Convexity properties for interior operator games
- 48 Downloads
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.
KeywordsCooperative game Antimatroid Interior operator Convexity
Unable to display preview. Download preview PDF.
- Driessen, T. S. H. (1988). Cooperative games. Solutions and applications. Dordrecht: Kluwer Academic. Google Scholar
- Driessen, T. S. H. (1991). k-Convexity of big boss games and clan games. Methods of Operations Research, 64, 267–275. Google Scholar
- Gillies, D. B. (1953). Some theorems on n -person games. Ph.D. thesis, Princeton University Press, Princeton. Google Scholar
- Goecke, O., Korte, B., & Lovász, L. (1986). Examples and algorithmic properties of greedoids. In B. Simeone (Ed.), Combinatorial optimization. Berlin: Springer. Google Scholar
- Jiménez-Losada, A. (1998). Valores para juegos sobre estructuras combinatorias. Ph.D. thesis, University of Seville, Spain. Google Scholar
- Korte, B., Lóvasz, L., & Schrader, R. (1991). Greedoids. Berlin: Springer. Google Scholar
- Muto, S., Nakayama, M., Potters, J., & Tijs, S. (1987). On big boss games. Economic Studies Quarterly, 39, 303–321. Google Scholar
- Tijs, S. (1981). Bounds for the core and the τ-value. In O. Moeschlin & D. Pallaschke (Eds.), Game theory and mathematical economics (pp. 123–132). Amsterdam: North-Holland. Google Scholar