This work is devoted to repeated (dynamic) contextual auctions with a single strategic buyer. In its setting, knowledge on buyer prior distribution is not available to the seller. The objective of the seller is to find a pricing (policy) that minimize cumulative regret. The key important differences of the considered scenario to previous works are: - The revenue of the seller is compared with respect to a dynamic benchmark; - Specific stochastic dependence of valuations on features of a good. The paper contains a lot of statements, and, thus, seems to be based on a huge work done before. The main issues of the work: 1. Key related work is missed: a. The authors study repeated auction scenario where a buyer is strategic up to a certain accuracy factor. Such buyer behavior has been modeled earlier in [Mohri&Medina, NIPS’2015]. Can the authors position their study with respect to this prior art? b. The authors have references to non-state-of-the-art results. For instance, they discuss the scenario of non-contextual auctions (in Lines 73-77), where optimal solutions with tight regret bound in O(log log T) have been found in [Drutsa, WWW’2017; Drutsa, ICML’2018]. However, the authors provide references to suboptimal algorithms from [Mohri&Medina, NIPS’2014] and [Amin et al, NIPS’2013] only. c. Robust dynamic pricing for repeated contextual auction has been also studied in the work [Cohen,EC’2016]. >Comment after rebuttal: I love the detailed and clear comparison with the missed related work. Please, add this discussion to the next revision of your work. 2. In the introduction, in Lines 38-41: “[Medina & Mohri, 2014] … only provide guarantees against the revenue-optimal static benchmark, which does not take advantage of auction state across time and whose revenue can be arbitrarily smaller than the optimal dynamic benchmark”. It is quite unclear, since in [Medina & Mohri, 2014], the benchmark is the best possible one, because it is equal exactly to the valuation of the buyer and, hence, generate the maximal revenue each round. So, even any dynamic pricing cannot provide higher revenue than this one. The same issue occurs in Lines 81-83. >Comment after rebuttal: I got the answer in general. I hope, the authors will improve clearness in the lines that I have indicated above. 3. The setup in Lines 89-99 is very difficult to follow. For example, reading it three times, I still don’t understand which of the setup is considered: stochastic valuations, or parametric ones, or both simultaneously. >Comment after rebuttal: Thank you for your clarification. I would suggest you to possibly include some scheme of how the workflow of the sampling process looks like and how it relates to the information knowledge both of the buyers and the seller. 4. Most statements in the paper are given without proofs or even their sketches. The proofs are deferred to Appendix. It’d be great to have more insights on the proofs in the main text of the paper. >Comment after rebuttal: Thank you. I hope more pages will allow you to include more insights and intuitions behind their proofs. The key results of the paper rely on the assumption that the seller knows up front that the buyer utilizes linear model to derive his valuation. Does this assumption realistic? Does the results of the study holds for non-linear dependences? >Comment after rebuttal: I got the answer. One suggestion for paper improvement: is it possible to make your algorithms workable in the scenario of kernel models? (see how [Cohen et al.,EC’2016] and [Amin et al, NIPS’2014] easily extended their results from linear models to kernel ones) Also, I noted that there are no conclusions. The paper terminates at Theorem statement. It seems like the paper is not ready for publication in the current version. >Comment after rebuttal: Thank you for your response. I raise the score believing and hoping that you will update the presentation according to your answer. References: Mohri, Mehryar, and Andres Munoz. "Revenue optimization against strategic buyers." Advances in Neural Information Processing Systems. 2015. Cohen, Maxime C., Ilan Lobel, and Renato Paes Leme. "Feature-based Dynamic Pricing." Proceedings of the 2016 ACM Conference on Economics and Computation. ACM, 2016. Drutsa, Alexey. "Horizon-independent optimal pricing in repeated auctions with truthful and strategic buyers." Proceedings of the 26th International Conference on World Wide Web. International World Wide Web Conferences Steering Committee, 2017. Drutsa, Alexey. "Weakly Consistent Optimal Pricing Algorithms in Repeated Posted-Price Auctions with Strategic Buyer." International Conference on Machine Learning. 2018.

This paper studies the problem of a revenue maximizing seller setting prices in a repeated dynamic contextual setting for a utility maximizing buyer. The core of the mechanism starts with the NonClairvoyantBalance mechanism in [Mirrokni et al 2018], which is a mixture of 3 auctions: free, Myerson pricing, and posted price w/ fee. In that work, the mechanism is run in each stage based off of the seller's knowledge of buyers distributions, and is parameterized by a "bank balance" that keeps track of the utility accrued over time. One auction is added here - a random posted price, and this auction is used to get the robust non-clairvoyant dynamic mechanism. More discussion would be helpful surrounding the exact benefit from adding the additional auction into the mix. The paper then tackles the contextual dynamic auctions problem in non-clairvoyant environments, combining a variant of the first 4-auction mechanism with a learning policy from Golrezaei et al 2018. In the process, the posted price auction becomes much more intricate (Def D.1, Hybrid Posted Price w/ Extra fee). The exposition covers directly what is happening, but lacks motivation for why things are happening. This combination of approaches builds very materially on prior work. [After response: Thank you, please include in final version]. The paper is generally well written, though more motivation throughout for decisions would be helpful (e.g., the choices of auctions constituting the mechanism - why these four, and why can't we drop any?). [After response - Thank you, please include in final version]

The presentation is good overall, but sometimes a bit hard to follow due to the level of technicality and the number of different settings/results. - As a non-expert I had a bit a problem following the presentation, sometimes an intuition could help, e.g. for fee or the balance (it becomes more clear later though). - And sometimes it would be nice to also have a formal proof for prop3.1 (e.g., that (BI) in fact holds). (Even though it may be trivial, it would be nice to see it explicitly.) Similarly, for the two main Theorems of the paper. I was surprised that there are not proofs for them. I guess it trivially follows form the lemmas but please (easily) save the reader some time here. - This being said, the paper overall, incl. supplement, looks rather good and polished. In terms of originality/significance, they do seem to strongly build on previous work in this direction, but seem to have enough interesting own contributions. - In particular, from how it looks, they build on Mirrokni [2019]. Mirrokni also treats non-clairvoyant mechanisms (i.e., future type (=valuation) distributions are unknown), but Mirrokni does not *learn* the (current) type distribution. In terms of quality, in the proofs they seem to be knowing what they are doing, but I didn't check the details. Looks interesting though. This is a purely theoretical paper without experiments but this is fine with me. After author response: Thanks for the response!