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Lexicographic Composition of Simple Games


  • Barry O'Neill
  • Bezalel Peleg


A two-house legislature can often be modelled as a proper simple game whose outcome depends on whether a coalition wins, blocks or loses in two smaller proper simple games. It is shown that there are exactly five ways to combine the smaller games into a larger one. This paper focuses on one of the rules, lexicographic composition, where a coalition wins in G1 => G2 when it either wins in G1, or blocks in G1 and wins in G2. It is the most decisive of the five. A lexicographically decomposable game is one that can be represented in this way using components whose player sets partition the whole set. Games with veto players are not decomposable, and anonymous games are decomposable if and only if they are decisive and have two or more players. If a player's benefit is assessed by any semi-value, then for two isomorphic games a player is better off from having a role in the first game than having the same role in the second. Lexicographic decomposability is sometimes compatible with equality of roles. A relaxation of it is suggested for its practical benefits.

Suggested Citation

  • Barry O'Neill & Bezalel Peleg, 2006. "Lexicographic Composition of Simple Games," Discussion Paper Series dp415, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
  • Handle: RePEc:huj:dispap:dp415

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    References listed on IDEAS

    1. Pradeep Dubey & Abraham Neyman & Robert James Weber, 1981. "Value Theory Without Efficiency," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 122-128, February.
    2. Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
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    Cited by:

    1. Yokote, Koji & Funaki, Yukihiko & Kamijo, Yoshio, 2016. "A new basis and the Shapley value," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 21-24.
    2. Rudolf Berghammer & Agnieszka Rusinowska & Harrie de Swart, 2009. "A Relation-algebraic Approach to Simple Games," Working Papers 0913, Groupe d'Analyse et de Théorie Economique Lyon St-Étienne (GATE Lyon St-Étienne), Université de Lyon.

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    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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