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Cooperative games in permutational structure

Listed author(s):
  • Gerard van der Laan

    (Department of Econometrics and Tinbergen Institute, Free University, De Boelelaan 1105, NL-1081 HV Amsterdam, THE NETHERLANDS)

  • Zaifu Yang

    (Institute of Socio-Economic Planning, The University of Tsukuba, Tsukuba, Ibaraki 305, JAPAN)

  • Dolf Talman

    (Department of Econometrics and CentER, Tilburg University, P.O. Box 90153, NL-5000 LE Tilburg, THE NETHERLANDS)

By a cooperative game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a set of payoffs for each coalition being a nonempty subset of the grand coalition of all players. It is well-known that balancedness is a sufficient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. In this paper we consider non-transferable utility games in which for any coalition the set of payoffs depends on a permutation or ordering upon any partition of the coalition into subcoalitions. We call such a game a cooperative game in permutational structure or shortly permutational game. Doing so we extend the scope of the standard cooperative game theory in dealing with economic or political problems. Next we define the concept of core for such games. By introducing balancedness for ordered partitions of coalitions, we prove the nonemptiness of the core of a balanced non-transferable utility permutational game. Moreover we show that the core of a permutational game coincides with the core of an induced game in coalitional structure, but that balancedness of the permutational game need not imply balancedness of the corresponding coalitional game. This leads to a weakening of the conditions for the existence of a nonempty core of a game in coalitional structure, induced by a game in permutational structure. Furthermore, we refine the concept of core for the class of permutational games. We call this refinement the balanced-core of the game and show that the balanced-core of a balanced permutational game is a nonempty subset of the core. The proof of the nonemptiness of the core of a permutational game is based on a new intersection theorem on the unit simplex, which generalizes the well-known intersection theorem of Shapley.

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Article provided by Springer & Society for the Advancement of Economic Theory (SAET) in its journal Economic Theory.

Volume (Year): 11 (1998)
Issue (Month): 2 ()
Pages: 427-442

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Handle: RePEc:spr:joecth:v:11:y:1998:i:2:p:427-442
Note: Received: October 31, 1995; revised version: February 5, 1997
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  1. van der Laan, G. & Talman, A.J.J. & Yang, Z., 1994. "Intersection theorems on polytopes," Discussion Paper 1994-20, Tilburg University, Center for Economic Research.
  2. Nowak Andrzej S. & Radzik Tadeusz, 1994. "The Shapley Value for n-Person Games in Generalized Characteristic Function Form," Games and Economic Behavior, Elsevier, vol. 6(1), pages 150-161, January.
  3. Kamiya, K. & Talman, D., 1990. "Variable Dimension Simplicial Algorithm For Balanced Games," Papers 9025, Tilburg - Center for Economic Research.
  4. Ichiishi, Tatsuro & Idzik, Adam, 1991. "Closed Covers of Compact Convex Polyhedra," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(2), pages 161-169.
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