Constrained core solutions for totally positive games with ordered players
- 383 Downloads
In many applications of cooperative game theory to economic allocation problems, such as river-, polluted river- and sequencing games, the game is totally positive (i.e., all dividends are nonnegative), and there is some ordering on the set of the players. A totally positive game has a nonempty core. In this paper we introduce constrained core solutions for totally positive games with ordered players which assign to every such a game a subset of the core. These solutions are based on the distribution of dividends taking into account the hierarchical ordering of the players. The Harsanyi constrained core of a totally positive game with ordered players is a subset of the core of the game and contains the Shapley value. For special orderings it coincides with the core or the Shapley value. The selectope constrained core is defined for acyclic orderings and yields a subset of the Harsanyi constrained core. We provide a characterization for both solutions.
KeywordsTotally positive TU-game Digraph Harsanyi dividends Core Shapley value Harsanyi set Selectope Polluted river games
JEL ClassificationC71 (Cooperative games)
This research has been done while the third author was visiting the Tinbergen Institute, VU University Amsterdam, on NWO-Grant 047.017.017 within the framework of Dutch-Russian Cooperation. This author also appreciates financial support from the Russian Leading Scientific Schools Fund (Grant 4113.2008.6) and Russian Fund for Basic Research (Grant 13-06-00311). We thank two anonymous referees and the associate editor for their valuable comments.
- Gillies DB (1953) Some theorems on \(n\)-person games. PhD Thesis, Princeton UniversityGoogle Scholar
- Graham DA, Marshall RC, Richard J-F (1990) Differential payments within a bidder coalition and the Shapley value. Am Econ Rev 80:493–510Google Scholar
- Hammer PL, Peled UN, Sorensen S (1977) Pseudo-Boolean functions and game theory. I. Core elements and Shapley value. Cah CERO 19:159–176Google Scholar
- Harsanyi JC (1959) A bargaining model for cooperative n-person games. In: Tucker AW, Luce RD (eds) Contributions to the theory of games IV. Princeton University Press, Princeton, pp 325–355Google Scholar
- Kilgour DM, Dinar A (1995) Are stable agreements for sharing international river water now possible? Policy Research Working Paper 1474. World Bank, WashingtonGoogle Scholar
- Shapley LS (1953) A value for N-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games II. Princeton University Press, Princeton, pp 307–317Google Scholar
- Vasil’ev VA (1975) The Shapley value for cooperative games of bounded polynomial variation. Optim Vyp 17:5–27 (in Russian)Google Scholar
- Vasil’ev VA (1978) Support function of the core of a convex game. Optim Vyp 21:30–35 (in Russian)Google Scholar
- Vasil’ev VA (1981) On a class of imputations in cooperative games. Sov Math Dokl 23:53–57Google Scholar
- Vasil’ev VA, van der Laan G (2002) The Harsanyi set for cooperative TU-games. Siberian Adv Math 12:97–125Google Scholar