, In the following, we prove the regret bound of the PDTFP dynamics (without the vanishing regret condition) in smooth auctions

, Observe that for any reserve price vector r, |REV(r, b) ? REV(r, b)| ? m · 2 ?M where r is a reserve price vector such that r ij is the largest multiple of 2 ?M smaller than r ij for every i, j. Therefore, one can approximate the revenue up to any arbitrary precision by restricting the reserve price values in the discretization

, Note that K ? L = L. We slightly abuse notation by denoting REV j (1 S , b) as REV j (r, b), Consider the lattice L = { · 2 ?M : 0 ? ? 2 M } n×m

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