Matching markets with bundle discounts: computing efficient, stable and fair solutions

We model a matching market in which nonstrategic vendors sell items of different types and offer discounted prices on the bundle of all items triggered by demand volumes. Each buyer acts strategically in order to maximize her utility, given by the difference between product valuations and price paid. In a market with transferable utility, buyers might be willing to cooperate by exchanging side-payments, to induce discounts. The core of the market is in general empty, therefore we consider a notion of stability that looks at unilateral deviations. We show that efficient matchings – the ones maximizing the social welfare – can be stabilized by transfers between buyers that enjoy desirable properties of rationality (according to which a buyer subsidizes only buyers who contribute to the activation of the desired discount) and fairness (according to which transfers amounts are balanced between buyers). Building on this existence results, and letting N , M and c be the numbers of buyers, vendors and product types, we propose a O(N + NM) algorithm that, given an efficient matching, computes stabilizing transfers that are rational and fair, by first determining transfers between groups of buyers with an equal product choice, and then between single buyers. Our results show that if cooperation is allowed then social efficiency and stability can coexists in a market presenting subtle externalities, and determining the right amount of cooperation is computationally tractable.