

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Statist. Sci., Volume 17, Issue 4 (2002), 383-404.

First available in Project Euclid: 10 April 2003

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Voting rules elections participation rational choice spatial models cost-benefit models Florida 2000 election


Gill, Jeff; Gainous, Jason. Why does voting get so complicated? A review of theories for analyzing democratic participation. Statist. Sci. 17 (2002), no. 4, 383--404. doi:10.1214/ss/1049993199.

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  • ARROW, K. J. (1951). Social Choice and Individual Values. Wiley, New York.
  • BLACK, D. (1948). On the rationale of group decision making. Journal of Political Economy 56 23-34.
  • BLACK, D. (1958). The Theory of Committees and Elections. Cambridge Univ. Press.
  • BOGDANOR, V. (1984). What is Proportional Representation? A Guide to the Issues. Martin Robertson, Oxford, UK.
  • BORDA, J.-C. (1781). Mémoire sur les Èlections au Scrutin. Histoire de l'Académie Roy ale des Sciences, Paris.
  • BRAMS, S. J. (1975). Game Theory and Politics. Free Press, New York.
  • BRAMS, S. J. and FISHBURN, P. C. (1978). Approval voting. American Political Science Review 72 831-847.
  • BRAMS, S. J. and FISHBURN, P. C. (1983). Approval Voting. Birkhäuser, Boston.
  • BUCHANAN, J. M. (1983). The public choice perspective. Economica delle Scelte Pubbliche 1 7-15.
  • BURDEN, B. C. (1997). Deterministic and probabilistic voting models. American Journal of Political Science 41 1150-1169.
  • CALVERT, R. L. (1985). Robustness of the multidimensional voting model: Candidate motivations, uncertainty, and convergence. American Journal of Political Science 29 69-95.
  • CALVERT, R. L. (1986). Models of Imperfect Information in Politics. Harwood, London.
  • CAMPBELL, D. E. (1977). Computational criteria for voting sy stems. British Journal of Political Science 7 85-98.
  • CONDORCET, J. A. N. DE CARITAT, MARQUIS DE. (1785). Essai sur l'Application de l'Analy se à la Probabilité des Décisions Rendues à la Pluralité des Voix. Imprimerie Roy ale, Paris.
  • COOMBS, C. (1964). A Theory of Data. Wiley, New York.
  • COUGHLIN, P. J. (1982). Pareto optimality of policy proposals with probabilistic voting. Public Choice 39 427-433.
  • COUGHLIN, P. J. (1992). Probabilistic Voting Theory. Cambridge Univ. Press.
  • COUGHLIN, P. J. and NITZAN, S. (1981). Election outcomes with probabilistic voting and Nash social welfare maxima. Journal of Public Economics 15 113-122.
  • COX, G. W. (1987). The uncovered set and the core. American Journal of Political Science 31 408-422.
  • COX, G. W. (1997). Making Votes Count: Strategic Coordination in the World's Electoral Sy stems. Cambridge Univ. Press.
  • DAHL, R. A. (1971). Poly archy: Participation and Opposition. Yale Univ. Press.
  • DAVIS, O. A. and HINICH, M. J. (1966). A mathematical model of policy formation in a democratic society. In Mathematical Applications in Political Science (J. L. Bernd, ed.) 2 175-208. Southern Methodist Univ. Press, Dallas.
  • DAVIS, O. A. and HINICH, M. J. (1967). Some results related to a mathematical model of policy formulation in a democratic society. In Mathematical Applications in Political Science (J. L. Bernd, ed.) 3. Univ. Virginia Press, Charlottesville.
  • DAVIS, O., HINICH, M. J. and ORDESHOOK, P. (1970). An expository development of a mathematical model of the electoral process. American Political Science Review 64 426- 448.
  • DENZAU, A. and KATS, A. (1977). Expected plurality voting equilibrium and social choice functions. Rev. Econom. Stud. 44 227-233.
  • DION, D. (1992). The robustness of the structure-induced equilibrium. American Journal of Political Science 36 462-482.
  • DOWNS, A. (1957). An economic theory of political action in a democracy. Journal of Political Economy 65 135-150.
  • DUMMETT, M. A. E. (1984). Voting Procedures. Oxford Univ. Press.
  • DUVERGER, M. (1963). Political Parties: Their Organization and Activity in the Modern State, 2nd ed. Science Editions, New York.
  • ENELOW, J. M., ENDERSBY, J. W. and MUNGER, M. C. (1993). A revised spatial model of elections: theory and evidence. In Information, Participation, and Choice: An Economic Theory of Democracy in Perspective (B. Grofman, ed.). Univ. Michigan Press, Ann Arbor.
  • ENELOW, J. and HINICH, M. (1984). The Spatial Theory of Voting: An Introduction. Cambridge Univ. Press.
  • ENELOW, J. and HINICH, M., eds. (1990). Advances in the Spatial Theory of Voting. Cambridge Univ. Press.
  • FARQUHARSON, R. (1969). Theory of Voting. Yale Univ. Press.
  • FARRELL, D. M. (2001). Electoral Sy stems: A Comparative Introduction. St. Martin's Press, New York.
  • FELSENTHAL, D. S., MAOT, Z. and RAPOPORT, A. (1993). An empirical evaluation of six voting procedures: Do they really
  • FEREJOHN, J. A., MCKELVEY, R. D. and PACKEL, E. W. (1984). Limiting distributions for continuous state Markov voting models. Social Choice and Welfare 1 45-67.
  • FIORINA, M. (1981). Retrospective Voting in American National Elections. Yale Univ. Press.
  • FIORINA, M. (1995). Rational choice and the new(?) institutionalism. Polity 28 107-115.
  • FIORINA, M. P. and PLOTT, C. R. (1978). Committee decisions under majority rule: An experimental study. American Political Science Review 72 575-598.
  • FISHBURN, P. C. (1971). A comparative analysis of group decision methods. Behavioral Science 16 538-544.
  • FISHBURN, P. C. and BRAMS, S. J. (1981). Approval voting, Condorcet's principle, and runoff elections. Public Choice 36 89-114.
  • FRANKLIN, C. H. (1991). Eschewing obfuscation? Campaigns and the perception of U.S. Senate incumbents. American Political Science Review 85 1193-1214.
  • FRIEDMAN, J., ed. (1996). The Rational Choice Controversy. Yale Univ. Press.
  • GELMAN, A., KING, G. and BOSCARDIN, W. J. (1998). Estimating the probability of events that have never occurrred: When is your vote decisive? J. Amer. Statist. Assoc. 93 1-9.
  • GREEN, D. P. and SHAPIRO, I. (1994). Pathologies of Rational Choice Theory. Yale Univ. Press.
  • GROFMAN, B. (1989). The comparative analysis of coalition formation and duration: Distinguishing between-country and within-country effects. British Journal of Political Science 19 291-302.
  • (1987). Stability and centrality of legislative choice in the spatial context. American Political Science Review 81 539- 553.
  • GUINIER, L. (1994). The Ty ranny of the Majority: Fundamental Fairness in Representative Democracy. Free Press, New York.
  • HARE, T. (1859). A Treatise on the Election of Representatives, Parliamentary and Municipal. Longman, Brown, Green, Longman and Roberts, London.
  • HINICH, M. (1977). Equilibrium in spatial voting: The median voter is an artifact. J. Econom. Theory 16 208-219.
  • HINICH, M., LEDy ARD, J. and ORDESHOOK, P. (1972). Nonvoting and the existence of equilibrium under majority rule. J. Econom. Theory 4 144-153.
  • HINICH, M., LEDy ARD, J. and ORDESHOOK, P. (1973). A theory of electoral equilibrium: A spatial analysis based on the theory of games. Journal of Politics 35 154-193.
  • HINICH, M. and MUNGER, M. C. (1994). Ideology and the Theory of Political Choice. Univ. Michigan Press, Ann Arbor.
  • HOTELLING, H. (1929). Stability in competition. Economic Journal 39 41-57.
  • HUNTINGTON, S. P. (1991). The Third Wave: Democratization in the Late Twentieth Century. Univ. Oklahoma Press, Norman.
  • LUCE, R. D. and RAIFFA, H. (1957). Games and Decisions. Wiley, New York.
  • MCKELVEY, R. D. (1976). Intransitivities in multidimensional voting models and some implications for agenda control. J. Econom. Theory 12 472-482.
  • MCKELVEY, R. D. (1986). Covering, dominance, and institutionfree properties of social choice. American Journal of Political Science 30 283-314.
  • MCKELVEY, R. and NIEMI, R. (1978). A multistage game representation of sophisticated voting for binary procedures. J. Econom. Theory 18 1-22, 19 568.
  • MERRILL, S., III. (1984). A comparison of efficiency of multicandidate electoral sy stems. American Journal of Political Science 28 23-48.
  • MILLER, N. R. (1980). A new solution set for tournaments and majority voting: Further graph-theoretical approaches to the theory of voting. American Journal of Political Science 24 68- 96.
  • MUELLER, D. C. (1989). Public Choice 2. Cambridge Univ. Press.
  • NANSON, E. (1882). Methods of election to fill one seat. Paper delivered to the Roy al Society of Victoria.
  • NISKANEN, W. (1971). Bureaucracy and Representative Government. Aldine, Chicago.
  • NISKANEN, W. (1975). Bureaucrats and politicians. Journal of Law and Economics 18 617-643.
  • NURMI, H. (1983). Voting procedures: A summary analysis. British Journal of Political Science 13 181-208.
  • OLSON, M. (1965). The Logic of Collective Action: Public Goods and the Theory of Groups. Harvard Univ. Press.
  • ORDESHOOK, P. C. (1986). Game Theory and Political Theory. Cambridge Univ. Press.
  • PAGE, B. I. (1978). Choices and Echoes in Presidential Elections. Univ. Chicago Press.
  • PLOTT, C. R. (1967). A notion of equilibrium and its possibility under majority rule. American Economic Review 57 787-806.
  • RAE, D. (1971). The Political Consequences of Electoral Laws. Yale Univ. Press.
  • RIKER, W. (1982). Liberalism Against Populism. Freeman, San Francisco.
  • RIKER, W. (1986). Duverger's law revisited. In Electoral Laws and Their Political Consequences (B. Grofman and A. Lijphart, eds.). Agathon, New York.
  • RIKER, W. and ORDESHOOK, P. (1968). A theory of the calculus of voting. American Political Science Review 62 25-42.
  • SAARI, D. G. (2001). Chaotic Elections: A Mathematician Looks at Voting. Amer. Math. Soc., Providence, RI.
  • SEN, A. K. (1984). Collective Choice and Social Welfare. NorthHolland, Amsterdam.
  • SHEPSLE, K. A. (1972). The strategy of ambiguity: Uncertainty and electoral competition. American Political Science Review 66 555-568.
  • SHEPSLE, K. A. (1991). Models of Multiparty Electoral Competition. Harwood, Chur, Switzerland.
  • SMITHIES, A. (1941). Optimum location in spatial competition. Journal of Political Economy 49 423-439.
  • STRAFFIN, P. D. (1980). Topics in the Theory of Voting. Birkhäuser, Boston.
  • TAy LOR, M. J. (1971). Mathematical political theory. British Journal of Political Science 1 339-382.
  • TULLOCK, G. (1967). Toward a Mathematics of Politics. Univ. Michigan Press, Ann Arbor.
  • TULLOCK, G. (1981). Why so much stability. Public Choice 37 189-202.
  • WEBER, R. J. (1995). Approval voting. Journal of Economic Perspectives 9(1) 39-49.
  • WICKSELL, K. (1896). A new principle of just taxation. Finanztheoretische Untersuchungen. [Reprinted in R. A. Musgrave and A. T. Peacock (1967) Classics in the Theory of Public Finance 72-118. St. Martin's, New York.]
  • YILMAZ, M. R. (1999). Can we improve upon approval voting? European Journal of Political Economy 15 89-100.
  • YOUNG, H. P. (1988). Condorcet's theory of voting. American Political Science Review 82 1231-1244.
  • ZAKARIA, F. (1997). The rise of illiberal democracy. Foreign Affairs 76 22-43.