Printed in U.S.A. A NOTE ON AN AXIOMATIZATION OF THE CORE OF MARKET GAMES

As shown by Peleg (1993), the core of market games is characterized by nonemptiness, individual rationality, superadditivity, the weak reduced game property, the converse reduced game property, and weak symmetry. It was not known whether weak symmetry was logically independent. With the help of a certain transitive 4-person TUgame, it is shown that weak symmetry is redundant in this result. Hence, the core on market games is axiomatized by the remaining five properties, if the universe of players contains at least four members. In this note we solve an open problem of the theory of cooperative games that arises in a natural way in the context of the characterization of the core of market games due to Peleg (1989, 1993). Theorem 2 of Peleg (1993) shows that the core of market games is characterized by nonemptiness (NE), individual rationality (IR), superadditivity (SUPA), the weak reduced game property (WRGP), the converse reduced game property (CRGP), and weak symmetry (WS). (Precise definitions of properties are recalled below.) The assertion without the assumption of WS was formulated in Peleg (1989), but it turned out that WS, which is a weak variant of anonymity, was needed in addition in the proof of the uniqueness part. The problem whether WS is logically independent was mentioned in Peleg (1993)