Assignment Situations with Multiple Ownership and Their Games

An assignment situation can be considered as a two-sided market consisting of two disjoint sets of objects. A non-negative reward matrix describes the profit if an object of one group is assigned to an object of the other group. Assuming that each object is owned by a different agent, Shapley and Shubnik (1972) introduced a class of assignment games arising from these assignement situations. This paper introduces assignment situations with multiple ownership. In these situations, each object can be owned by several agents and each agent can participate in the ownership of more than one object. In this paper, we study simply assignment games and relaxations that arise from assignment situations with multiple ownership. First, necessary and sufficient conditions are provided for balanced assignment situations with multiple ownership. An assignment situation with multiple ownership is balanced if for any choice of the reward matrix the corresponding simple assignment game is balanced. Second, balancedness results are obtained for relaxations of simple assignments games