On the Power and Limits of Dynamic Pricing in Combinatorial Markets

Abstract

We study the power and limits of optimal dynamic pricing in combinatorial markets; i.e., dynamic pricing that leads to optimal social welfare. Previous work by demonstrated the existence of optimal dynamic prices for unit-demand buyers, and showed a market with coverage valuations that admits no such prices. However, finding the frontier of markets (i.e., valuation functions) that admit optimal dynamic prices remains an open problem. In this work we establish positive and negative results that narrow the existing gap.On the positive side, we provide tools for handling markets beyond unit-demand valuations. In particular, we characterize all optimal allocations in multi-demand markets. This characterization allows us to partition the items into equivalence classes according to the role they play in achieving optimality. Using these tools, we provide a poly-time optimal dynamic pricing algorithm for up to 3 multi-demand buyers.On the negative side, we establish a maximal domain theorem, showing that for every nongross substitutes valuation, there exist unit-demand valuations such that adding them yields a market that does not admit an optimal dynamic pricing. This result is reminiscent of the seminal maximal domain theorem by Gul and Stacchetti [JET'99] for Walrasian equilibrium. Yang [JET'17] discovered an error in their original proof, and established a different, incomparable version of their maximal domain theorem. En route to our maximal domain theorem for optimal dynamic pricing, we provide the first complete proof of the original theorem by Gul and Stacchetti.

A Two-Step Approach to Optimal Dynamic Pricing in Multi-Demand Combinatorial Markets

“…Our goal is to find a dynamic pricing that guarantees social welfare optimization, despite the fact that each individual buyer always maximizes their own utility. We introduce the useful concepts of legal items, legal bundles, and legal allocations as in [5].…”

“…To formalize the main results we use the notion of legality introduced in [5]. An item x is legal for a player i if there exists an optimal allocation in which x is allocated to i.…”

“…A very powerful Theorem 2.2 by Berger et al states that optimal allocations and legal allocations are equivalent [5]. Consequently, we use these terms interchangeably.…”

“…The base cases are markets with at most three items, which implies that there are at most three players. Such markets have been studied in [5]. From the induction hypothesis, we may assume that a dynamic pricing can be found in any multi-demand market with less than m items and at most four players.…”

Market Pricing for Matroid Rank Valuations

“…Ideally, the prices are set in such a way that there exists an allocation of the items to buyers in which the market clears and everyone receives a bundle that maximizes her utility. A pair of pricing and allocation possessing these properties is called a Walrasian equilibrium 1 , while we will refer to the price vector itself as Walrasian pricing. The fundamental notion of Walrasian equilibrium first appeared in [38], and the definition immediately implies that the allocation in a Walrasian equilibrium maximizes social welfare.…”