List of Figures. 5.1 Saari Triangle of the 33-Voter Profile 52

Transcription

1 List of Figures 4.1 Vote Share Simplex Two Areas of Electoral Outcomes Saari Triangle Condorcet's Paradox as a Saari Triangle General Condorcet's Paradox Triangle Example: Starting Triangle Example: Condorcet Portion Example: Reversal Portion Saari Triangle of the 33-Voter Profile The 6-Voter Subgroup Profile The 15-Voter Subgroup Profile The Combined Group Profile 94

2 List of Tables 2.1 A Preference Profile A Pairwise Comparison Matrix A Tournament Matrix Preference Profile of Voters 1 and Preference Profile of Voters 3 and A Preference Profile Corresponding Table Pairwise Comparison Matrix of Table Borda's Paradox Outranking Matrix for Table Condorcet's Paradox Example of Condorcet's Solution Outranking Matrix for Table Outranking Matrix for Unanimous Collective Ranking ABC A Preference Profile for Illustration of Condorcet's Solutions Outranking Matrix of Table Nanson's Example The Incompatibility of Borda's and Condorcet's Intuitions Outranking Matrix Table Condorcet's Paradox in Impartial Cultures No Condorcet Winner and No Condorcet Paradox The Frequency of Borda's Paradox Voter Calculus: Payoffs 47

3 132 Voting Paradoxes and How to Deal with Them 5.2 Voter Calculus: Regrets 5.3 No-Show Paradox 5.4 A 33-Voter Profile 5.5 A 26-Voter Profile Plurality Runoff and Additional Support Paradox Coombs' Procedure and Additional Support Paradox Nanson's Procedure and Additional Support Paradox Dodgson's Procedure and Additional Support Paradox Monotonicity and Vulnerability to No-Show Paradox Truncation Paradox and Alternative Vote Truncation Paradox and Copeland's Procedure Tournament Matrix of Table Modified Tournament Matrix of Table Truncation Paradox and Strong Condorcet Condition Truncation Paradox and the Borda Count Ostrogorski's Paradox Ostrogorski's Paradox: Strict Version Anscombe's Paradox Ostrogorski's Paradox Which Is Not Anscombe's Paradox Paradox of Multiple Elections Strong Paradox of Multiple Elections Referendum Paradox Simpson's Paradox Party Support and Simpson's Paradox Intertemporal Simpson's Paradox Gorman's Example of Anscombe's Paradox Modified Anscombe's Paradox Amendment Procedure and Pareto Criterion Inconsistency of the Plurality Runoff Borda Count Fails on Q Plurality Voting Fails on Q Uncovered Set and Condition Q: Example 1 103

4 List of Tables Uncovered Set and Condition a : Dominance Matrix Uncovered Set and Condition a: Dominance Matrix Uncovered Set and Condition a: Example The Alabama Paradox The Population Paradox The New States Paradox The Paradox of Redistribution Schwartz' Paradox: An Example Some Features of Paradoxes Probabilistic Choice Example 127

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7 Bibliography 137 Brams, S. J., Kilgour, D. M., Zwicker, W. S. (1998): The Paradox of Multiple Elections. Social Choice and Welfare 15, Cohen, M. R., Nagel, E. (1934): Introduction to Logic and Scientific Method. Routledge and Kegan Paul, London Daudt, H., Rae, D. (1978): Social Contract and the Limits of Majority Rule. In Birnbaum, P., Lively, J., Parry, G. (Eds.): Democracy, Consensus (3 Social Contract. SAGE Publications, London Beverly Hills De Grazia, A. (1953): Mathematical Derivation of an Election System. Isis 44, Deb, R., Kelsey, D. (1987): On Constructing a Generalized Ostrogorski Paradox: Necessary and Sufficient Conditions. Mathematical Social Sciences 14, Doron, G. (1979): The Hare System Is Inconsistent. Political Studies 27, Downs, A. (1957): An Economic Theory of Democracy. Harper, New York Epstein, D. (1998): Uncovering Some Subtleties of the Uncovered Set: Social Choice Theory and Distributive Politics. Social Choice and Welfare 15, Ferejohn, J. A., Fiorina, M. P. (1974): The Paradox of Not Voting: A Decision Theoretic Analysis, The American Political Science Review 68, Fishburn, P. C. (1977): Condorcet Social Choice Functions. SIAM Journal of Applied Mathematics 33, Fishburn, P. C. (1981): Inverted Orders for Monotone Scoring Rules. Discrete Applied Mathematics 3, Fishburn, P. C. (1982): Monotonicity Paradoxes in the Theory of Voting. Discrete Applied Mathematics 4,

8 138 Voting Paradoxes and How to Deal with Them Fishburn, P. C., Brams, S. J. (1983): Paradoxes of Preferential Voting. Mathematics Magazine 56, Fishburn, P. C., Brams, S. J. (1984): Manipulability of Voting by Sincere Truncation of Preferences. Public Choice 44, Gehrlein, W.V. (1983): Condorcet's Paradox. Theory and Decision 15, Gehrlein, W.V. (1997): Condorcet's Paradox and the Condorcet Efficiency of Voting Rules. Mathematica Japonica 45, Gehrlein, W. V., Fishburn, P.C. (1976): The Probability of Paradox of Voting: A Computable Solution. Journal of Economic Theory 13, Gehrlein, W. V., Lepelley, D. (1997): The Condorcet Efficiency of Approval Voting and the Probability of Electing the Condorcet Loser. Mimeo, Department of Business Administration, University of Delaware Goodman, L. A. (1959): Some Alternatives to Ecological Correlations. American Journal of Sociology 44, Gorman, J. L. (1978): A Problem in the Justification of Democracy. Analysis 39, Intriligator, M.D. (1973): A Probabilistic Model of Social Choice. Review of Economic Studies 40, Kiersky, J. H., Caste, N. J. (1995): Thinking Critically. Techniques for Logical Reasoning. West Publishing Company, Minneapolis/St. Paul New York Los Angeles San Francisco Kelly, J. S. (1974): Voting Anomalies, the Number of Voters, and the Number of Alternatives. Econometrica 42, Kelly, J. S. (1978): Press, New York Arrow Impossibility Theorems. Academic

9 Bibliography 139 Kemeny, J. G. (1959): Mathematics without Numbers. Daedalus 88, Kemeny, J.G., Snell, J.L. (1962): Mathematical Models m the Social Sciences. Blaisdell, New York Toronto London Kramer, G. H. (1977): A Dynamical Model of Political Equilibrium. Journal of Economic Theory 16, Lagerspetz, E. (1986): Pufendorf on Collective Decisions. Public Choice 49, Lagerspetz, E. (1995): Paradoxes and Representation. Electoral Studies 15, Laslier, J.-F. (1997): Tournament Solutions and Majority Voting. Springer, Berlin Heidelberg New York McGarvey, D. C. (1953): A Theorem on the Construction of Voting Paradoxes. Econometrica 21, McKelvey, R. D., Niemi, R. G. (1978): A Multistage Game Representation of Sophisticated Voting for Binary Procedures. Journal of Economic Theory 18, 1-22 McKelvey, R. D., Schofield, N. (1986): Generalized Symmetry Conditions at a Core Point. Econometrica 55, McLean, I., Urken, A. B. (Eds.) (1995): Classics of Social Choice. The University of Michigan Press, Ann Arbor Meehl, P. E. (1977): The Selfish Voter Paradox and the Thrown Away Vote Argument. The American Political Science Review 71, Michaud, P. (1985): Hommage a Condorcet (version integrale pour le bicentenaire de l'essai de Condorcet). Etude F.094, Novembre Compagnie IBM France, Centre scientifique de Paris

10 140 Voting Paradoxes and How to Deal with Th em Miller '.N. R. (1977): Graph-Theoretical Approaches to the Theory of Voting. American Journal of Political Science 21, 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, Miller, N. R. (1986): Information, Electorates, and Democracy: Some Extensions and Intepretations of the Condorcet Jury Theorem. In Grofman, B., Owen, G. (Eds.): Information Pooling and Group Decision Making. JAI Press, Greenwich, CT Miller, N. R. (1995): Committees, Agendas, and Voting. Harwood Academic Publishers, Chur Miller, W. E. ( ): Presidential Coattails: A Study of Political Myth and Methodology: Public Opinion Quarterly 19, Moulin, H. (1986): Choosing from a Tournament. Social Choic e and Welfare 3, Moulin, H. (1988): Condorcet's Principle Implies the No Show Paradox. Journal of Economic Theory 45, Nanson, E.J. (1882): Methods of Election. Transactions and Proceedings of the Royal Society of Victoria XIX, Niemi, R. G., Weisberg, H. F. (1968): A Mathematical Solution for the Probability of the Paradox of Voting. Behavioral Science 13, Niou, E. M. S. (1987): A Note on Nanson's Rule. Public Choice 54, Nurmi, H. (1984): On Taking Preferences Seriously. In Anckar, D., Berndtson, E. (Eds.): Essays on Democratic Theory. The Finnish Political Science Association, Helsinki

11 Bibliography 141 Nurmi, H. (1986): Mathematical Models of Elections and Their Relevance for Institutional Design. Electoral Studies 5, Nurmi, H. (1987): Comparing Voting Systems. D. Reidel, Dordrecht Nurmi, H. (1988): Inferential Modes in Applying Social Choice Theory. In Munier, B. R., Shakun, M. F. (Eds.): Compromise, Negot iation and Group Decision. D. Reidel, Dordrecht Nurmi, H. (1989): On Nanson's Method. In Paastela, J. (Ed.), Democracy in the Modern World. Acta Universitatis Tamperensis, ser A vol 260, Tampere Nurmi, H. (1997a): Referendum Design: An Exercise in Applied Social Choice Theory. Scandinavian Political Studies 20, Nurmi, H. (1997b) : Compound Majority Paradoxes and Proportional Representation. European Journal of Political Economy 13, Nurmi, H. (1997c): It's Not Just the Lack of Monotonicity. Representation 34, Nurmi, H. (1998): Voting Paradoxes and Referenda. Social Choice and Welfare 15, Nurmi, H., Meskanen, T. (1997): Voting Paradoxes and MCDM. Group Decision and Negotiation, forthcoming Nurmi, H., Uusi-Heikkila, Y. (1985): Computer Simulations of Approval and Plurality Voting: The Frequency of Weak Pareto Violations and Condorcet Loser Choices in Impartial Cultures. European Journal of Political Economy 2, Ostrogorski, M. (1970): Democracy and the Organization of Political Parties, Vol. I-II. Haskell House Publishers, New

12 142 Voting Paradoxes and How to Deal with Them York (Original: Ostrogorski, M. (1902) : La democratic et l'organisation des partis politiques. Calmann-Levy, Paris) Rae, D., Daudt, H. (1976) : The Ostrogorski Paradox: A Peculiarity of Compound Majority Decision. European Journal of Political Research 4, Rasch, B. E. (1995): Parliamentary Voting Procedures. In Doring, H. (Ed.): Parliaments and Majority Rule in Western Europe. Campus Verlag, Frankfurt Richelson, J. T. (1979): A Comparative Analysis of Social Choice Functions I, II, III: A Summary. Behavioral Science 24, 355 Richelson, J. T. (1981) : Majority Rule and Collective Choice. Mimeo. Riker, W. H. (1982) : Liberalism against Populism. A Confrontation Between the Theory of Democracy and the Theory of Social Choice. W. H. Freeman, San Francisco Riker, W. H., Ordeshook, P. C. (1968) : A Theory of the Calculus of Voting. The American Political Science Review 62, Riker, W. H., Ordshook, P. C. (1973): An Introduction to Positive Political Theory. Prentice-Hall, Englewood Cliffs Robinson, W. S. (1950): Ecological Correlations and the Behavior of Individuals. American Sociological Review 15, Saari, D. G.(1988): Symmetry, Voting and Social Choice. The Mathematical Intelligencer 10, Saari, D. G. (1989) : A Dictionary of Voting Paradoxes. Journal of Economic Theory 48, Saari, D. G. (1990) : Consistency of Decision Processes. Annals of Operat ions Research 23, Saari, D. G. (1995): Basic Geometry of Voting. Springer, Berlin Heidelberg New York

13 Bibliography 143 Saari, D. G. (1997): The Generic Existence of a Core for q-rules. Economic Theory 9, Savage, L. (1951): The Theory of Statistical Decision. Journal of the American Statistical Association 46, Scarsini, M. (1998): A Strong Paradox of Multiple Elections. Social Choice and Welfare 15, Schotter, A. (1982): The Paradox of Redistribution: Some Theoretical and Empirical Results. In Holler, M. J. (Ed.): Power, Voting, and Voting Power. Physica Verlag, Wiirzburg Schwartz, Th. (1986) The Logic of Collective Choice. Columbia University Press, New York Schwartz, Th. (1995) The Paradox of Representation. The Journal of Politics 57, Sen, A. K. (1970): Collective Choice and Social Welfare. Holden Day, San Francisco Sen, A. K. (1977): Social Choice Theory: A Re-Examination. Econometrica 45, Simpson, E. H. (1951): The Interpretation of Interaction in Contingency Tables. Journal of the Royal Statistical Society B 13, Straffin, Ph. D., Jr. (1980): Topics m the Theory of Voting. Birkhauser, Boston Todhunter, 1. (1949): A History of the Mathematical Theory of Probability from the Time of Pascal to That of Laplace. Chelsea, New York Tsebelis, G. (1989): The Abuse of Probability in Political Analysis: The Robinson Crusoe Fallacy. American Political Science Review 83, 77-91

14 144 Voting Paradoxes and How to Deal with Them Tullock, G. (1968): Toward a Mathematics of Politics. University of Michigan Press, Ann Arbor Van Deemen, A. M. A. (1997): Coalition Formation and Social Choice. Kluwer Academic Publishers, Boston Dordrecht London Wagner, C. (1983): Anscombe's Paradox and the Rule of Three Fourths. Theory and Decision 15, Wagner, C. (1984): Avoiding Anscombe's Paradox. Theory and Decision 16, von Wright, G. H. (1971): Explanation and Understanding. Cornell University Press, Ithaca Young, H. P. (1975): Social Choice Scoring Functions. SIAM Journal of Applied Mathematics 28, Young, H. P. (1988): Condorcet's Theory of Voting. American Political Science Review 82, Young, H. P. (1994): Equity in Theory and Practice. Princeton University Press, Princeton.

15 Index Achen, Ch. H., 78, 135 Affuso, P. J., 113, 136 Alabama paradox, 109 Aleskerov, F., 121, 135 alternative vote, 49, 56, 60, 67, 119, 122 amendment system, 60, 88 anonymity, 91 Anscombe's paradox, 73, 77, 82, 126 Anscombe, G. E. M., 73, 135 approval voting, 86, 90, 95 Arrow, K. J., 95, 121, 125, 135 Balinski, M. L., 109, 112, 113, 135 Banks chain, 106, 107 set, 106, 107 Banks, J. S., 82, 106, 135 Barry, B., 44, 135 Berg, S., 21, 26, 28, 62, 136 Bezembinder, Th., 71, 72, 136 binary fairness, 111, 118

16 146 Voting Paradoxes and How to Deal with Them bipolar culture, 29 Birkhoff, G., 111, 115, 136 Birnbaum, P., 137 Bjurulf, B., 26, 136 Black, D., 18, 136 Boland, J., 26, 136 Borda count, 12, 15, 39, 54, 57, 68, 86, 90, score, 12-14, 36,57, 96, 126 winner, 13, 35, 37, 93 Borda's paradox, 3, 11, 28, 120 Borda, J. - C., 11, 12,22 Brams, S. J., 49, 63, 65, 74,86, 91, 113, 136, 138 Caste, N. J., 2, 138 Chernoff property, 95 Cohen, M. R., 79, 137 combination voting, 86 condition a, 95, 96, 98, 102, 104, 107 Condorcet condition moderate, 65 strong, 65 extension, 51, 62, 67, 68 loser, 12-15, 28, 96 paradox, 92, 94 portion, 36, 37, 39, 53 winner, 12, 13, 15, 22, 25, 26, 28, 35-37, 50-52, 57, 60, 65, 92, 100, 102, 127 Condorcet's maximal agreement, 18, 21, 23 paradox, 3, 26, 35, 50, 68, 85, 101, 120

17 Index 147 practical method, 21 successive reversal, 18, 20, 21, 23 Condorcet, Marquis de, 15, 17, 22 consistency, 89, 101 binary, 97 strict, 90 weak, 90 contraction consistency, 95 Coombs' method, 57,60 Copeland's method, 62, 67, 98 core, 65 covering, 101, 104 cross-level fallacy, 78 Cusanus, N., 11 Daudt, H., 71, 72, 137, 142 Daunou, P. C. F., 14, 24 De Grazia, A., 11, 137 Deb, R., 84, 137 deductive argument, 42 dictionary, 97 divisor method, 118 Dodgson's method, 60, 62 Dodgson, J. L., 60 dominance matrix, 7 Doring, H., 142 Doron, G., 91, 137 Downs, A., 43, 137 ecological fallacy, 78

18 148 Voting Paradoxes and How to Deal with Them Epstein, D., 104, 137 Euclidean space, 31 exact quota, 111 Ferejohn, J. A., 43, 46, 137 Fiorina, M. P., 43, 46, 137 Fishburn, P. C., 24, 26, 49, 56, 59, 60, 62, 63, 65, 91, 96, 137, 138 Gehrlein, W. V., 26, 28, 30, 138 Goodman, L. A., 78, 138 Gorman, J. L., 82, 138 Grofman, B., 140 Hamilton's method, 110, 115, 118 Holler, M. J., 143 homogeneity, 61 impartial culture, 26, 29 inter-profile paradox, 87, 121 intra-profile paradox, 87, 121 intransitivity, 15 Intriligator, M. D., 125, 127, 138 Kelly, J. S., 27, 28, 121, 138 Kelsey, D., 84, 137 Kemeny's method, 18, 21, 23, 60, 127 Kemeny, J. G., 18, 139 Kiersky, J. H., 2, 138 Kilgour, D. M., 74, 136 Kramer, G. H., 99, 139

19 Index 149 Lagerspetz, E., 11, 71, 72, 139 Laslier, J. - F., 95, 101, 139 Lepelley, D., 30, 138 Lively, J., 137 majority cycle, 99 rule, 15 McGarvey, D. C., 8, 139 McKelvey, R. D., 82, 106, 139 McLean, 1., 11, 14, 21, 22, 24, 139 Meehl, P. E., 43, 139 merit degree of, 13 order of, 12 Meskanen, T., 73, 141 method of largest remainders, 110 Michaud, P., 18, 139 Miller, N. R., 26, 101, 103, 106, 140 Miller, W. E., 78, 140 minimax regret, 48 monotonicity, 55, 60, 121 house, 111 paradox, 55, 68 population, 111 Moulin, H., 49, 51, 53, 101, 140 Munier, B. R., 141 Nagel, E., 79, 137 Nanson's method, 24, 57, 60 Nanson, E. J., 18, 21, 24, 140

20 150 Voting Paradoxes and How to Deal with Them neutrality, 91 Niemi, R. G., 26, 106, 139, 140 Niou, E. M. S., 24, 140 no-show paradox, 49, 53, 55, 62, 68, 118 Nurmi, H., 21, 26, 29, 56, 62, 64, 71, 73, 77, 90, 95, 122, 136, 140, 141 Ordeshook, P. C., 43-45, 142 Ostrogorski's paradox, 70, 77, 84, 121, 126 Ostrogorski, M., 70, 142.outranking matrix, 7, 14, 16, 19, 24 Owen, G., 140 Paastela, J., 141 pairwise comparison, 7, 10, 99, 106 paradox of liar, 2, 44 of multiple elections, 74, 85 of new states, 113 of redistribution, 114 Pareto criterion, 88 set, 104 violation, 88, 99 Parry, G., 137 path-independence, 121 Pliny the Younger, 11 plurality method, 119 runoff, 49, 56, 60, 68, 91, 119 system, 12, 15, 22, 28, 35, 54, 90, winner, 36, 37

21 Index 151 population paradox, 112 positional domination, 59 positional procedure, 15, 97, 98 practical syllogism, 41 preference cycle, 17 profile, 6-8 ranking, 5 relation, 15 collective, 16, 17 truncation, 63 Pufendorf, S., 11 q-rule, 82, 99 quota property, 115 Rae, D., 71, 72, 137, 142 ranking region, 33 Rasch, B. E., 89, 142 referendum paradox, 76 regret, 48 relation asymmetric, 8 binary, 5 complete, 5, 8, 16, 97 transitive, 5, 8, 16, 97 reversal portion, 36, 39 Richelson, J. T., 24, 56, 142 Riker, W. H., 2, 43-45, 142 risk-loving, 52 Robinson Crusoe fallacy, 43 Robinson, W. S., 78, 142

22 152 Voting Paradoxes and How to Deal with Them Saari triangle, 34, 35, 91 Saari, D. G., 31, 32, 34, 36, 82, 97, 142 Savage, L., 48, 143 Scarsini, M., 75, 143 Schofield, N., 82, 139 Schotter, A., 114, 143 Schwartz's paradox, 115, 118 Schwartz, Th., 24, 115, 117, 118, 143 Seaton, C. W., 110 Sen, A. K., 95, 143 separability, 76, 85, 127 Shakun, M. F., 141 Shively, W. Ph., 78, 135 Simpson's paradox, 79, 86, 112 Simpson, E. H., 79, 143 single transferable vote (STV), 91, 122 Snell, J. 1., 18, 139 Straffin, Ph. D., 56, 58, 143 successive system, 89 Todhunter, I., 18, 143 tournament matrix, 7, 8, 10, 62, 64, 101 truncation paradox, 63, 68, 69 truth-value, 2 Tsebelis, G., 43, 143 Tullock, G., 43, 144 uncovered set, 102, 104 unipolar culture, 29

23 Index 153 unit simplex, 32 Urken, A. B., 11, 14, 21, 22, 24, 139 Uusi-Heikkila, Y., 29, 141 Van Acker, P., 71, 72, 136 Van Deemen, A. M. A., 18, 144 von Wright, G. H., 144 Wagner, C., 73, 82, 84, 144 Weisberg, H. F., 26, 140 Young, H. P., 18, 20, 91, 109, 112, 113, 115, 135, 144 Zwicker, W. S., 74, 136