Stability and competitive equilibria in multi-unit trading networks with discrete concave utility functions

  • Yoshiko T. Ikebe
  • Yosuke SekiguchiEmail author
  • Akiyoshi Shioura
  • Akihisa Tamura
Original Paper Area 1


Hatfield, Kominers, Nichifor, Ostrovsky, and Westkamp showed the existence of stable outcomes and competitive equilibria in a model of trading networks under the assumption that all agents’ preferences satisfy a condition called the full substitutes condition. In this paper, we extend their model by using discrete concave utility functions called twisted \(\hbox {M}^{\natural }\)-concave functions. We show that a valuation function of an agent is twisted \(\hbox {M}^{\natural }\)-concave if and only if the agent’s preference satisfies the generalized variant of the full substitutes condition. We also show that under the generalized full substitutes condition, there exist stable outcomes and competitive equilibria in the extended model and the set of competitive equilibrium price vectors forms a lattice. In addition, we discuss the connection among competitive equilibria, stability, and efficiency. Finally, we investigate the relationship among stability, strong group stability, and chain stability and verify these three stability concepts are equivalent as long as valuation functions of all agents are twisted \(\hbox {M}^{\natural }\)-concave.


Stability Competitive equilibria Efficiency Lattice Twisted \(\hbox {M}^{\natural }\)-concave functions Generalized full substitutes condition 

Mathematics Subject Classification

91B50 91B68 


  1. 1.
    Danilov, V., Koshevoy, G., Murota, K.: Discrete convexity and equilibria in economies with indivisible goods and money. Math. Soc. Sci. 41, 251–273 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Eguchi, A., Fujishige, S., Tamura, A.: A generalized Gale–Shapley algorithm for a discrete-concave stable-marriage model. In: Proceedings of the 14th International Symposium on Algorithms and Computation (ISAAC 2003). Lecture Notes in Computer Science, vol. 2906, pp. 495–504 (2003)Google Scholar
  3. 3.
    Fujishige, S., Tamura, A.: A two-sided discrete-concave market with possibly bounded side payments: an approach by discrete convex analysis. Math. Oper. Res. 32, 136–155 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fujishige, S., Yang, Z.: A note on Kelso and Crawford’s gross substitutes condition. Math. Oper. Res. 28, 463–469 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69, 9–15 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gul, F., Stacchetti, E.: Walrasian equilibrium with gross substitutes. J. Econ. Theory 87, 95–124 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gul, F., Stacchetti, E.: The English auction with differentiated commodities. J. Econ. Theory 92, 66–95 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hatfield, J.W., Kominers, S.D.: Contract Design and Stability in Matching Markets. Harvard Business School, Mimeo (2010)Google Scholar
  9. 9.
    Hatfield, J.W., Kominers, S.D.: Matching in networks with bilateral contracts. Am. Econ. J. Microecon. 4, 176–208 (2012)Google Scholar
  10. 10.
    Hatfield, J.W., Kominers, S.D., Nichifor, A., Ostrovsky, M., Westkamp, A.: Stability and competitive equilibrium in trading networks. J. Polit. Econ. 121, 966–1005 (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hatfield, J.W., Milgrom, P.R.: Matching with contracts. Am. Econ. Rev. 95, 913–935 (2005)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ikebe, Y.T., Tamura, A.: Stability in supply chain networks: an approach by discrete convex analysis. In: Proceedings of the 8th Japanese–Hungarian Symposium on Discrete Mathematics and Its Applications, pp. 243–250 (2013)Google Scholar
  13. 13.
    Kelso Jr, A.S., Crawford, V.P.: Job matching, coalition formation, and gross substitutes. Econometrica 50, 1483–1504 (1982)CrossRefzbMATHGoogle Scholar
  14. 14.
    Lehmann, B., Lehmann, D., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. Games Econ. Behav. 55, 270–296 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Murota, K.: Discrete convex analysis. Math. Program. 83, 313–371 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Murota, K.: Discrete Convex Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2003)Google Scholar
  17. 17.
    Murota, K., Shioura, A.: M-convex function on generalized polymatroid. Math. Oper. Res. 24, 95–105 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Murota, K., Shioura, A.: Extension of M-convexity and L-convexity to polyhedral convex functions. Adv. Appl. Math. 25, 352–427 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ostrovsky, M.: Stability in supply chain network. Am. Econ. Rev. 98, 897–923 (2008)CrossRefGoogle Scholar
  20. 20.
    Roth, A.E., Sotomayor, M.A.O.: Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  21. 21.
    Shapley, L.S., Shubik, M.: The assignment game I: the core. Int. J. Game Theory 1, 111–130 (1972)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Shioura, A., Yang, Z.: Equilibrium, auction, multiple substitutes and complements. In: Discussion papers from Department of Economics, University of York (2013)Google Scholar
  23. 23.
    Sotomayor, M.: The lattice structure of the set of stable outcomes of the multiple partners assignment game. Int. J. Game Theory 28, 567–583 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sun, N., Yang, Z.: Equilibria and indivisibilities: gross substitutes and complements. Econometrica 74, 1385–1402 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sun, N., Yang, Z.: A double-track adjustment process for discrete markets with substitutes and complements. Econometrica 77, 933–952 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2015

Authors and Affiliations

  1. 1.Department of Management ScienceTokyo University of ScienceTokyoJapan
  2. 2.Department of MathematicsKeio UniversityYokohamaJapan
  3. 3.Graduate School of Information SciencesTohoku UniversitySendaiJapan

Personalised recommendations