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Covering relations, closest orderings and hamiltonian bypaths in tournaments


We show that the Slater's set of a tournament, i.e. the set of the top elements of the closest orderings, is a subset of the top cycle of the uncovered set of the tournament. We also show that the covering relation is related to the hamiltonian bypaths of a strong tournament in that if x covers y, then there exists an hamiltonian bypath from x to y.

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  • Banks JS (1985) Sophisticated voting outcomes and agenda control. Soc Choice W elfare 1: 295–306

    Google Scholar 

  • Banks JS, Bordes G (1988) Voting games, indifference, and consistent sequential choice rules. Soc Choice Welfare 5: 31–44

    Google Scholar 

  • Bermond JC (1972) Ordres a distance minimum d'un tournoi et graphes partiels sans circuits maximaux. Math Sci Hum 37: 5–25

    Google Scholar 

  • Bermond JC, Thomassen C (1981) Cycles in digraphs — A survey. J Graph Theory 5: 1–43

    Google Scholar 

  • Bordes G (1983) On the possibility of reasonable consistent majoritarian choice: some positive results. J Econ Theory 31: 122–132

    Google Scholar 

  • Camion P (1959) Chemins et circuits hamiltoniens des graphes complets. CR Acad Sci Paris 249: 2151–2152

    Google Scholar 

  • Dutta B (1988) Covering sets and a new Condorcet choice correspondence. J Econ Theory 44: 63–80

    Google Scholar 

  • Fishburn PC (1977) Condorcet social choice functions. SIAM J Math 33: 469–489

    Google Scholar 

  • Gillies DB (1959) Solutions to general non-zero sum games. In: Tucker AW, Luce RD (eds) Contributions to the theory of games, IV (Ann Math Stud, No. 40) Princeton University Press, Princeton, NJ

    Google Scholar 

  • Jacquet-Lagreze E (1969) L'agrégation des opinions individuelles. Inform Sci Hum 4: 1–21

    Google Scholar 

  • Kemeny JG (1959) Mathematics without numbers. Dædalus 88: 577–591

    Google Scholar 

  • Laffond G, Laslier J-F (1991) Slater's winners of a tournament may not be in the Banks set. Soc Choice Welfare 8: 365–369

    Google Scholar 

  • Miller NR (1980) A new solution set for tournament and majority voting: further graphtheoritical approaches to the theory of voting. Am J Polit Sci 24: 68–96

    Google Scholar 

  • Monjardet B (1984) Unpublished lecture notes

  • Moulin H (1986) Choosing from a tournament. Soc Choice Welfare 3: 271–291

    Google Scholar 

  • Richelson JT (1981) Majority rule and collective choice. Mimeo

  • Schwartz T (1990) Cyclic tournaments and cooperative majority voting: a solution. Soc Choice Welfare 7: 19–29

    Google Scholar 

  • Slater P (1961) Inconsistencies in a schedule of paired comparisons. Biometrica 48: 303–312

    Google Scholar 

  • Thomassen C (1980) Hamiltonian connected tournaments. J Combinat Theory B 28: 142–163

    Google Scholar 

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We thank B. Monjardet and an anonymous editor for helpful suggestions.

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Banks, J.S., Bordes, G. & Le Breton, M. Covering relations, closest orderings and hamiltonian bypaths in tournaments. Soc Choice Welfare 8, 355–363 (1991).

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  • Covering Relation
  • Close Ordering
  • Strong Tournament