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.
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- 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
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)
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)
Awasthi, P., Blum, A., Sheffet, O.: Center-based clustering under perturbation stability. Inf. Process. Lett. 112(1–2), 49–54 (2012)
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)
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)
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)
Blumrosen, L., Nisan, N.: On the computational power of demand queries. SIAM J. Comput. 39(4), 1372–1391 (2009)
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
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
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)
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)
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
Dobzinski, S., Nisan, N., Schapira, M.: Truthful randomized mechanisms for combinatorial auctions. J. Comput. Syst. Sci. 78(1), 15–25 (2012)
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)
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)
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)
Feige, U.: On maximizing welfare when utility functions are subadditive. SIAM J. Comput. 39(1), 122–142 (2009)
Feige, U., Vondrák, J.: The submodular welfare problem with demand queries. Theor. Comput. 6(1), 247–290 (2010)
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)
Fikioris, G., Fotakis, D.: Mechanism design for perturbation stable combinatorial auctions. CoRR abs/2006.09889 (2020). https://arxiv.org/abs/2006.09889
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)
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)
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
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
Lehmann, B., Lehmann, D., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. Games Econ. Behav. 55(2), 270–296 (2006)
Milgrom, P.: Putting Auction Theory to Work. Churchill Lectures in Economics. Cambridge University Press, Cambridge (2004)
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)
Cramton, P., Shoham, Y., Steinberg, R.: Combinatorial Auctions. MIT Press, Cambridge (2006)
Rassenti, S., Smith, V., Bulfin, R.: A combinatorial auction mechanism for airport time slot allocation. Bell J. Econ. 13(2), 402–417 (1982)
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)
Roughgarden, T.: Beyond worst-case analysis. Commun. ACM 62(3), 88–96 (2019)
Roughgarden, T., Syrgkanis, V., Tardos, É.: The price of anarchy in auctions. J. Artif. Intell. Res. 59, 59–101 (2017)
Roughgarden, T., Talgam-Cohen, I.: Approximately optimal mechanism design. CoRR abs/1812.11896 (2018). http://arxiv.org/abs/1812.11896
Spielman, D., Teng, S.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)
Spielman, D., Teng, S.: Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Commun. ACM 52(10), 76–84 (2009)
Syrgkanis, V., Tardos, É.: Composable and efficient mechanisms. In: Proceedings of the 45th Symposium on Theory of Computing (STOC 2013), pp. 211–220 (2013)
Thaler, R.: Toward a positive theory of consumer choice. J. Econ. Behav. Organ. 1(1), 39–60 (1980)
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)
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.
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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
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