Earning Limits in Fisher Markets with Spending-Constraint Utilities
- 771 Downloads
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.
- 1.Anari, N., Mai, T., Gharan, S.O., Vazirani, V.: Nash social welfare for indivisible items under separable, piecewise-linear concave utilities (2016). CoRR abs/1612.05191Google Scholar
- 2.Bei, X., Garg, J., Hoefer, M.: Ascending-price algorithms for unknown markets. In: Proceedings of 17th Conference Economics and Computation (EC), p. 699 (2016)Google Scholar
- 3.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)Google Scholar
- 4.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)Google Scholar
- 5.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)Google Scholar
- 6.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)Google Scholar
- 8.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)Google Scholar
- 14.Orlin, J.: Improved algorithms for computing Fisher’s market clearing prices. In: Proceedings of 42nd Symposium Theory of Computing (STOC), pp. 291–300 (2010)Google Scholar