# Applications of \(\alpha \)-Strongly Regular Distributions to Bayesian Auctions

## Abstract

Two classes of distributions that are widely used in the analysis of Bayesian auctions are the Monotone Hazard Rate (MHR) and Regular distributions. They can both be characterized in terms of the rate of change of the associated virtual value functions: for MHR distributions the condition is that for values \(v < v'\), \(\phi (v') - \phi (v) \ge v' - v\), and for regular distributions, \(\phi (v') - \phi (v) \ge 0\). Cole and Roughgarden introduced the interpolating class of \(\alpha \)-Strongly Regular distributions (\(\alpha \)-SR distributions for short), for which \(\phi (v') - \phi (v) \ge \alpha (v' - v)\), for \(0 \le \alpha \le 1\). In this paper, we investigate five distinct auction settings for which good expected revenue bounds are known when the bidders’ valuations are given by MHR distributions. In every case, we show that these bounds degrade gracefully when extended to \(\alpha \)-SR distributions. For four of these settings, the auction mechanism requires knowledge of these distribution(s) (in the other setting, the distributions are needed only to ensure good bounds on the expected revenue). In these cases we also investigate what happens when the distributions are known only approximately via samples, specifically how to modify the mechanisms so that they remain effective and how the expected revenue depends on the number of samples.

## Notes

### Acknowledgements

We thank the referees for their thoughtful comments.

## References

- 1.Bhattacharya, S., Goel, G., Gollapudi, S., Munagala, K.: Budget constrained auctions with heterogeneous items. In: STOC, pp. 379–388 (2010)Google Scholar
- 2.Chawla, S., Malec, D.L., Malekian, A.: Bayesian mechanism design for budget-constrained agents. In: EC, pp. 253–262 (2011)Google Scholar
- 3.Cole, R., Roughgarden, T.: The sample complexity of revenue maximization. In: Symposium on Theory of Computing, STOC 2014, New York, NY, USA, pp. 243–252, 31 May–03 June 2014Google Scholar
- 4.Dhangwatnotai, P., Roughgarden, T., Yan, Q.: Revenue maximization with a single sample. In: EC, pp. 129–138 (2010)Google Scholar
- 5.Hartline, J.D.: Mechanism design and approximation. Book draft (2014)Google Scholar
- 6.Hartline, J.D., Karlin, A.: Profit maximization in mechanism design. In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V.V. (eds.) Algorithmic Game Theory, chapt. 13, pp. 331–362. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
- 7.Hartline, J.D., Roughgarden, T.: Simple versus optimal mechanisms. In: EC, pp. 225–234 (2009)Google Scholar
- 8.Myerson, R.B.: Optimal auction design. Math. Oper. Res.
**6**(1), 58–73 (1981)MathSciNetCrossRefzbMATHGoogle Scholar

## Copyright information

Open Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.