Reconciling marginalism with the core in two-sided markets with money

Abstract

In two-sided markets with money, core stability and marginalism are often in conflict. We reconcile them with two main results. First, we show that in the assignment game (Shapley & Shubik, 1972), the Banzhaf value is core stable if and only if the game is exact. This is surprising for two reasons: (i) the Banzhaf value is generally not efficient, and (ii) although exactness suffices for the Shapley value to be stable, it is not necessary. Consequently, stability of the Banzhaf value implies stability of the Shapley value, but not vice versa. Second, we consider a family of intra-sector Shapley and Banzhaf values by applying each value separately to the game on the set of buyers assuming all sellers are available and to the game on the set of sellers assuming all buyers are available, and then taking any convex combination. We prove that all such convex combinations lie in the core if and only if the valuation matrix has a dominant diagonal. Under this condition, the equal-weight intra-sector Shapley and Banzhaf values coincide with the fair-division point. Together, these results deliver simple criteria under which marginalist solutions assign stable payoff vectors in the original market.