SAGT 2018: Algorithmic Game Theory pp 1-11

# On Revenue Monotonicity in Combinatorial Auctions

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11059)

## Abstract

Along with substantial progress made recently in designing near-optimal mechanisms for multi-item auctions, interesting structural questions have also been raised and studied. In particular, is it true that the seller can always extract more revenue from a market where the buyers value the items higher than another market? In this paper we obtain such a revenue monotonicity result in a general setting. Precisely, consider the revenue-maximizing combinatorial auction for m items and n buyers in the Bayesian setting, specified by a valuation function v and a set F of nm independent item-type distributions. Let REV(vF) denote the maximum revenue achievable under F by any incentive compatible mechanism. Intuitively, one would expect that $$REV(v, G)\ge REV(v, F)$$ if distribution G stochastically dominates F. Surprisingly, Hart and Reny (2012) showed that this is not always true even for the simple case when v is additive. A natural question arises: Are these deviations contained within bounds? To what extent may the monotonicity intuition still be valid? We present an approximate monotonicity theorem for the class of fractionally subadditive (XOS) valuation functions v, showing that $$REV(v, G)\ge c\,REV(v, F)$$ if G stochastically dominates F under v where $$c>0$$ is a universal constant. Previously, approximate monotonicity was known only for the case $$n=1$$: Babaioff et al. (2014) for the class of additive valuations, and Rubinstein and Weinberg (2015) for all subaddtive valuation functions.

## Keywords

Mechanism design Subadditive valuation Maximum revenue

## References

1. 1.
Babaioff, M., Immorlica, N., Lucier, B., Weinberg, S.M.: A simple and approximately optimal mechanism for an additive buyer. In: Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2014)Google Scholar
2. 2.
Cai, Y., Devanur, N.R., Weinberg, S.M.: A duality based unified approach to Bayesian mechanism design. In: Proceedings of the 52th Annual ACM Symposium on Theory of Computing (STOC) (2016)Google Scholar
3. 3.
Cai, Y., Zhao, M.: Simple mechanisms for subadditive buyers via duality. In: Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC) (2017)Google Scholar
4. 4.
Chawla, S., Hartline, J.D., Malec, D.L., Sivan, B.: Multi-parameter mechanism design and sequential posted pricing. In: Proceedings of the 42th Annual ACM Symposium on Theory of Computing (STOC), pp. 311–320 (2010)Google Scholar
5. 5.
Chawla, S., Malec, D.L., Sivan, B.: The power of randomness in Bayesian optimal mechanism design. Games Econ. Behav. 91, 297–317 (2015)
6. 6.
Dobzinski, S., Nisan, N., Schapira, M.: Approximation algorithms for combinatorial auctions with complement-free bidders. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC) (2005)Google Scholar
7. 7.
Feldman, M., Gravin, N., Lucier, B.: Combinatorial auctions via posted price. In: Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 123–135 (2015)Google Scholar
8. 8.
Gonczarowski, Y., Nisan, N.: Efficient empirical revenue maximization in single-parameter auction environments. In: Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC), pp. 856–868 (2017)Google Scholar
9. 9.
Hart, S., Nisan, N.: Approximate revenue maximization with multiple items. In: Proceedings of the 13th ACM Conference on Electric Commerce (EC), p. 656 (2012)Google Scholar
10. 10.
Hart, S., Reny, P.: Maximizing revenue with multiple goods: nonmonotonicity and other observations. Theor. Econ. 10, 893–992 (2015)
11. 11.
Hartline, J.D., Karlin, A.R.: Profit maximization in mechanism design. In: Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V. (eds.) Algorithmic Game Theory, Chap. 13, pp. 331–361. Cambridge University Press (2007)Google Scholar
12. 12.
Kleinberg, R., Weinberg, S.M.: Matroid prophet inequalities. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC) (2012)Google Scholar
13. 13.
Myerson, R.B.: Optimal auction design. Math. Oper. Res. 6(1), 58–73 (1981)
14. 14.
Rubinstein, A., Weinberg, S.M.: Simple mechanisms for a subadditive buyer and applications to revenue monotonicity. In: Proceedings of the 16th ACM Conference on Electronic Commerce, pp. 377–394 (2015)Google Scholar
15. 15.
Yao, A.C.: An n-to-1 bidder reduction for multi-item auctions and its applications. In: Proceedings of the 25th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 92–109 (2015)Google Scholar