Voting games and acyclic collective choice rules. (English) Zbl 0886.90023
Summary: The subject of this paper is the representation of collective choice rules by voting games and the acyclicity of these rules. A collective choice rule is a function that associates a collective preference with every profile of individual preferences. Such a rule is acyclic if it always yields an acyclic collective preference. A voting game is either a simple game or a non-neutral version of such a game called a binary game in constitutional form. Both are special forms of cooperative games that simply specify the structure of power in a society or an organization. The power structure conferred by certain collective choice rules takes the form of a voting game. The paper classifies collective choice rules that can or cannot be represented by a voting game. Conditions for the acyclicity of the collective choice rules that can be represented by a voting game are then obtained from the structure of their corresponding voting games. These results are applied to a large class of voting rules defined by quotas.
References:
| [1] | Andjiga, N. G.; Moulen, J., Binary games in constitutional form and collective choice, Math. Soc. Sci., 16, 189-201 (1988) · Zbl 0699.90006 |
| [2] | Blau, J. H.; Deb, R., Social decision function and the veto, Econometrica, 45, 871-879 (1977) · Zbl 0362.90002 |
| [3] | Blair, D. H.; Pollak, R. A., Acyclic collective choice rules, Econometrica, 50, 931-943 (1982) · Zbl 0483.90009 |
| [4] | Bloomfield, S. D., A social choice interpretation of the von Neumann-Morgenstern game, Econometrica, 44, 105-114 (1976) · Zbl 0382.90117 |
| [5] | Brown, J. D., Aggregation of preferences, Quart. J. Econom., 89, 3, 456-469 (1975) |
| [6] | Deb, R., \(k\)-Monotone social decision functions and the veto, Econometrica, 49, 899-909 (1981) · Zbl 0486.90007 |
| [7] | Deslierres, M.; Truchon, M., Monotonic voting rules, mimeo (1994) |
| [8] | Ferejohn, J. A.; Fishburn, P. C., Representations of binary decision rules by generalized decisiveness structures, J. Econom. Theory, 21, 28-45 (1979) · Zbl 0413.90005 |
| [9] | Ferejohn, J. A.; Grether, D. M., On a class of rational social decision procedures, J. Econom. Theory, 8, 471-482 (1974) |
| [10] | Gibbard, A., Social choice and the Arrow condition, mimeo (1979) |
| [11] | Kelsey, D., Acyclic choice and group veto, Soc. Choice and Welfare, 2, 131-137 (1985) · Zbl 0575.90006 |
| [12] | Le Breton, M., On the core of voting games, Soc. Choice and Welfare, 4, 295-305 (1987) · Zbl 0636.90099 |
| [13] | Mas-Colell, A.; Sonnenschein, H., General possibility theorems for group decision functions, Rev. Econom. Studies, 39, 185-192 (1972) · Zbl 0239.90003 |
| [14] | Moulin, H., From social welfare ordering to acyclic aggregation of preferences, Math. Soc. Sci., 9, 1-17 (1985) · Zbl 0577.90005 |
| [15] | Moulin, H., Axioms of Cooperative Decision Making (1988), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0699.90001 |
| [16] | Nakamura, K., The vetoers in a simple game with ordinal preferences, Int. J. Game Theory, 8, 55-61 (1979) · Zbl 0415.90087 |
| [17] | Peleg, B., Representations of simple games by social choice functions, Int. J. Game Theory, 7, 81-94 (1978) · Zbl 0391.90007 |
| [18] | Peleg, B., Game Theoretic Analysis of Voting in Committees (1984), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0563.90002 |
| [19] | Truchon, M., Acyclicity and decisiveness structures, (Cahier de recherche, 9316 (1993), Département d’économique, Université Laval) · Zbl 0856.90006 |
| [20] | Von Neumann, J.; Morgenstern, O., Theory of Games and Economic Behavior (1953), Princeton University Press: Princeton University Press Princeton, NJ, Ch. X, Simple Games · Zbl 0053.09303 |
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