Stability of Generalized Two-sided Markets with Transaction Thresholds

We study a type of generalized two-sided markets where a set of sellers, each with multiple units of the same divisible good, trade with a set of buyers. Possible trade of each unit between a buyer and a seller generates a given welfare (to be split among them), indicated by the weight of the edge between them. What makes the markets interesting is a special type of constraints, called transaction threshold constraints, which essentially mean that the amount of goods traded between a pair of agents can either be zero or above a certain edge-specific threshold. This constraints has originally been motivated from the water-right market domain by Liu et. al. where minimum thresholds must be imposed to mitigate administrative and other costs. The same constraints have been witnessed in several other market domains. Without the threshold constraints, it is known that the seminal result by Shapley and Shubick holds: the social welfare maximizing assignments between buyers and sellers are in the core. In other words, by algorithmically optimizing the market, one can obtain desirable incentive properties for free. This is no longer the case for markets with threshold constraints: the model considered in this paper. We first demonstrate a counterexample where no optimal assignment (with respect to any way to split the trade welfare) is in the core. Motivated by this observation, we study the stability of the optimal assignments from the following two perspectives: 1) by relaxing the definition of core; 2) by restricting the graph structure. For the first line, we show that the optimal assignments are pairwise stable, and no coalition can benefit twice as large when they deviate. For the second line, we exactly characterize the graph structure for the nonemptyness of core: the core is nonempty if and only if the market graph is a tree. Last but not least, we compliment our previous results by quantitatively measuring the welfare loss caused by the threshold constraints: the optimal welfare without transaction thresholds is no greater than constant times of that with transaction thresholds. We evaluate and confirm our theoretical results using real data from a water-right market.