Abstract
We investigate revenue-maximizing mechanisms in settings where bidders’ utility functions are characterized by convex costs. Such costs arise, for instance, in procurement auctions for energy, and when bidders borrow money at non-linear interest rates. We provide a 1 / 16e approximation guarantee for a prior-free randomized mechanism when bidders’ values are drawn from MHR distributions, and their costs are polynomial. Additionally, we propose two heuristics that allocate proportionally, using either bidders’ values or virtual values. Perhaps surprisingly, in the convex cost setting, it is preferable to allocate to multiple relatively high bidders, rather than only to bidders with the highest (virtual) value, as is optimal in the traditional quasi-linear utility setting.
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- 1.
A notable exception is [8], who study prior-free auctions for risk-averse agents, which are modelled by a very specific form of capped quasi-linear utilities.
- 2.
Multiplying \(\mathfrak {u_{}}_{i}\) by \(v_{i}\) yields a familiar utility function, that of the forward setting, with utility measured in units of power, rather than money: \(v_{i} \mathfrak {u_{}}_{i} = u_{i} = v_{i} x_{i} - c_{i}(p_{i})\).
- 3.
For example, it is more expensive to convert bitumen into synthetic crude oil than it is to drill and pump conventional crude oil.
- 4.
In the convex cost setting, if we interpret \(v_{}(q_{})\) as a posted cost, rather than a posted price (i.e., payment) then \(R(q_{})\) can be wrongly interpreted as an expected cost function.
References
Alaei, S., Fu, H., Haghpanah, N., Hartline, J.: The simple economics of approximately optimal auctions. In: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pp. 628–637. IEEE (2013)
Border, K.C.: Implementation of reduced form auctions: a geometric approach. Econometrica 59(4), 1175–1187 (1991). http://www.jstor.org/stable/2938181
Bulow, J., Klemperer, P.: Auctions versus negotiations. Am. Econ. Rev. 86(1), 180–194 (1996)
Cai, Y., Daskalakis, C., Weinberg, S.M.: Understanding incentives: mechanism design becomes algorithm design. In: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pp. 618–627, October (2013). https://doi.org/10.1109/FOCS.2013.72
Chawla, S., Hartline, J.D., Sivan, B.: Optimal crowdsourcing contests. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 856–868. SIAM (2012)
Dhangwatnotai, P., Roughgarden, T., Yan, Q.: Revenue maximization with a single sample. Games Econ. Behav. 91, 318–333 (2015)
DiPalantino, D., Vojnovic, M.: Crowdsourcing and all-pay auctions. In: Proceedings of the 10th ACM Conference on Electronic Commerce, pp. 119–128. ACM (2009)
Fu, H., Hartline, J., Hoy, D.: Prior-independent auctions for risk-averse agents. In: Proceedings of the Fourteenth ACM Conference on Electronic Commerce, pp. 471–488. EC ’13, ACM, New York, NY, USA (2013). http://doi.acm.org/10.1145/2482540.2482551
Gossen, H.H.: Entwickelung der gesetze des menschlichen verkehrs, und der daraus fliessenden regeln für menschliche handeln. F. Vieweg (1854)
Greenwald, A., Oyakawa, T., Syrgkanis, V.: Simple vs optimal contests with convex costs. In: Champin, P., Gandon, F.L., Lalmas, M., Ipeirotis, P.G. (eds.) Proceedings of the 2018 World Wide Web Conference on World Wide Web, WWW 2018, Lyon, France, 23–27 April 2018, pp. 1429–1438. ACM (2018). http://doi.acm.org/10.1145/3178876.3186048
Hartline, J.D.: Mechanism design and approximation. Book draft. October (2015)
Hartline, J.D., Roughgarden, T.: Simple versus optimal mechanisms. In: Proceedings of the 10th ACM Conference on Electronic Commerce, pp. 225–234. EC ’09, ACM, New York, NY, USA (2009). http://doi.acm.org/10.1145/1566374.1566407
Maskin, E., Riley, J.: Optimal auctions with risk averse buyers. Econometrica 52(6), 1473–1518 (1984). http://www.jstor.org/stable/1913516
Myerson, R.B.: Optimal auction design. Math. Oper. Res. 6(1), 58–73 (1981)
Pai, M.M., Vohra, R.: Optimal auctions with financially constrained buyers. J. Econ. Theory 150, 383–425 (2014). https://doi.org/10.1016/j.jet.2013.09.015. http://www.sciencedirect.com/science/article/pii/S0022053113001701
Roughgarden, T., Talgam-cohen, I., Yan, Q.: Robust auctions for revenue via enhanced competition (2016)
Sakurai, Y., Saito, Y., Iwasaki, A., Yokoo, M.: Beyond quasi-linear utility: strategy/false-name-proof multi-unit auction protocols. In: IEEE/WIC/ACM International Conference on Web Intelligence and Intelligent Agent Technology, 2008. WI-IAT’08, vol. 2, pp. 417–423. IEEE (2008)
Samuel-Cahn, E.: Comparison of threshold stop rules and maximum for independent nonnegative random variables. Ann. Probab. 12(4), 1213–1216 (1984)
Singer, Y.: Budget feasible mechanism design. SIGecom Exch. 12(2), 24–31 (2014). http://doi.acm.org/10.1145/2692359.2692366
Vickrey, W.: Counterspeculation, auctions, and competitive sealed tenders. J. Financ. 16(1), 8–37 (1961)
Wilson, R.: Incentive efficiency of double auctions. Econometrica J. Econ. Soc. 53, 1101–1115 (1985)
Wilson, R.: Game-theoretic approaches to trading processes. In: Advances in Economic Theory: Fifth World Congress, pp. 33–77 (1987)
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This research was supported by NSF Grant #1217761 and Microsoft Research.
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Greenwald, A., Oyakawa, T., Syrgkanis, V. (2018). On Revenue-Maximizing Mechanisms Assuming Convex Costs. In: Deng, X. (eds) Algorithmic Game Theory. SAGT 2018. Lecture Notes in Computer Science(), vol 11059. Springer, Cham. https://doi.org/10.1007/978-3-319-99660-8_11
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