Abstract
We consider a fundamental pricing problem in combinatorial auctions. We are given a set of indivisible items and a set of buyers with randomly drawn monotone valuations over subsets of items. A decision maker sets item prices and then the buyers make sequential purchasing decisions, taking their favorite set among the remaining items. We parametrize an instance by d, the size of the largest set a buyer may want. Our main result asserts that there exist prices such that the expected (over the random valuations) welfare of the allocation they induce is at least a factor \(1/(d+1)\) times the expected optimal welfare in hindsight. Moreover we prove that this bound is tight. Thus, our result not only improves upon the \(1/(4d-2)\) bound of Dütting et al., but also settles the approximation that can be achieved by using item prices. We further show how to compute our prices in polynomial time. We provide additional results for the special case when buyers’ valuations are known (but a posted-price mechanism is still desired).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
That is, for all sets A with \(|A|>d\), \(v(A):=\max _{B \subset A, |B| \le d} \{v(B)\}\).
- 2.
And quite hard to approximate [22].
- 3.
That is, each buyer has a fixed set T, and values all sets S at \(v(S)\,:=\,I(T \subseteq S) \cdot v(T)\).
- 4.
Throughout the paper M is actually a set and refers to the set of different items.
- 5.
Note that different copies of the same item need to get the same price.
- 6.
In some of the constructions in Sect. 4 we break ties conveniently but all the results hold by slightly tweaking the instances.
References
Assadi, S., Kesselheim, T., Singla, S.: Improved truthful mechanisms for subadditive combinatorial auctions: breaking the logarithmic barrier. In: Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 653–661. SIAM (2021)
Assadi, S., Singla, S.: Improved truthful mechanisms for combinatorial auctions with submodular bidders. In: 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), pp. 233–248. IEEE (2019)
Babaioff, M., Lucier, B., Nisan, N., Paes Leme, R.: On the efficiency of the Walrasian mechanism. In: Proceedings of the Fifteenth ACM Conference on Economics and Computation, pp. 783–800 (2014)
Baldwin, E., Klemperer, P.: Understanding preferences:“demand types’’, and the existence of equilibrium with indivisibilities. Econometrica 87(3), 867–932 (2019)
Cai, Y., Zhao, M.: Simple mechanisms for subadditive buyers via duality. In: Hatami, H., McKenzie, P., King, V. (eds.) Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, 19–23 June 2017, pp. 170–183. ACM (2017)
Chawla, S., Hartline, J.D., Malec, D.L., Sivan, B.: Multi-parameter mechanism design and sequential posted pricing. In: Proceedings of the Forty-second ACM Symposium on Theory of Computing, pp. 311–320 (2010)
Chawla, S., Miller, J.B.: Mechanism design for subadditive agents via an ex ante relaxation. In: Conitzer, V., Bergemann, D., Chen, Y. (eds.) Proceedings of the 2016 ACM Conference on Economics and Computation, EC 2016, Maastricht, The Netherlands, 24–28 July 2016, pp. 579–596. ACM (2016)
Dobzinski, S.: Breaking the logarithmic barrier for truthful combinatorial auctions with submodular bidders. In: Wichs, D., Mansour, Y. (eds.) Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, 18–21 June 2016, pp. 940–948. ACM (2016)
Dobzinski, S., Nisan, N., Schapira, M.: Approximation algorithms for combinatorial auctions with complement-free bidders. In: Gabow, H.N., Fagin, R. (eds.) Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, 22–24 May 2005, pp. 610–618. ACM (2005)
Dutting, P., Feldman, M., Kesselheim, T., Lucier, B.: Prophet inequalities made easy: stochastic optimization by pricing nonstochastic inputs. SIAM J. Comput. 49(3), 540–582 (2020)
Dütting, P., Kesselheim, T., Lucier, B.: An o(log log m) prophet inequality for subadditive combinatorial auctions. In: Irani, S. (ed.) 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, 16–19 November 2020, pp. 306–317. IEEE (2020)
Ezra, T., Feldman, M., Gravin, N., Tang, Z.G.: Online stochastic max-weight matching: prophet inequality for vertex and edge arrival models. In: Proceedings of the 21st ACM Conference on Economics and Computation, pp. 769–787 (2020)
Feldman, M., Gravin, N., Lucier, B.: Combinatorial auctions via posted prices. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 123–135. SIAM (2014)
Feldman, M., Gravin, N., Lucier, B.: Combinatorial walrasian equilibrium. SIAM J. Comput. 45(1), 29–48 (2016)
Feldman, M., Svensson, O., Zenklusen, R.: Online contention resolution schemes. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1014–1033. SIAM (2016)
Gravin, N., Wang, H.: Prophet inequality for bipartite matching: merits of being simple and non adaptive. In: Proceedings of the 2019 ACM Conference on Economics and Computation, pp. 93–109 (2019)
Hall, M.: Combinatorial Theory, vol. 71. Wiley, Hoboken (1998)
Kleinberg, R., Weinberg, S.M.: Matroid prophet inequalities. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, pp. 123–136 (2012)
Krysta, P., Vöcking, B.: Online mechanism design (randomized rounding on the fly). In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012. LNCS, vol. 7392, pp. 636–647. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31585-5_56
Leme, R.P., Wong, S.C.W.: Computing walrasian equilibria: fast algorithms and structural properties. Math. Program. 179(1), 343–384 (2020)
Samuel-Cahn, E.: Comparison of threshold stop rules and maximum for independent nonnegative random variables. Ann. Probab. 12(4), 1213–1216 (1984)
Trevisan, L.: Non-approximability results for optimization problems on bounded degree instances. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 453–461 (2001)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Correa, J., Cristi, A., Fielbaum, A., Pollner, T., Weinberg, S.M. (2022). Optimal Item Pricing in Online Combinatorial Auctions. In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_10
Download citation
DOI: https://doi.org/10.1007/978-3-031-06901-7_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-06900-0
Online ISBN: 978-3-031-06901-7
eBook Packages: Computer ScienceComputer Science (R0)