The Nucleolus of a Matrix Game and Other Nucleoli

We define the nucleolus of a continuous convex map F: Π → R^m on a convex compact set Π ⊂ Rⁿ. As special cases we obtain known notions as nucleolus, prenucleolus and weighted nucleolus of a TU-game with (or without) coalition structure. Also the nucleolus of a matrix game turns out to be an interesting special case. It appears that the nucleolus of a matrix game coincides with the set of Dresher optimal strategy pairs of the game. This implies, in particular, that the nucleolus consists of the proper equilibria of the matrix game. To each (zero-normalized) TU-game one can construct a matrix game—the excess game—such that the nucleolus of the TU-game coincides with the unique proper optimal strategy of player II in the excess game. Also for other nucleoli of TU-games a suitable matrix game can be constructed where the nucleolus under consideration is related to the nucleolus of the matrix game in an analogous way. A balancedness condition is given characterizing nucleolus elements of a matrix game. It is shown that this balancedness result implies the known balancedness characterizations of Kohlberg, Sobolev, Owen and Wallmeier.