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Consider a single-product revenue-maximization problem where the sellermonotonically decreases the price in $n$ rounds with an unknown demand modelcoming from a given family. Without monotonicity, the minimax regret is $\tildeO(n^{2/3})$ for the Lipschitz demand family and $\tilde O(n^{1/2})$ for ageneral class of parametric demand models. With monotonicity, the minimaxregret is $\tilde O(n^{3/4})$ if the revenue function is Lipschitz andunimodal. However, the minimax regret for parametric families remained open. Inthis work, we provide a complete settlement for this fundamental problem. Weintroduce the crossing number to measure the complexity of a family of demandfunctions. In particular, the family of degree-$k$ polynomials has a crossingnumber $k$. Based on conservatism under uncertainty, we present (i) a policywith an optimal $\Theta(\log^2 n)$ regret for families with crossing number$k=0$, and (ii) another policy with an optimal $\tilde \Theta(n^{k/(k+1)})$regret when $k\ge 1$. These bounds are asymptotically higher than the $\tildeO(\log n)$ and $\tilde \Theta(\sqrt n)$ minimax regret for the same familieswithout the monotonicity constraint.

Markdown Pricing Under an Unknown Parametric Demand Model