  Dynamic pricing schemes were introduced as an alternative to posted-price
mechanisms. In contrast to static models, the dynamic setting allows 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
multi-demand 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
multi-demand markets up to three buyers, thus giving new interpretations of
results of Cohen-Addad et al. and Berger et al., respectively. Furthermore, we
provide an optimal dynamic pricing scheme for bi-demand valuations with an
arbitrary number of buyers. In all cases, our proofs also provide efficient
algorithms for determining the optimal dynamic prices.
