Abstract
Earning limits are an interesting novel aspect in the classic Fisher market model. Here sellers have bounds on their income and can decide to lower the supply they bring to the market if income exceeds the limit. Beyond several applications, in which earning limits are natural, equilibria of such markets are a central concept in the allocation of indivisible items to maximize Nash social welfare.
In this paper, we analyze earning limits in Fisher markets with linear and spending-constraint utilities. We show a variety of structural and computational results about market equilibria. The equilibrium price vectors form a lattice, and the spending of buyers is unique in non-degenerate markets. We provide a scaling-based algorithm that computes an equilibrium in time \(O(n^3\ell \log (\ell + nU))\), where n is the number of agents, \(\ell \ge n\) a bound on the segments in the utility functions, and U the largest integer in the market representation. Moreover, we show how to refine any equilibrium in polynomial time to one with minimal prices, or one with maximal prices (if it exists). Finally, we discuss how our algorithm can be used to obtain in polynomial time a 2-approximation for Nash social welfare in multi-unit markets with indivisible items that come in multiple copies.
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Notes
- 1.
Consider the example of a linear market with one buyer and one good. The utility is \(u_{11} > 0\), the buyer has a budget \(m_1 = 2\), the good has an earning limit \(d_1 = 1\).
References
Anari, N., Mai, T., Gharan, S.O., Vazirani, V.: Nash social welfare for indivisible items under separable, piecewise-linear concave utilities (2016). CoRR abs/1612.05191
Bei, X., Garg, J., Hoefer, M.: Ascending-price algorithms for unknown markets. In: Proceedings of 17th Conference Economics and Computation (EC), p. 699 (2016)
Bei, X., Garg, J., Hoefer, M., Mehlhorn, K.: Computing equilibria in markets with budget-additive utilities. In: Proceedings of 24th European Symposium Algorithms (ESA), pp. 8:1–8:14 (2016)
Birnbaum, B., Devanur, N., Xiao, L.: Distributed algorithms via gradient descent for Fisher markets. In: Proceedings of 12th Conference Electronic Commerce (EC), pp. 127–136 (2011)
Cole, R., Devanur, N., Gkatzelis, V., Jain, K., Mai, T., Vazirani, V., Yazdanbod, S.: Convex program duality, Fisher markets, and Nash social welfare. In: Proceedings of 18th Conference Economics and Computation (EC) (2017, to appear)
Cole, R., Gkatzelis, V.: Approximating the Nash social welfare with indivisible items. In: Proceedings of 47th Symposium Theory of Computing (STOC), pp. 371–380 (2015)
Devanur, N., Papadimitriou, C., Saberi, A., Vazirani, V.: Market equilibrium via a primal-dual algorithm for a convex program. J. ACM 55(5), 22:1–22:18 (2008)
Devanur, N., Vazirani, V.: The spending constraint model for market equilibrium: algorithmic, existence and uniqueness results. In: Proceedings of 36th Symposium Theory of Computing (STOC), pp. 519–528 (2004)
Duan, R., Mehlhorn, K.: A combinatorial polynomial algorithm for the linear Arrow-Debreu market. Inf. Comput. 243, 112–132 (2015)
Eisenberg, E., Gale, D.: Consensus of subjective probabilities: the Pari-Mutuel method. Ann. Math. Stat. 30(1), 165–168 (1959)
Hochbaum, D., Shanthikumar, G.: Convex separable optimization is not much harder than linear optimization. J. ACM 37(4), 843–862 (1990)
Jain, K.: A polynomial time algorithm for computing the Arrow-Debreu market equilibrium for linear utilities. SIAM J. Comput. 37(1), 306–318 (2007)
Karzanov, A., McCormick, T.: Polynomial methods for separable convex optimization in unimodular linear spaces with applications. SIAM J. Comput. 26(4), 1245–1275 (1997)
Orlin, J.: Improved algorithms for computing Fisher’s market clearing prices. In: Proceedings of 42nd Symposium Theory of Computing (STOC), pp. 291–300 (2010)
Shmyrev, V.: An algorithm for finding equilibrium in the linear exchange model with fixed budgets. J. Appl. Indust. Math. 3(4), 505–518 (2009)
Vazirani, V.: Spending constraint utilities with applications to the adwords market. Math. Oper. Res. 35(2), 458–478 (2010)
Végh, L.: Concave generalized flows with applications to market equilibria. Math. Oper. Res. 39(2), 573–596 (2014)
Végh, L.: Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives. SIAM J. Comput. 45(5), 1729–1761 (2016)
Ye, Y.: A path to the Arrow-Debreu competitive market equilibrium. Math. Prog. 111(1–2), 315–348 (2008)
Zhang, L.: Proportional response dynamics in the Fisher market. Theoret. Comput. Sci. 412(24), 2691–2698 (2011)
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Bei, X., Garg, J., Hoefer, M., Mehlhorn, K. (2017). Earning Limits in Fisher Markets with Spending-Constraint Utilities. In: Bilò, V., Flammini, M. (eds) Algorithmic Game Theory. SAGT 2017. Lecture Notes in Computer Science(), vol 10504. Springer, Cham. https://doi.org/10.1007/978-3-319-66700-3_6
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DOI: https://doi.org/10.1007/978-3-319-66700-3_6
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