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

List of Figures 4.1 Vote Share Simplex 32 4.2 Two Areas of Electoral Outcomes 33 4.3 Saari Triangle 34 4.4 Condorcet's Paradox as a Saari Triangle 35 4.5 General Condorcet's Paradox Triangle 35 4.6 Example: Starting Triangle 37 4.7 Example: Condorcet Portion 38 4.8 Example: Reversal Portion 38 5.1 Saari Triangle of the 33-Voter Profile 52 8.1 The 6-Voter Subgroup Profile 92 8.2 The 15-Voter Subgroup Profile 93 8.3 The Combined Group Profile 94

List of Tables 2.1 A Preference Profile 6 2.2 A Pairwise Comparison Matrix 7 2.3 A Tournament Matrix 7 2.4 Preference Profile of Voters 1 and 2 9 2.5 Preference Profile of Voters 3 and.4 9 2.6 A Preference Profile Corresponding Table 2.3 10 2.7 Pairwise Comparison Matrix of Table 2.6 10 3.1 Borda's Paradox 12 3.2 Outranking Matrix for Table 3.1 14 3.3 Condorcet's Paradox 16 3.4 Example of Condorcet's Solution 17 3.5 Outranking Matrix for Table 3.4 17 3.6 Outranking Matrix for Unanimous Collective Ranking ABC 20 3.7 A Preference Profile for Illustration of Condorcet's Solutions 20 3.8 Outranking Matrix of Table 3.7 21 3.9 Nanson's Example 22 3.10 The Incompatibility of Borda's and Condorcet's Intuitions 22 3.11 Outranking Matrix Table 3.9 23 3.12 Condorcet's Paradox in Impartial Cultures 27 3.13 No Condorcet Winner and No Condorcet Paradox 28 3.14 The Frequency of Borda's Paradox 29 5.1 Voter Calculus: Payoffs 47

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 48 49 51 53 6.1 Plurality Runoff and Additional Support Paradox 57 6.2 Coombs' Procedure and Additional Support Paradox 58 6.3 Nanson's Procedure and Additional Support Paradox 59 6.4 Dodgson's Procedure and Additional Support Paradox 61 6.5 Monotonicity and Vulnerability to No-Show Paradox 63 6.6 Truncation Paradox and Alternative Vote 63 6.7 Truncation Paradox and Copeland's Procedure 64 6.8 Tournament Matrix of Table 6.7 64 6.9 Modified Tournament Matrix of Table 6.7 65 6.10 Truncation Paradox and Strong Condorcet Condition 66 6.11 Truncation Paradox and the Borda Count 68 7.1 Ostrogorski's Paradox 72 7.2 Ostrogorski's Paradox: Strict Version 72 7.3 Anscombe's Paradox 73 7.4 Ostrogorski's Paradox Which Is Not Anscombe's Paradox 74 7.5 Paradox of Multiple Elections 74 7.6 Strong Paradox of Multiple Elections 76 7.7 Referendum Paradox 77 7.8 Simpson's Paradox 79 7.9 Party Support and Simpson's Paradox 80 7.10 Intertemporal Simpson's Paradox 81 7.11 Gorman's Example of Anscombe's Paradox 83 7.12 Modified Anscombe's Paradox 83 8.1 Amendment Procedure and Pareto Criterion 88 8.2 Inconsistency of the Plurality Runoff 91 8.3 Borda Count Fails on Q 96 8.4 Plurality Voting Fails on Q 98 8.5 Uncovered Set and Condition Q: Example 1 103

List of Tables 133 8.6 Uncovered Set and Condition a : Dominance Matrix 1 103 8.7 Uncovered Set and Condition a: Dominance Matrix 2 104 8.8 Uncovered Set and Condition a: Example 2 105 9.1 The Alabama Paradox 110 9.2 The Population Paradox 112 9.3 The New States Paradox 113 9.4 The Paradox of Redistribution 115 9.5 Schwartz' Paradox: An Example 117 10.1 Some Features of Paradoxes 123 11.1 Probabilistic Choice Example 127

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Bibliography 143 Saari, D. G. (1997): The Generic Existence of a Core for q-rules. Economic Theory 9, 219-260 Savage, L. (1951): The Theory of Statistical Decision. Journal of the American Statistical Association 46, 55-67 Scarsini, M. (1998): A Strong Paradox of Multiple Elections. Social Choice and Welfare 15, 237-238 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, 309-323 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, 53-89 Simpson, E. H. (1951): The Interpretation of Interaction in Contingency Tables. Journal of the Royal Statistical Society B 13, 238-241 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

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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

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, 95-97 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

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

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

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

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, 96-98 winner, 36, 37

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

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

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