A Dual Approach for Dynamic Pricing in Multidemand Markets.

Dynamic pricing schemes were introduced as an alternative to posted-price mechanisms. In contrast to static models, the dynamic setting allows us to update the prices between buyer-arrivals based on the remaining sets of items and buyers, and so it is capable of maximizing social welfare without the need for a central coordinator. In this paper, we study the existence of optimal dynamic pricing schemes in combinatorial markets. In particular, we concentrate on multidemand valuations, a natural extension of unit-demand valuations. The proposed approach is based on computing an optimal dual solution of the maximum social welfare problem with distinguished structural properties. Our contribution is twofold. By relying on an optimal dual solution, we show the existence of optimal dynamic prices in unit-demand markets and in multidemand markets up to three buyers, thus giving new interpretations of results of Cohen-Addad et al. [Proceedings of the ACM Conference on Economics and Computation, 2016, pp. 383-400] and Berger, Eden, and Feldman [Proceedings of the International Conference on Web and Internet Economics, Springer, 2020, pp. 206-219], respectively. Furthermore, we provide an optimal dynamic pricing scheme for bidemand valuations with an arbitrary number of buyers. In all cases, our proofs also provide efficient algorithms for determining the optimal dynamic prices.