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 Cohen-Addad et al. [EC'16] 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 non-gross 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 the dynamic pricing equivalent 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.

中文翻译：

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组合市场中动态定价的力量与局限

我们研究了组合市场中最优动态定价的力量和限制；即，导致最佳社会福利的动态定价。Cohen-Addad 等人以前的工作。[EC'16] 证明了单位需求买家的最优动态价格的存在，并展示了一个市场，其覆盖估值不承认这样的价格。然而，找到承认最优动态价格的市场边界（即估值函数）仍然是一个悬而未决的问题。在这项工作中，我们建立了缩小现有差距的积极和消极结果。从积极的方面来看，我们提供了处理超出单位需求估值的市场的工具。特别是，我们描述了多需求市场中的所有最优配置。这种表征允许我们根据项目在实现最优性中所扮演的角色将它们划分为等价类。使用这些工具，我们为高达 3 美元的多需求买家提供了多时间最优动态定价算法。在消极方面，我们建立了一个最大域定理，表明对于每个非总替代品估值，都存在单位需求估值，使得将它们相加会产生一个不允许最优动态定价的市场。这个结果是 Gul 和 Stacchetti [JET'99] 对瓦尔拉斯均衡的开创性最大域定理的动态定价等价物。Yang [JET'17] 在他们的原始证明中发现了一个错误，并建立了他们最大域定理的一个不同的、无与伦比的版本。在使用我们的最大域定理以获得最优动态定价的过程中，