Shapely and Shubik (1972) provide the concept of a core of stable and feasible allocation in which players wouldn’t block any trade in an assignment game for bilateral trading market with multiple buyers and sellers. One of the important contributions of Shapely and Shubik (1972) is the existence of a core in such an assignment game. The geometric feature of the core suggests that the system of disagreement payoff is consistent with the lowest payoffs of allocation in the core for each player on both sides of the market. There are some examples provided in this paper explaining that such payoffs are usually stipulated to be zero-payoff for many buyers and sellers. It is difficult to find a non-degenerated system of disagreement payoffs as a general equilibrium.
One goal of this paper is to contend with the difficulty by refining the core of Shapely and Shubik (1972). The refinement is that players can enforce a coalition of buyers to buy out some other buyers from the market and a coalition of sellers to buy out some other sellers from the market. By the concept of the refined core, we can show that not all buyers or sellers have degenerated disagreement payoffs. As another goal, we show that the refined core obtained by buying out at once is without loss of generality. An alternative approach is buying out sequentially. We show that they are indifferent between these two approaches of refinement of the core.