Sepehr Assadi
ABSTRACT
Multi-item revenue-optimal mechanisms are known to be extremely complex, often offering buyers randomized lotteries of goods. In the standard buy-one model, it is known that optimal mechanisms can yield revenue infinitely higher than that of any "simple" mechanism---the ones with size polynomial in the number of items---even with just two items and a single buyer [Briest et al. 2015; Hart and Nisan 2017].
We introduce a new parameterized class of mechanisms, buy-k mechanisms, which smoothly interpolate between the classical buy-one mechanisms and the recently studied buy-many mechanisms [Chawla et al. 2022, 2019, 2020a,b]. Buy-k mechanisms allow the buyer to buy up to k many menu options. We show that restricting the seller to the class of buy-n incentive-compatible mechanisms suffices to overcome the bizarre, infinite revenue properties of the buy-one model. Our main result is that the revenue gap with respect to bundling, an extremely simple mechanism, is bounded by O(n2) for any arbitrarily correlated distribution D over n items for the case of an additive buyer. Our techniques also allow us to prove similar upper bounds for arbitrary monotone valuations, albeit with an exponential factor in the approximation.
On the negative side, we show that allowing the buyer to purchase a small number of menu options does not suffice to guarantee sub-exponential approximations, even when we weaken the benchmark to the optimal buy-k deterministic mechanism. If an additive buyer is only allowed to buy k = Θ(n1/2--ε) many menu options, the gap between the revenue-optimal deterministic buy-k mechanism and bundling may be exponential in n. In particular, this implies that no "simple" mechanism can obtain a sub-exponential approximation in this regime. As a complementary result, we show that when [EQUATION], bundling recovers a poly(n) fraction of the optimal deterministic buy-k mechanism's revenue.
- Moshe Babaioff, Yannai A. Gonczarowski, and Noam Nisan. 2017. The menu-size complexity of revenue approximation. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19--23, 2017. 869--877. Google Scholar
Digital Library
- Moshe Babaioff, Nicole Immorlica, Brendan Lucier, and S. Matthew Weinberg. 2014. A Simple and Approximately Optimal Mechanism for an Additive Buyer. In the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS).Google Scholar
- Patrick Briest, Shuchi Chawla, Robert Kleinberg, and S. Matthew Weinberg. 2010. Pricing Randomized Allocations. In the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA).Google ScholarDigital Library
- Patrick Briest, Shuchi Chawla, Robert Kleinberg, and S. Matthew Weinberg. 2015. Pricing lotteries. J. Economic Theory 156 (2015), 144--174. Google ScholarCross Ref
- Yang Cai, Nikhil Devanur, and S. Matthew Weinberg. 2016. A Duality Based Unified Approach to Bayesian Mechanism Design. In Proceedings of the 48th ACM Conference on Theory of Computation(STOC).Google Scholar
- Yang Cai and Mingfei Zhao. 2017. Simple mechanisms for subadditive buyers via duality. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19--23, 2017. 170--183. Google ScholarDigital Library
- Shuchi Chawla, Jason D. Hartline, and Robert D. Kleinberg. 2007. Algorithmic Pricing via Virtual Valuations. In the 8th ACM Conference on Electronic Commerce (EC).Google Scholar
- Shuchi Chawla, Jason D. Hartline, David L. Malec, and Balasubramanian Sivan. 2010. Multi-Parameter Mechanism Design and Sequential Posted Pricing. In the 42nd ACM Symposium on Theory of Computing (STOC).Google Scholar
- Shuchi Chawla, David L. Malec, and Balasubramanian Sivan. 2015. The power of randomness in Bayesian optimal mechanism design. Games and Economic Behavior 91 (2015), 297--317. Google ScholarCross Ref
- Shuchi Chawla and J. Benjamin Miller. 2016. Mechanism Design for Subadditive Agents via an Ex Ante Relaxation. In Proceedings of the 2016 ACM Conference on Economics and Computation, EC '16, Maastricht, The Netherlands, July 24--28, 2016. 579--596. Google ScholarDigital Library
- Shuchi Chawla, Rojin Rezvan, Yifeng Teng, and Christos Tzamos. 2022. Pricing ordered items. In STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 -- 24, 2022, Stefano Leonardi and Anupam Gupta (Eds.). ACM, 722--735. Google ScholarDigital Library
- Shuchi Chawla, Yifeng Teng, and Christos Tzamos. 2019. Buy-Many Mechanisms Are Not Much Better than Item Pricing. In Proceedings of the 2019 ACM Conference on Economics and Computation (Phoenix, AZ, USA) (EC '19). Association for Computing Machinery, New York, NY, USA, 237--238. Google ScholarDigital Library
- Shuchi Chawla, Yifeng Teng, and Christos Tzamos. 2020a. Buy-Many Mechanisms: What Are They and Why Should You Care? SIGecom Exch. 18, 1 (dec 2020), 12--18. Google ScholarDigital Library
- Shuchi Chawla, Yifeng Teng, and Christos Tzamos. 2020b. Menu-Size Complexity and Revenue Continuity of Buy-Many Mechanisms. In Proceedings of the 21st ACM Conference on Economics and Computation (Virtual Event, Hungary) (EC '20). Association for Computing Machinery, New York, NY, USA, 475--476. Google ScholarDigital Library
- Constantinos Daskalakis. 2015. Multi-Item Auctions Defying Intuition? SIGecom Exch. 14, 1 (nov 2015), 41--75. Google ScholarDigital Library
- Zoltán Füredi. 1996. Onr-Cover-Free Families. J. Comb. Theory Ser. A 73, 1 (jan 1996), 172--173. Google ScholarDigital Library
- Sergiu Hart and Noam Nisan. 2012. Approximate Revenue Maximization with Multiple Items. In the 13th ACM Conference on Electronic Commerce (EC).Google Scholar
- Sergiu Hart and Noam Nisan. 2013. The menu-size complexity of auctions. In the 14th ACM Conference on Electronic Commerce (EC).Google Scholar
- Sergiu Hart and Noam Nisan. 2017. Approximate revenue maximization with multiple items. J. Economic Theory 172 (2017), 313--347. Google ScholarCross Ref
- Sergiu Hart and Philip J. Reny. 2015. Maximizing Revenue with Multiple Goods: Nonmonotonicity and Other Observations. Theoretical Economics 10, 3 (2015), 893--922.Google ScholarCross Ref
- W. Kautz and R. Singleton. 1964. Nonrandom binary superimposed codes. IEEE Transactions on Information Theory 10, 4 (1964), 363--377. Google ScholarDigital Library
- Pravesh Kothari, Sahil Singla, Divyarthi Mohan, Ariel Schvartzman, and S. Matthew Weinberg. 2019. Approximation Schemes for a Unit-Demand Buyer with Independent Items via Symmetries. In 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9--12, 2019, David Zuckerman (Ed.). IEEE Computer Society, 220--232. Google ScholarCross Ref
- Xinye Li and Andrew Chi-Chih Yao. 2013. On Revenue Maximization for Selling Multiple Independently Distributed Items. Proceedings of the National Academy of Sciences 110, 28 (2013), 11232--11237.Google ScholarCross Ref
- Roger B. Myerson. 1981. Optimal Auction Design. Mathematics of Operations Research 6, 1 (1981), 58--73.Google ScholarDigital Library
- Alexandros Psomas, Ariel Schvartzman, and S. Matthew Weinberg. 2019. Smoothed Analysis of Multi-Item Auctions with Correlated Values. In Proceedings of the 2019 ACM Conference on Economics and Computation (Phoenix, AZ, USA) (EC '19). Association for Computing Machinery, New York, NY, USA, 417--418. Google ScholarDigital Library
- Alexandros Psomas, Ariel Schvartzman, and S. Matthew Weinberg. 2022. On Infinite Separations Between Simple and Optimal Mechanisms: A Converse of a Theorem of Hart and Nisan. Manuscript (2022).Google Scholar
- Aviad Rubinstein and S. Matthew Weinberg. 2015. Simple Mechanisms for a subadditive buyer and applications to revenue monotonicity. In Proceedings of the 16th ACM Conference on Electronic Commerce.Google Scholar
- Andrew Chi-Chih Yao. 2015. An n-to-1 bidder reduction for multi-item auctions and its applications. In the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA).Google Scholar
Index Terms
- Fine-Grained Buy-Many Mechanisms Are Not Much Better Than Bundling
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