Skip to main content
SpringerLink
Log in

Mechanism Design for Perturbation Stable Combinatorial Auctions

  • Conference paper
  • First Online:
Algorithmic Game Theory (SAGT 2020)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12283))

Included in the following conference series:

  • 597 Accesses

Abstract

Motivated by recent research on combinatorial markets with endowed valuations by (Babaioff et al., EC 2018) and (Ezra et al., EC 2020), we introduce a notion of perturbation stability in Combinatorial Auctions (CAs) and study the extend to which stability helps in social welfare maximization and mechanism design. A CA is \(\gamma \)-stable if the optimal solution is resilient to inflation, by a factor of \(\gamma \ge 1\), of any bidder’s valuation for any single item. On the positive side, we show how to compute efficiently an optimal allocation for 2-stable subadditive valuations and that a Walrasian equilibrium exists for 2-stable submodular valuations. Moreover, we show that a Parallel 2nd Price Auction (P2A) followed by a demand query for each bidder is truthful for general subadditive valuations and results in the optimal allocation for 2-stable submodular valuations. To highlight the challenges behind optimization and mechanism design for stable CAs, we show that a Walrasian equilibrium may not exist for 2-stable XOS valuations, that a polynomial-time approximation scheme does not exist for \((2-\varepsilon )\)-stable submodular valuations, and that any DSIC mechanism that computes the optimal allocation for stable CAs and does not use demand queries must use exponentially many value queries. We conclude with analyzing the Price of Anarchy of P2A and Parallel 1st Price Auctions (P1A) for CAs with stable submodular and XOS valuations. Our results indicate that the quality of equilibria of simple non-truthful auctions improves only for \(\gamma \)-stable instances with \(\gamma \ge 3\).

This work was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant”, project BALSAM, HFRI-FM17-1424.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 64.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 84.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For a better understanding of the two conditions at a technical level, we note that a (technically very useful) necessary condition for a valuations profile \(\textit{\textbf{v}}\) to be \(\gamma \)-stable is that for the optimal allocation \((O_1, \ldots , O_n)\), any bidders \(i \ne k\) and any item \(j \in O_i\),

    $$\begin{aligned} v_i(O_i) - v_i(O_i \setminus \{ j \}) > v_k(O_k \cup \{ j \}) - v_k(O_k) + (\gamma - 1) v_k(\{ j \}) \ge (\gamma - 1) v_k(\{ j \}) \,. \end{aligned}$$

    For this condition, we use (local) optimality of \((O_1, \ldots , O_n)\) for both \(\textit{\textbf{v}}\) and its \(\gamma \)-perturbation on bidder k and item j (see also Lemma 1).

    A similar (technically useful) condition satisfied by any valuations profile \(\textit{\textbf{v}}\) that has resulted from the \(\alpha \)-endowment of an optimal (or locally optimal) solution \((O_1, \ldots , O_n)\) to an initial valuations profile \(\textit{\textbf{x}}\) is that for any bidders \(i \ne k\) and any item \(j \in O_i\),

    $$\begin{aligned} v_i(O_i) - v_i(O_i \setminus \{ j \}) \ge \alpha \big (v_k(O_k \cup \{ j \}) - v_k(O_k) \big )\,. \end{aligned}$$

    For this condition, we use local optimality of \((O_1, \ldots , O_n)\) for \(\textit{\textbf{x}}\), multiply the resulting inequality by \(\alpha \), and observe that \(v_i(O_i) - v_i(O_i \setminus \{ j \}) = \alpha \big (x_i(O_i) - x_i(O_i \setminus \{ j \})\big )\) and that \(v_k(O_k \cup \{ j \}) - v_k(O_k) = x_k(O_k \cup \{ j \}) - x_k(O_k)\).

References

  1. Angelidakis, H., Makarychev, K., Makarychev, Y.: Algorithms for stable and perturbation-resilient problems. In: Proceedings of the 49th ACM Symposium on Theory of Computing (STOC 2017), pp. 438–451 (2017)

    Google Scholar 

  2. Assadi, S., Singla, S.: Improved Truthful Mechanisms for Combinatorial Auctions with Submodular Bidders. In: Proceedings of the 60th IEEE Symposium on Foundations of Computer Science (FOCS 2019), pp. 233–248 (2019)

    Google Scholar 

  3. Awasthi, P., Blum, A., Sheffet, O.: Center-based clustering under perturbation stability. Inf. Process. Lett. 112(1–2), 49–54 (2012)

    Article  MathSciNet  Google Scholar 

  4. Babaioff, M., Dobzinski, S., Oren, S.: Combinatorial auctions with endowment effect. In: Proceedings of the 2018 ACM Conference on Economics and Computation (EC 2018), pp. 73–90 (2018)

    Google Scholar 

  5. Balcan, M., Haghtalab, N., White, C.: \(k\)-center clustering under perturbation resilience. In: Proc. of the 43rd International Colloquium on Automata, Languages and Programming (ICALP 2016). LIPIcs, vol. 55, pp. 68:1–68:14 (2016)

    Google Scholar 

  6. Bilu, Y., Linial, N.: Are stable instances easy? In: Proc. of the 1st Symposium on Innovations in Computer Science (ICS 2010), pp. 332–341. Tsinghua University Press (2010)

    Google Scholar 

  7. Blumrosen, L., Nisan, N.: On the computational power of demand queries. SIAM J. Comput. 39(4), 1372–1391 (2009)

    Article  MathSciNet  Google Scholar 

  8. Christodoulou, G., Kovács, A., Schapira, M.: Bayesian combinatorial auctions. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5125, pp. 820–832. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70575-8_67

    Chapter  Google Scholar 

  9. Dobzinski, S.: Two Randomized Mechanisms for Combinatorial Auctions. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) APPROX/RANDOM -2007. LNCS, vol. 4627, pp. 89–103. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74208-1_7

    Chapter  MATH  Google Scholar 

  10. Dobzinski, S.: An impossibility result for truthful combinatorial auctions with submodular valuations. In: Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC 2011), pp. 139–148 (2011)

    Google Scholar 

  11. Dobzinski, S.: Breaking the logarithmic barrier for truthful combinatorial auctions with submodular bidders. In: Proceedings of the 48th ACM Symposium on Theory of Computing (STOC 2016), pp. 940–948 (2016)

    Google Scholar 

  12. Dobzinski, S., Feldman, M., Talgam-Cohen, I., Weinstein, O.: Welfare and revenue guarantees for competitive bundling equilibrium. In: Markakis, E., Schäfer, G. (eds.) WINE 2015. LNCS, vol. 9470, pp. 300–313. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48995-6_22

    Chapter  MATH  Google Scholar 

  13. Dobzinski, S., Nisan, N., Schapira, M.: Truthful randomized mechanisms for combinatorial auctions. J. Comput. Syst. Sci. 78(1), 15–25 (2012)

    Article  MathSciNet  Google Scholar 

  14. Düetting, P., Feldman, M., Kesselheim, T., Lucier, B.: Prophet inequalities made easy: stochastic optimization by pricing non-stochastic inputs. In: Proceedings of the 58th Symposium on Foundations of Computer Science (FOCS 2017), pp. 540–551 (2017)

    Google Scholar 

  15. Englert, M., Röglin, H., Vöcking, B.: Smoothed analysis of the \(2\)-Opt algorithm for the general TSP. ACM Trans. Algorithms 13(1), 10:1–10:15 (2016)

    Article  MathSciNet  Google Scholar 

  16. Ezra, T., Feldman, M., Friedler, O.: A general framework for endowment effects in combinatorial markets. In: Proceedings of the 2020 ACM Conference on Economics and Computation (EC 2020) (2020)

    Google Scholar 

  17. Feige, U.: On maximizing welfare when utility functions are subadditive. SIAM J. Comput. 39(1), 122–142 (2009)

    Article  MathSciNet  Google Scholar 

  18. Feige, U., Vondrák, J.: The submodular welfare problem with demand queries. Theor. Comput. 6(1), 247–290 (2010)

    Article  MathSciNet  Google Scholar 

  19. Feldman, M., Gravin, N., Lucier, B.: Combinatorial auctions via posted prices. In: Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms, pp. 123–135 (2014)

    Google Scholar 

  20. Fikioris, G., Fotakis, D.: Mechanism design for perturbation stable combinatorial auctions. CoRR abs/2006.09889 (2020). https://arxiv.org/abs/2006.09889

  21. Fu, H., Kleinberg, R., Lavi, R.: Conditional equilibrium outcomes via ascending price processes with applications to combinatorial auctions with item bidding. In: Proceedings of the 13th ACM Conference on Electronic Commerce (EC 2012), p. 586 (2012)

    Google Scholar 

  22. Hershberger, J., Suri, S.: Vickrey prices and shortest paths: what is an edge worth? In: Proceedings of the 42nd Symposium on Foundations of Computer Science (FOCS 2001), pp. 252–259 (2001)

    Google Scholar 

  23. Khot, S., Lipton, R.J., Markakis, E., Mehta, A.: Inapproximability results for combinatorial auctions with submodular utility functions. In: Deng, X., Ye, Y. (eds.) WINE 2005. LNCS, vol. 3828, pp. 92–101. Springer, Heidelberg (2005). https://doi.org/10.1007/11600930_10

    Chapter  Google Scholar 

  24. 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

    Chapter  Google Scholar 

  25. Lehmann, B., Lehmann, D., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. Games Econ. Behav. 55(2), 270–296 (2006)

    Article  MathSciNet  Google Scholar 

  26. Milgrom, P.: Putting Auction Theory to Work. Churchill Lectures in Economics. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  27. Mirrokni, V., Schapira, M., Vondrák, J.: Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions. In: Proceedings 9th ACM Conference on Electronic Commerce (EC 2008), pp. 70–77 (2008)

    Google Scholar 

  28. Cramton, P., Shoham, Y., Steinberg, R.: Combinatorial Auctions. MIT Press, Cambridge (2006)

    MATH  Google Scholar 

  29. Rassenti, S., Smith, V., Bulfin, R.: A combinatorial auction mechanism for airport time slot allocation. Bell J. Econ. 13(2), 402–417 (1982)

    Article  Google Scholar 

  30. Roughgarden, T.: Barriers to near-optimal equilibria. In: Proceedings of the 55th IEEE Symposium on Foundations of Computer Science (FOCS 2014), pp. 71–80 (2014)

    Google Scholar 

  31. Roughgarden, T.: Beyond worst-case analysis. Commun. ACM 62(3), 88–96 (2019)

    Article  Google Scholar 

  32. Roughgarden, T., Syrgkanis, V., Tardos, É.: The price of anarchy in auctions. J. Artif. Intell. Res. 59, 59–101 (2017)

    Article  MathSciNet  Google Scholar 

  33. Roughgarden, T., Talgam-Cohen, I.: Approximately optimal mechanism design. CoRR abs/1812.11896 (2018). http://arxiv.org/abs/1812.11896

  34. Spielman, D., Teng, S.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)

    Article  MathSciNet  Google Scholar 

  35. Spielman, D., Teng, S.: Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Commun. ACM 52(10), 76–84 (2009)

    Article  Google Scholar 

  36. Syrgkanis, V., Tardos, É.: Composable and efficient mechanisms. In: Proceedings of the 45th Symposium on Theory of Computing (STOC 2013), pp. 211–220 (2013)

    Google Scholar 

  37. Thaler, R.: Toward a positive theory of consumer choice. J. Econ. Behav. Organ. 1(1), 39–60 (1980)

    Article  Google Scholar 

  38. Vondrák, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: Proceedings of the 40th ACM Symposium on Theory of Computing (STOC 2008), pp. 67–74 (2008)

    Google Scholar 

Download references

Acknowledgements

We wish to thank Kyriakos Lotidis and Grigoris Velegkas for many helpful discussions on combinatorial markets with endowed valuations and on the possibility of exploiting endowed valuations in mechanism design.

Author information

Authors and Affiliations

  1. School of Electrical and Computer Engineering, National Technical University of Athens, 15780, Athens, Greece

    Giannis Fikioris & Dimitris Fotakis

Authors
  1. Giannis Fikioris
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dimitris Fotakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dimitris Fotakis .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, Augsburg University, Augsburg, Germany

    Tobias Harks

  2. Institute of Mathematics, Technical University Berlin, Berlin, Germany

    Max Klimm

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Fikioris, G., Fotakis, D. (2020). Mechanism Design for Perturbation Stable Combinatorial Auctions. In: Harks, T., Klimm, M. (eds) Algorithmic Game Theory. SAGT 2020. Lecture Notes in Computer Science(), vol 12283. Springer, Cham. https://doi.org/10.1007/978-3-030-57980-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-57980-7_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-57979-1

  • Online ISBN: 978-3-030-57980-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation