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.
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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