The lexicogfraphic prekernel

The lexicographic prekernel of a cooperative game with transferable utilities (TU) is a subset of the payoff vectors lexicographically minimizing the vector of maximal surpluses of one player over another one. This solution is non-empty for every TU game, it is efficient, is contained both in the prekernel and in the least core, and may not contain the prenucleolus [9]. A combinatorial characterization of the lexicographic prekernel being a weak analog of the known characterization of the prenucleolus by Kohlberg [4] with the help of balanced collections of coalitions is given. The difference consists in sets of vectors to be lexicographic minimized: the prenucleolus deals with excess vectors, and the lexicographi prekernel deals with vectors of maximal surpluses. It is shown that finding the lexicographic prekernel comes to solving a finite set (not more than the number of players) of optimization and of combinatorial problems.