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The Lovász Extension of Market Games

Author

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  • E. Algaba
  • J.M. Bilbao
  • J.R. Fernández
  • A. Jiménez

Abstract

The multilinear extension of a cooperative game was introduced by Owen in 1972. In this contribution we study the Lovász extension for cooperative games by using the marginal worth vectors and the dividends. First, we prove a formula for the marginal worth vectors with respect to compatible orderings. Next, we consider the direct market generated by a game. This model of utility function, proposed by Shapley and Shubik in 1969, is the concave biconjugate extension of the game. Then we obtain the following characterization: The utility function of a market game is the Lovász extension of the game if and only if the market game is supermodular. Finally, we present some preliminary problems about the relationship between cooperative games and combinatorial optimization. Copyright Kluwer Academic Publishers 2004

Suggested Citation

  • E. Algaba & J.M. Bilbao & J.R. Fernández & A. Jiménez, 2004. "The Lovász Extension of Market Games," Theory and Decision, Springer, vol. 56(1), pages 229-238, April.
    • Handle: RePEc:kap:theord:v:56:y:2004:i:1:p:229-238
    DOI: 10.1007/s11238-004-5650-6
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    References listed on IDEAS

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    1. Kannai, Yakar, 1992. "The core and balancedness," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 12, pages 355-395, Elsevier.
    2. Shapley, Lloyd S. & Shubik, Martin, 1969. "On market games," Journal of Economic Theory, Elsevier, vol. 1(1), pages 9-25, June.
    3. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
    4. Einy, Ezra & Wettstein, David, 1996. "Equivalence between Bargaining Sets and the Core in Simple Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 25(1), pages 65-71.
    5. Ichiishi, Tatsuro, 1981. "Super-modularity: Applications to convex games and to the greedy algorithm for LP," Journal of Economic Theory, Elsevier, vol. 25(2), pages 283-286, October.
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    Cited by:

    1. Faigle, U. & Grabisch, M. & Heyne, M., 2010. "Monge extensions of cooperation and communication structures," European Journal of Operational Research, Elsevier, vol. 206(1), pages 104-110, October.
    2. Marichal, Jean-Luc & Kojadinovic, Ivan, 2008. "Distribution functions of linear combinations of lattice polynomials from the uniform distribution," Statistics & Probability Letters, Elsevier, vol. 78(8), pages 985-991, June.
    3. Casajus, André & Kramm, Michael & Wiese, Harald, 2020. "Asymptotic stability in the Lovász-Shapley replicator dynamic for cooperative games," Journal of Economic Theory, Elsevier, vol. 186(C).

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