Bounds for mixed strategy equilibria and the spatial model of elections. (English) Zbl 1022.91001

Authors’ summary: We prove that the support of mixed strategy equilibria of two-player, symmetric, zero-sum games lies in the uncovered set, a concept originating in the theory of tournaments, and the spatial theory of politics. We allow for uncountably infinite strategy spaces, and as a special case, we obtain a lone-standing claim to the same effect, due to R. McKelvey [Am. J. Polit. Sci. 30, 283-314 (1986)] in the political science literature. Further, we prove the nonemptiness of the uncovered set under quite general assumptions, and we establish, under various assumptions, the coanalyticity and measurability of this set. In the concluding section, we indicate how the inclusion result may be extended to multiplayer, non-zero-sum games.


91A05 2-person games
91A10 Noncooperative games
91C99 Social and behavioral sciences: general topics
Full Text: DOI Link


[1] Austen-Smith, D.; Banks, J., Cycling of simple rules in the spatial model, Soc. Choice Welfare, 16, 663-672 (1999) · Zbl 1066.91510
[2] Banks, J., Singularity theory and core existence in the spatial model, J. Math. Econ., 24, 523-536 (1995) · Zbl 0843.90006
[3] J. Banks, J. Duggan, and, M. Le Breton, The spatial model of elections with arbitrary distributions of voters, mimeo, 1998.; J. Banks, J. Duggan, and, M. Le Breton, The spatial model of elections with arbitrary distributions of voters, mimeo, 1998.
[4] Bernheim, D., Rationalizable strategic behavior, Econometrica, 52, 1007-1028 (1984) · Zbl 0552.90098
[5] Bordes, G.; Le Breton, M.; Salles, M., Gillies and Miller’s subrelations of a relation over an infinite set of alternatives: General results and applications to voting games, Math. Op. Res., 17, 509-518 (1992) · Zbl 0760.90003
[6] Cohen, L., Cycle sets in multidimensional voting models, J. Econ. Theory, 20, 1-12 (1979) · Zbl 0404.90004
[7] Cox, G., Non-collegial simple games and the nowhere denseness of the set of preference profiles having a core, Soc. Choice Welfare, 1, 159-164 (1984) · Zbl 0587.90105
[8] Cox, G., The uncovered set and the core, Amer. J. Polit. Sci., 31, 408-422 (1987)
[9] Dellacherie, C.; Meyer, P.-A, Probabilities and Potential (1978), North-Holland: North-Holland New York · Zbl 0494.60001
[10] Duggan, J.; Le Breton, M., Mixed refinements of Shapley’s saddles and weak tournaments, CORE Discussion Paper 9921 (1999) · Zbl 1069.91503
[11] Duggan, J.; Le Breton, M., Mixed refinements of Shapley’s saddles and weak tournaments, Soc. Choice Welfare, 18, 65-78 (2001) · Zbl 1069.91503
[12] Dutta, B., Covering sets and a new condorcet choice correspondence, J. Econ. Theory, 44, 63-80 (1988) · Zbl 0652.90013
[13] Dutta, B.; Laslier, J.-F, Comparison functions and choice correspondences, Soc. Choice Welfare, 16, 513-532 (1999) · Zbl 1066.91535
[14] D. Fisher, and, J. Ryan, Optimal strategies for a generalized “scissors, paper, and stone” game, mimeo, 1991.; D. Fisher, and, J. Ryan, Optimal strategies for a generalized “scissors, paper, and stone” game, mimeo, 1991. · Zbl 0776.05047
[15] Fishburn, P., Condorcet social choice functions, SIAM J. Appl. Math., 33, 469-489 (1977) · Zbl 0369.90002
[16] Gilles, D., Solutions to general non-zero-sum games, (Tucker, A.; Luce, R., Contributions to the Theory of Games IV. Contributions to the Theory of Games IV, Annals of Mathematics Studies, 40 (1959), Princeton Univ. Press: Princeton Univ. Press Princeton) · Zbl 0085.13106
[17] Glicksberg, I., A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points, Proc. Amer. Math. Soc., 3, 170-174 (1952) · Zbl 0046.12103
[18] Kramer, G., Existence of electoral equilibrium, (Ordeshook, P., Game Theory and Political Science (1978), New York Univ. Press: New York Univ. Press New York)
[19] Laffond, G.; Laslier, J.; Le Breton, M., The bipartisan set of a tournament game, Games Econ. Behav., 5, 182-201 (1993) · Zbl 0770.90080
[20] Larman, D., Projecting and uniformising Borel sets with \(K_σ\)-sections II, Mathematika, 20, 233-246 (1973) · Zbl 0287.54040
[21] Le Breton, M., On the core of voting games, Soc. Choice Welfare, 4, 295-305 (1987) · Zbl 0636.90099
[22] McKelvey, R., Intransitivities in multi-dimensional voting models and some implications for agenda control, J. Econ. Theory, 12, 472-482 (1976) · Zbl 0373.90003
[23] McKelvey, R., General conditions for global intransitivities in formal voting models, Econometrica, 47, 1086-1112 (1979) · Zbl 0411.90009
[24] McKelvey, R., Covering, dominance, and institution-free properties of social choice, Amer. J. Polit. Sci., 30, 283-314 (1986)
[25] McKelvey, R.; Schofield, N., Generalized symmetry conditions at a core point, Econometrica, 55, 923-934 (1987) · Zbl 0617.90004
[26] Miller, N., A new solution set for tournaments and majority voting: Further graph-theoretical approaches to the theory of voting, Amer. J. Polit. Sci., 24, 68-96 (1980)
[27] Moulin, H., Choosing from a tournament, Soc. Choice Welfare, 3, 271-291 (1986) · Zbl 0618.90004
[28] Pearce, D., Rationalizable strategic behavior and the problem of perfection, Econometrica, 52, 1029-1050 (1984) · Zbl 0552.90097
[29] Plott, C., A notion of equilibrium and its possibility under majority rule, Amer. Econ. Rev., 57, 787-806 (1967)
[30] Riker, W., Implications from the disequilibrium of majority rule for the study of institutions, Amer. Polit. Sci. Rev., 74, 432-446 (1980)
[31] Rubinstein, A., A note on the nowhere denseness of societies having an equilibrium under majority rule, Econometrica, 47, 511-514 (1979) · Zbl 0416.90006
[32] Rudin, W., Real and Complex Analysis (1966), McGraw-Hill: McGraw-Hill New York · Zbl 0148.02904
[33] Saari, D., The generic existence of a core for \(q\)-rules, Econ. Theory, 9, 219-260 (1997) · Zbl 0886.90043
[34] Schofield, N., Generic instability of majority rule, Rev. Econ. Stud., 50, 695-705 (1983) · Zbl 0521.90011
[35] Stinchcombe, M.; White, H., Some measurability results for extrema of random functions over random sets, Rev. Econ. Stud., 59, 495-512 (1992) · Zbl 0752.60008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.