Competitive Equilibria for Non-quasilinear Bidders in Combinatorial Auctions

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10123)


quasilinearity is a ubiquitous and questionable assumption in the standard study of Walrasian equilibria. Quasilinearity implies that a buyer’s value for goods purchased in a Walrasian equilibrium is always additive with goods purchased with unspent money. It is a particularly suspect assumption in combinatorial auctions, where buyers’ complex preferences over goods would naturally extend beyond the items obtained in the Walrasian equilibrium.

We study Walrasian equilibria in combinatorial auctions when quasilinearity is not assumed. We show that existence can be reduced to an Arrow-Debreu style market with one divisible good and many indivisible goods, and that a “fractional” Walrasian equilibrium always exists. We also show that standard integral Walrasian equilibria are related to integral solutions of an induced configuration LP associated with a fractional Walrasian equilibrium, generalizing known results for both quasilinear and non-quasilnear settings.


Competitive Equilibrium Price Vector Combinatorial Auction Assignment Game Complementary Slackness 
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  1. Alaei, S., Jain, K., Malekian, A.: Competitive equilibrium in two sided matching markets with general utility functions. ACM SIGecom Exchanges 10(2), 34–36 (2011)CrossRefzbMATHGoogle Scholar
  2. Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica: J. Econom. Soc. 22, 265–290 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bevia, C., Quinzii, M., Silva, J.A.: Buying several indivisible goods. Math. Soc. Sci. 37(1), 1–23 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bikhchandani, S., Mamer, J.W.: Competitive equilibrium in an exchange economy with indivisibilities. J. Econ. Theor. 74(2), 385–413 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Clarke, E.H.: Multipart pricing of public goods. Public Choice 11(1), 17–33 (1971)CrossRefGoogle Scholar
  6. Cramton, P.C., Shoham, Y., Steinberg, R., et al.: Combinatorial Auctions, vol. 475. MIT press, Cambridge (2006)zbMATHGoogle Scholar
  7. Demange, G., Gale, D.: The strategy structure of two-sided matching markets. Econometrica: J. Econom. Soc. 53, 873–888 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Echenique, F., Oviedo, J.: A theory of stability in many-to-many matching markets (2004)Google Scholar
  9. Feldman, M., Gravin, N., Lucier, B.: Combinatorial walrasian equilibrium. SIAM J. Comput. 45(1), 29–48 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gale, D.: Equilibrium in a discrete exchange economy with money. Int. J. Game Theor. 13(1), 61–64 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Groves, T.: Incentives in teams. Econometrica: J. Econom. Soc. 41, 617–631 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Gul, F., Stacchetti, E.: Walrasian equilibrium with gross substitutes. J. Econ. Theor. 87(1), 95–124 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kakutani, S., et al.: A Generalization of Brouwer’s Fixed Point Theorem. Duke University Press, Durham (1941)zbMATHGoogle Scholar
  14. Kaneko, M., Yamamoto, Y.: The existence and computation of competitive equilibria in markets with an indivisible commodity. J. Econ. Theor. 38(1), 118–136 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kelso Jr., A.S., Crawford, V.P.: Job matching, coalition formation, gross substitutes. Econometrica: J. Econom. Soc. 50, 1483–1504 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Maskin, E.S.: On the fair allocation of indivisible goods. In: Feiwel, G.R. (ed.) Arrow and the Foundations of the Theory of Economic Policy, pp. 341–349. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  17. Murota, K., Tamura, A.: Computation of Competitive Equilibria of Indivisible Commodities Via M-convex Submodular Flow Problem. Kyoto University, Research Institute for Mathematical Sciences, Kyoto (2001)zbMATHGoogle Scholar
  18. Niazadeh, R., Wilkens, C.A.: Competitive equilibria for non-quasilinear bidders in combinatorial auctions. CoRR, abs/1606.06846 (2016)Google Scholar
  19. Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic Game Theory, vol. 1. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  20. Quinzii, M.: Core and competitive equilibria with indivisibilities. Int. J. Game Theor. 13(1), 41–60 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Schrijver, A.: Short proofs on the matching polyhedron. J. Comb. Theor. Ser. B 34(1), 104–108 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Shapley, L.S., Shubik, M.: The assignment game I: the core. Int. J. Game Theor. 1(1), 111–130 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Svensson, L.-G.: Competitive equilibria with indivisible goods. J. Econ. 44(4), 373–386 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Vickrey, W.: Counterspeculation, auctions, and competitive sealed tenders. J. Finan. 16(1), 8–37 (1961)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceCornell UniversityIthacaUSA
  2. 2.Yahoo ResearchSanta ClaraUSA

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