ON WEAKLY BALANCED GAMES AND DUALITY THEORY

Abstract : The notion of a 'weakly balanced game' under very general conditions involving no topology whatever is defined. The work of Schmeidler is extended by establishing duality results for a pair of (possibly) infinite dimensional linear programming problems arising from a generalized game. A necessary and sufficient condition is given in order that a separating hyperplane argument can be employed to prove the existence of a candidate core member for a weakly balanced game. This candidate is shown to be in the core if and only if the game is balanced. No use is made of topological ideas, but conditions are given under which the core member takes on values in a bounded set. The Charnes-Kortanek M-operator is used to characterize the redundancy of certain coalition values in restriction core membership.