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Dynamic pricing schemes were introduced as an alternative to posted-pricemechanisms. In contrast to static models, the dynamic setting allows to updatethe prices between buyer-arrivals based on the remaining sets of items andbuyers, and so it is capable of maximizing social welfare without the need fora central coordinator. In this paper, we study the existence of optimal dynamicpricing schemes in combinatorial markets. In particular, we concentrate onmulti-demand valuations, a natural extension of unit-demand valuations. Theproposed approach is based on computing an optimal dual solution of the maximumsocial welfare problem with distinguished structural properties.  Our contribution is twofold. By relying on an optimal dual solution, we showthe existence of optimal dynamic prices in unit-demand markets and inmulti-demand markets up to three buyers, thus giving new interpretations ofresults of Cohen-Addad et al. and Berger et al., respectively. Furthermore, weprovide an optimal dynamic pricing scheme for bi-demand valuations with anarbitrary number of buyers. In all cases, our proofs also provide efficientalgorithms for determining the optimal dynamic prices.

A dual approach for dynamic pricing in multi-demand markets