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Convexity properties for interior operator games

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  • J. Bilbao
  • C. Chacón
  • A. Jiménez-Losada
  • E. Lebrón

Abstract

Interior operator games arose by abstracting some properties of several types of cooperative games (for instance: peer group games, big boss games, clan games and information market games). This reason allow us to focus on different problems in the same way. We introduced these games in Bilbao et al. (Ann. Oper. Res. 137:141–160, 2005 ) by a set system with structure of antimatroid, that determines the feasible coalitions, and a non-negative vector, that represents a payoff distribution over the players. These games, in general, are not convex games. The main goal of this paper is to study under which conditions an interior operator game verifies other convexity properties: 1-convexity, k-convexity (k≥2 ) or semiconvexity. But, we will study these properties over structures more general than antimatroids: the interior operator structures. In every case, several characterizations in terms of the gap function and the initial vector are obtained. We also find the family of interior operator structures (particularly antimatroids) where every interior operator game satisfies one of these properties. Copyright Springer Science+Business Media, LLC 2008

Suggested Citation

  • J. Bilbao & C. Chacón & A. Jiménez-Losada & E. Lebrón, 2008. "Convexity properties for interior operator games," Annals of Operations Research, Springer, vol. 158(1), pages 117-131, February.
    • Handle: RePEc:spr:annopr:v:158:y:2008:i:1:p:117-131:10.1007/s10479-007-0244-7
    DOI: 10.1007/s10479-007-0244-7
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    1. Rodica Brânzei & Vito Fragnelli & Stef Tijs, 2002. "Tree-connected peer group situations and peer group games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 55(1), pages 93-106, March.
    2. Gilles, Robert P & Owen, Guillermo & van den Brink, Rene, 1992. "Games with Permission Structures: The Conjunctive Approach," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(3), pages 277-293.
    3. J. Bilbao & A. Jiménez-Losada & E. Lebrón & C. Chacón, 2005. "Values for Interior Operator Games," Annals of Operations Research, Springer, vol. 137(1), pages 141-160, July.
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    2. Keyzer, Michiel & van Wesenbeeck, Cornelia, 2011. "Optimal coalition formation and surplus distribution: Two sides of one coin," European Journal of Operational Research, Elsevier, vol. 215(3), pages 604-615, December.

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