Abstract
We provide two new characterizations of exact games. First, a game is exact if and only if it is exactly balanced; and second, a game is exact if and only if it is totally balanced and overbalanced. The condition of exact balancedness is identical to the one of balancedness, except that one of the balancing weights may be negative, while for overbalancedness one of the balancing weights is required to be non-positive and no weight is put on the grand coalition. Exact balancedness and overbalancedness are both easy to formulate conditions with a natural game-theoretic interpretation and are shown to be useful in applications. Using exact balancedness we show that exact games are convex for the grand coalition and we provide an alternative proof that the classes of convex and totally exact games coincide. We provide an example of a game that is totally balanced and convex for the grand coalition, but not exact. Finally we relate classes of balanced, totally balanced, convex for the grand coalition, exact, totally exact, and convex games to one another.
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References
Azrieli Y, Lehrer E (2005) Concavification and convex games. Working Paper
Biswas AK, Parthasarathy T, Potters JAM, Voorneveld M (1999) Large cores and exactness. Games Econ Behav 28: 1–12
Bondareva ON (1963) Some applications of linear programming methods to the theory of cooperative games (in Russian). Problemy Kybernetiki 10: 119–139
Branzei R, Tijs S, Zarzuelo J (2009) Convex multi-choice games: characterizations and monotonic allocation schemes. Eur J Oper Res 198: 571–575
Calleja P, Borm P, Hendrickx R (2005) Multi-issue allocation situations. Eur J Oper Res 164: 730–747
Carpente L, Casas-Méndez B, García-Jurado I, van den Nouweland A (2010) The truncated core for games with upper bounds. Int J Game Theory 39: 645–656
Casas-Méndez B, García-Jurado I, van den Nouweland A, Vázquez-Brage M (2003) An extension of the τ-value to games with coalition structures. Eur J Oper Res 148: 494–513
Csóka P, Herings PJJ, Kóczy LÁ (2009) Stable allocations of risk. Games Econ Behav 67(1): 266–276
Csóka P, Herings PJJ, Kóczy LÁ, Pintér M (2011) Convex and exact games with non-transferable utility. Eur J Oper Res 209: 57–62
Derks J, Reijnierse H (1998) On the core of a collection of coalitions. Int J Game Theory 27: 451–459
Dubey P, Shapley LS (1984) Totally balanced games arising from controlled programming problems. Mathematical Program 29: 245–267
Fang Q, Fleischer R, Li J, Sun X (2010) Algorithms for core stability, core largeness, exactness, and extendability of flow games. Front Math Chin 5: 47–63
Kalai E, Zemel E (1982a) Generalized network problems yielding totally balanced games. Oper Res 30: 998–1008
Kalai E, Zemel E (1982b) Totally balanced games and games of flow. Math Oper Res 7: 476–478
Legut J (1990) On totally balanced games arising from cooperation in fair division. Games Econ Behav 2: 47–60
Owen G (1975) On the core of linear production games. Math Program 9: 358–370
Predtetchinski A, Herings PJJ (2004) A necessary and sufficient condition for the non-emptiness of the core of a non-transferable utility game. J Econ Theory 116: 84–92
Schmeidler D (1972) Cores of exact games. J Math Anal Appl 40: 214–225
Shapley LS (1967) On balanced sets and cores. Nav Res Logist Q 14: 453–460
Shapley LS (1971) Cores of convex games. Int J Game Theory 1: 11–26
Shapley LS, Shubik M (1969) On market games. J Econ Theory 1: 9–25
Tijs S, Parthasarathy T, Potters J, Prassad VR (1984) Permutation games: another class of totally balanced games. OR Spektrum 6: 119–123
van Velzen B, Hamers H, Solymosi T (2008) Core stability in chain-component additive games. Games Econ Behav 62: 116–139
Acknowledgments
Péter Csóka thanks funding by the project TÁMOP-4.2.1/B-09/1/KMR-2010-0005 and the Hungarian Academy of Sciences under its Momentum programme (LD-004/2010). P. Jean-Jacques Herings would like to thank the Netherlands Organisation for Scientific Research (NWO) for financial support. László Á. Kóczy thanks funding by the European Commission under the Marie Curie Intra-European Fellowship MEIF-CT-2004-011537, the Reintegration Grant PERG-GA-2008-230879 as well as by the OTKA (Hungarian Scientific Research Fund) for the project NF-72610 and the support of the Hungarian Academy of Sciences under its Momentum programme (LD-004/2010).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Csóka, P., Herings, P.JJ. & Kóczy, L.Á. Balancedness conditions for exact games. Math Meth Oper Res 74, 41–52 (2011). https://doi.org/10.1007/s00186-011-0348-3
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DOI: https://doi.org/10.1007/s00186-011-0348-3