Abstract
This paper proposes and justifies a natural way to weaken the concept of covering relation defined on a finite tournament. Various weak covering relations, calledk-covering relations, are introduced. To eachk-covering relation corresponds a strong uncovered set containing all nonk-covered outcomes. It is proved that those strong uncovered sets may be empty. Moreover, the set of all tournaments having an empty strong uncovered set is characterized within two rather large classes of tournaments. Finally, we offer a complete study of the cases where the directed graph defined by ak-covering relation coincides with the initial tournament.
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References
Banks, J. S.: 1985, ‘Sophisticated Voting Outcomes and Agenda Control’,Social Choice and Welfare,4, 295–306.
Banks, J. S., Bordes, G., and Lebreton, M.: 1991, ‘Covering Relations, Closest Orderings and Hamiltonian Bypaths in Tournaments’,Social Choice and Welfare,8, 355–363.
Cox, G. W.: 1987, ‘The Uncovered Set and the Core’,American Journal of Political Science,31, 408–442.
Dutta, B.: 1988, ‘Covering Sets and a New Condorcet Choice Correspondence’,Journal of Economic Theory,44, 63–80.
Dutta, B.: 1990, ‘On the Tournament Equilibrium Set’,Social Choice and Welfare,7, 381–383.
Feld, S. L., Grofman, B., Hartley, R., Kilgour, M., Miller, N., and Noviello, N.: 1987, ‘The Uncovered Set in Spatial Voting Games’,Theory and Decision,23, 129–155.
Gillies, D. B.: 1959, ‘Solutions to General Non-Zero-Sum Games’, inContributions to the Theory of Games, Vol. IV, A. W. Tucker and R. D. Luce (Eds.), Princeton University Press.
Henriet, D.: 1985, ‘The Copeland Choice Function: An Axiomatic Formulation,Social Choice and Welfare,2, 49–63.
Laffond, G. and Laine J.: 1991a, ‘A Note on Weakly Transitive Strong Tournaments’,Keele University Working Paper Series.
Laffond, G., Laine, J., and Laslier, J. F.: 1993a, ‘On Regular Composed Tournaments’, forthcoming inModels and Experiments on Risk and Rationality, M. Machina and B. Munier (Eds.), Kluwer.
Laffond, G., Laine, J., and Laslier, J. F.: 1993b, ‘Composition Consistent Tournament Solutions and Social Choice Functions’,Keele University Working Paper Series.
Laffond, G. and Laslier, J. F.: 1991b, ‘Slater's Winner of a Tournament may not be the Banks Set’,Social Choice and Welfare,8, 365–369.
Laffond, G., Laslier, J. F., and Lebreton, M.: 1992, ‘Condorcet-consistent Choice Correspondences: A Set-theoretical Comparison’,CNAM Discussion Paper.
McGarvey, D. S.: 1953, ‘A Theorem on the Construction of Voting Paradoxes’,Econometrica,21, 608–610.
MacKelvey, R. D.: 1986, ‘Covering, Dominance and Institution-Free Properties of Social Choice’,American Journal of Political Sciences,30, 283–314.
Miller, N.: 1977, ‘Graph-Theoretical Approaches to the Theory of Voting’,American Journal of Political Sciences,21, 769–803.
Miller, N.: 1980, ‘A New Solution Set for Tournaments and Majority Voting: Further Graph Theoretical Approaches to the Theory of Voting’,American Journal of Political Sciences,24, 68–96.
Moon, J. W.: 1968, ‘Topics on Tournaments’, Holt, Rinehart and Winston.
Moulin, H.: 1986, ‘Choosing from a Tournament’,Social Choice and Welfare,2, 271–291.
Rubinstein, A.: 1980, ‘Ranking the Participants in a Tournament’,SIAM Journal of Applied Mathematics,98, 108–111.
Schwartz, T.: 1990, ‘Cyclic Tournaments and Cooperative Majority Voting: A Solution’,Social Choice and Welfare,7, 19–29.
Shepsle, K. and Weingast, B.: 1982, ‘Uncovered Sets and Sophisticated Voting Outcomes with Implications for Agenda Institutions’,American Journal of Political Sciences,28, 49–74.
Stearns, R.: 1959, ‘The Voting Problem’,American Mathematical Monthly,66, 761–763.
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Laffond, G., Laine, J. Weak covering relations. Theor Decis 37, 245–265 (1994). https://doi.org/10.1007/BF01079911
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DOI: https://doi.org/10.1007/BF01079911