Skip to main content
SpringerLink
Log in

On Symmetries in Multi-dimensional Mechanism Design

  • Conference paper
  • First Online:
Web and Internet Economics (WINE 2021)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 13112))

Included in the following conference series:

  • 1143 Accesses

Abstract

We consider a revenue-maximizing seller with multiple items for sale to a single population of buyers. Our main result shows that for a single population of additive buyers with independent (but not necessarily identically distributed) item values, bundling all items together achieves a constant-factor approximation to the revenue-optimal item-symmetric mechanism.

We further motivate this direction via fairness in ad auctions. In ad auction domains the items correspond to views from particular demographics, and recent works have therefore identified a novel fairness constraint: equally-qualified users from different demographics should be shown the same desired ad at equal rates. Prior work abstracts this to the following fairness guarantee: if an advertiser places an identical bid on two users, those two users should view the ad with the same probability [27, 34]. We first propose a relaxation of this guarantee from worst-case to Bayesian settings, which circumvents strong impossibility results from these works, and then study this guarantee through the lens of symmetries, as any item-symmetric auction is also fair (by this definition). Observe that in this domain, bundling all items together corresponds to concealing all demographic data [23].

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 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
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.

    The below paragraphs distinguish our model from [14, 27, 34]. We discuss other related works such as [11] in Sect. 1.2.

  2. 2.

    A valuation function \(v(\cdot )\) is additive if for all sets S of items, \(v(S)=\sum _{j \in S} v(\{j\})\).

  3. 3.

    To quickly see why the unlimited supply setting is equivalent to the single bidder setting: Because there are no supply constraints, it is feasible to pick any single-bidder mechanism and just use it for every bidder.

  4. 4.

    This follows as the optimal auction when all items are i.i.d. is in fact item-symmetric.

  5. 5.

    For example, it could be that \(D_i = D_{i'}\) for all \(i, i'\), and each bidder is drawn from the same population. This represents settings where the platform cannot price-discriminate based on properties of the advertiser. It could also be that \(D_i \ne D_{i'}\). In such settings, perhaps \(D_i\) is the population of ‘big’ advertisers, and \(D_{i'}\) is the population of ‘small’ advertisers, and the platform knows from which population each individual advertiser is drawn.

  6. 6.

    We use the standard notation \(\vec {v}_{-i}\) to refer to the vector of bids excluding bidder i, and \(D_{-i}\) to refer to the distribution over valuation profiles, excluding bidder i.

  7. 7.

    To see this, recall that \(\sigma (\vec {x}(\vec {v}_i;\vec {v}_{-i}))\) is a vector that puts \(x_{ij}(\vec {v}_i;\vec {v}_{-i}))\) in the \(i,\sigma (j)\) coordinate.

  8. 8.

    We also note that it is an interesting open direction to extend our main results from an additive bidder to a ’scaled additive’ bidder so that this connection holds even for non-uniform demographic distributions.

References

  1. Google translate’s gender problem (and bing translate’s, and systran’s) (2013). https://www.fastcompany.com/3010223/google-translates-gender-problem-and-bing-translates-and-systrans

  2. Google photos labeled black people ’gorillas’. USA Today (2015). https://www.usatoday.com/story/tech/2015/07/01/google-apologizes-after-photos-identify-black-people-as-gorillas/29567465/

  3. Ali, M., Sapiezynski, P., Bogen, M., Korolova, A., Mislove, A., Rieke, A.: Discrimination through optimization: how Facebook’s ad delivery can lead to biased outcomes. Proc. ACM Hum. Comput. Interact. 3(CSCW), 199:1–199:30 (2019). https://doi.org/10.1145/3359301

  4. Babaioff, M., Gonczarowski, Y.A., Nisan, N.: 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, 19–23 June 2017, pp. 869–877 (2017). https://doi.org/10.1145/3055399.3055426. http://doi.acm.org/10.1145/3055399.3055426

  5. Babaioff, M., Immorlica, N., Lucier, B., Weinberg, S.M.: A simple and approximately optimal mechanism for an additive buyer. In: 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, 18–21 October 2014, pp. 21–30 (2014). https://doi.org/10.1109/FOCS.2014.11

  6. Beyhaghi, H., Weinberg, S.M.: Optimal (and benchmark-optimal) competition complexity for additive buyers over independent items. In: Proceedings of the 51st ACM Symposium on Theory of Computing Conference (STOC) (2019)

    Google Scholar 

  7. Bolukbasi, T., Chang, K.W., Zou, J.Y., Saligrama, V., Kalai, A.T.: Man is to computer programmer as woman is to homemaker? Debiasing word embeddings. In: Advances in Neural Information Processing Systems, pp. 4349–4357 (2016)

    Google Scholar 

  8. Buolamwini, J., Gebru, T.: Gender shades: intersectional accuracy disparities in commercial gender classification. In: ACM Conference on Fairness, Accountability, Transparency (FAccT) (2018)

    Google Scholar 

  9. Cai, Y., Devanur, N., Weinberg, S.M.: A duality based unified approach to Bayesian mechanism design. In: Proceedings of the 48th ACM Conference on Theory of Computation (STOC) (2016)

    Google Scholar 

  10. Cai, Y., Zhao, M.: 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, 19–23 June 2017, pp. 170–183 (2017). https://doi.org/10.1145/3055399.3055465. http://doi.acm.org/10.1145/3055399.3055465

  11. Celis, L.E., Mehrotra, A., Vishnoi, N.K.: Toward controlling discrimination in online ad auctions. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning, ICML 2019, Long Beach, California, USA, 9–15 June 2019. Proceedings of Machine Learning Research, vol. 97, pp. 4456–4465. PMLR (2019). http://proceedings.mlr.press/v97/mehrotra19a.html

  12. Chawla, S., Hartline, J.D., Kleinberg, R.D.: Algorithmic pricing via virtual valuations. In: The 8th ACM Conference on Electronic Commerce (EC) (2007)

    Google Scholar 

  13. Chawla, S., Hartline, J.D., Malec, D.L., Sivan, B.: Multi-parameter mechanism design and sequential posted pricing. In: The 42nd ACM Symposium on Theory of Computing (STOC) (2010)

    Google Scholar 

  14. Chawla, S., Jagadeesan, M.: Fairness in ad auctions through inverse proportionality. CoRR abs/2003.13966 (2020). https://arxiv.org/abs/2003.13966

  15. Chawla, S., Malec, D.L., Sivan, B.: The power of randomness in Bayesian optimal mechanism design. In: The 11th ACM Conference on Electronic Commerce (EC) (2010)

    Google Scholar 

  16. Chawla, S., Miller, J.B.: Mechanism design for subadditive agents via an ex ante relaxation. In: Proceedings of the 2016 ACM Conference on Economics and Computation, EC 2016, Maastricht, The Netherlands, 24–28 July 2016, pp. 579–596 (2016). https://doi.org/10.1145/2940716.2940756. http://doi.acm.org/10.1145/2940716.2940756

  17. Chen, X., Diakonikolas, I., Orfanou, A., Paparas, D., Sun, X., Yannakakis, M.: On the complexity of optimal lottery pricing and randomized mechanisms. In: IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17–20 October 2015, pp. 1464–1479 (2015). https://doi.org/10.1109/FOCS.2015.93

  18. Cheng, Yu., Gravin, N., Munagala, K., Wang, K.: A simple mechanism for a budget-constrained buyer. In: Christodoulou, G., Harks, T. (eds.) WINE 2018. LNCS, vol. 11316, pp. 96–110. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-04612-5_7

    Chapter  Google Scholar 

  19. Chouldechova, A.: Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. arXiv preprint arXiv:1703.00056 (2017)

  20. Clarke, E.H.: Multipart Pricing of Public Goods. Public Choice 11(1), 17–33 (1971)

    Article  Google Scholar 

  21. Daskalakis, C., Deckelbaum, A., Tzamos, C.: The complexity of optimal mechanism design. In: The 25th ACM-SIAM Symposium on Discrete Algorithms (SODA) (2014)

    Google Scholar 

  22. Daskalakis, C., Deckelbaum, A., Tzamos, C.: Strong duality for a multiple-good monopolist. Econometrica 85(3), 735–767 (2017)

    Article  MathSciNet  Google Scholar 

  23. Daskalakis, C., Papadimitriou, C.H., Tzamos, C.: Does information revelation improve revenue? 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. 233–250. ACM (2016). https://doi.org/10.1145/2940716.2940789

  24. Daskalakis, C., Weinberg, S.M.: Symmetries and optimal multi-dimensional mechanism design. In: Proceedings of the 13th ACM Conference on Electronic Commerce, EC 2012, Valencia, Spain, 4–8 June 2012, pp. 370–387 (2012). https://doi.org/10.1145/2229012.2229042

  25. Datta, A., Tschantz, M.C., Datta, A.: Automated experiments on ad privacy settings. Proc. Priv. Enhancing Technol. 2015(1), 92–112 (2015). https://doi.org/10.1515/popets-2015-0007

  26. Dwork, C., Hardt, M., Pitassi, T., Reingold, O., Zemel, R.S.: Fairness through awareness. In: Goldwasser, S. (ed.) Innovations in Theoretical Computer Science 2012, Cambridge, MA, USA, 8–10 January 2012, pp. 214–226. ACM (2012). https://doi.org/10.1145/2090236.2090255

  27. Dwork, C., Ilvento, C.: Fairness under composition. In: Blum, A. (ed.) 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, San Diego, California, USA, 10–12 January 2019. LIPIcs, vol. 124, pp. 33:1–33:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019). https://doi.org/10.4230/LIPIcs.ITCS.2019.33

  28. Eden, A., Feldman, M., Friedler, O., Talgam-Cohen, I., Weinberg, S.M.: The competition complexity of auctions: a bulow-klemperer result for multi-dimensional bidders. In: Proceedings of the 2017 ACM Conference on Economics and Computation, EC 2017, Cambridge, MA, USA, 26–30 June 2017, p. 343 (2017). https://doi.org/10.1145/3033274.3085115. http://doi.acm.org/10.1145/3033274.3085115

  29. Eden, A., Feldman, M., Friedler, O., Talgam-Cohen, I., Weinberg, S.M.: A simple and approximately optimal mechanism for a buyer with complements: abstract. In: Proceedings of the 2017 ACM Conference on Economics and Computation, EC 2017, Cambridge, MA, USA, 26–30 June 2017, p. 323 (2017). https://doi.org/10.1145/3033274.3085116. http://doi.acm.org/10.1145/3033274.3085116

  30. Groves, T.: Incentives in teams. Econometrica 41(4), 617–631 (1973)

    Article  MathSciNet  Google Scholar 

  31. Hart, S., Nisan, N.: Approximate revenue maximization with multiple items. J. Economic Theory 172, 313–347 (2017). https://doi.org/10.1016/j.jet.2017.09.001

    Article  MathSciNet  MATH  Google Scholar 

  32. Hart, S., Reny, P.J.: Maximizing revenue with multiple goods: nonmonotonicity and other observations. Theor. Econ. 10(3), 893–922 (2015)

    Article  MathSciNet  Google Scholar 

  33. Hartline, J.D., Roughgarden, T.: Simple versus optimal mechanisms. In: ACM Conference on Electronic Commerce, pp. 225–234 (2009)

    Google Scholar 

  34. Ilvento, C., Jagadeesan, M., Chawla, S.: Multi-category fairness in sponsored search auctions. In: Hildebrandt, M., Castillo, C., Celis, E., Ruggieri, S., Taylor, L., Zanfir-Fortuna, G. (eds.) FAT* 2020: Conference on Fairness, Accountability, and Transparency, Barcelona, Spain, 27–30 January 2020, pp. 348–358. ACM (2020). https://doi.org/10.1145/3351095.3372848

  35. Kay, M., Matuszek, C., Munson, S.A.: Unequal representation and gender stereotypes in image search results for occupations. In: Proceedings of the 33rd Annual ACM Conference on Human Factors in Computing Systems, pp. 3819–3828. ACM (2015)

    Google Scholar 

  36. Kothari, P., Mohan, D., Schvartzman, A., Singla, S., Weinberg, S.M.: Approximation schemes for a buyer with independent items via symmetries. In: The 60th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2019)

    Google Scholar 

  37. Lambrecht, A., Tucker, C.: Algorithmic bias? An empirical study of apparent gender-based discrimination in the display of STEM career ads. Manag. Sci. 65(7), 2966–2981 (2019). https://doi.org/10.1287/mnsc.2018.3093

    Article  Google Scholar 

  38. Levin, S.: A beauty contest was judged by AI and the robots didn’t like dark skin, September 2016

    Google Scholar 

  39. Li, X., Yao, A.C.C.: On revenue maximization for selling multiple independently distributed items. Proc. Natl. Acad. Sci. 110(28), 11232–11237 (2013)

    Article  MathSciNet  Google Scholar 

  40. Liu, S., Psomas, C.: On the competition complexity of dynamic mechanism design. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, 7–10 January 2018, pp. 2008–2025 (2018). https://doi.org/10.1137/1.9781611975031.131

  41. Manelli, A.M., Vincent, D.R.: Multidimensional mechanism design: revenue maximization and the multiple-good monopoly. J. Econ. Theory 137(1), 153–185 (2007)

    Article  MathSciNet  Google Scholar 

  42. Myerson, R.B.: Optimal auction design. Math. Oper. Res. 6(1), 58–73 (1981)

    Article  MathSciNet  Google Scholar 

  43. Pavlov, G.: Optimal mechanism for selling two goods. B.E. J. Theor. Econ. 11(3) (2011)

    Google Scholar 

  44. Plaugic, L.: Faceapp’s creator apologizes for the app’s lightening ‘hot’ filter, April 2017

    Google Scholar 

  45. Rubinstein, A., Weinberg, S.M.: Simple mechanisms for a subadditive buyer and applications to revenue monotonicity. In: Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC 2015, Portland, OR, USA, 15–19 June 2015, pp. 377–394 (2015). https://doi.org/10.1145/2764468.2764510

  46. Ryu, H., Adam, H., Mitchell, M.: Inclusivefacenet: improving face attribute detection with race and gender diversity. In: Proceedings of FAT/ML (2018)

    Google Scholar 

  47. Thanassoulis, J.: Haggling over substitutes. J. Econ. Theory 117, 217–245 (2004)

    Article  MathSciNet  Google Scholar 

  48. Torralba, A., Efros, A.A.: Unbiased look at dataset bias. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2011)

    Google Scholar 

  49. Vickrey, W.: Counterspeculations, auctions, and competitive sealed tenders. J. Finance 16(1), 8–37 (1961)

    Article  MathSciNet  Google Scholar 

  50. Wang, A., Narayanan, A., Russakovsky, O.: Vibe: a tool for measuring and mitigating bias in image datasets. CoRR abs/2004.07999 (2020). https://arxiv.org/abs/2004.07999

  51. Yao, A.C.C.: 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) (2015)

    Google Scholar 

  52. Zemel, R., Wu, Y., Swersky, K., Pitassi, T., Dwork, C.: Learning fair representations. In: Proceedings of the 30th International Conference on Machine Learning (ICML 2013), pp. 325–333 (2013)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08540, USA

    Meryem Essaidi & S. Matthew Weinberg

Authors
  1. Meryem Essaidi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Matthew Weinberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Meryem Essaidi .

Editor information

Editors and Affiliations

  1. Tel Aviv University, Tel Aviv, Israel

    Michal Feldman

  2. Shanghai University of Finance and Economics, Shanghai, China

    Hu Fu

  3. Technion – Israel Institute of Technology, Haifa, Israel

    Inbal Talgam-Cohen

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Essaidi, M., Weinberg, S.M. (2022). On Symmetries in Multi-dimensional Mechanism Design. In: Feldman, M., Fu, H., Talgam-Cohen, I. (eds) Web and Internet Economics. WINE 2021. Lecture Notes in Computer Science(), vol 13112. Springer, Cham. https://doi.org/10.1007/978-3-030-94676-0_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-94676-0_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-94675-3

  • Online ISBN: 978-3-030-94676-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation