The Complexity of Determining the Uniqueness of Tarski’s Fixed Point under the Lexicographic Ordering

  • Chuangyin Dang
  • Yinyu Ye
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6484)


The well-known Tarski’s fixed point theorem asserts that an increasing mapping from a complete lattice into itself has a fixed point. This theorem plays an important role in the development of supermodular games for economic analysis. Let C be a finite lattice consisting of all integer points in an n-dimensional box and f be an increasing mapping from C into itself in terms of lexicographic ordering. It has been shown in the literature that, when f is given as an oracle, a fixed point of f can be found in polynomial time. The problem we consider in this paper is the complexity of determining whether or not f has a unique fixed point. We present a polynomial-time reduction of integer programming to an increasing mapping from C into itself. As a result of this reduction, we prove that, when f is given as an oracle, determining whether or not f has a unique fixed point is Co-NP hard.


Lexicographic Ordering Lattice Finite Lattice Increasing Mapping Fixed Point Integer Programming Co-NP Completeness Co-NP Hardness 


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  1. 1.
    Bernstein, F., Federgruen, A.: Decentralized supply chains with competing retailers under demand uncertainty. Management Science 51, 18–29 (2005)CrossRefMATHGoogle Scholar
  2. 2.
    Bernstein, F., Federgruen, A.: A general equilibrium model for industries with price and service competition. Operations Research 52, 868–886 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cachon, G.P.: Stcok wars: inventory competition in a two echelon supply chain. Operations Research 49, 658–674 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cachon, G.P., Lariviere, M.A.: Capacity choice and allocation: strategic behavior and supply chain performance. Management Sceince 45, 1091–1108 (1999)MATHGoogle Scholar
  5. 5.
    Chang, C.L., Lyuu, Y.D., Ti, Y.W.: The complexity of Tarski’s fixed point theorem. Theoretical Computer Science 401, 228–235 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Echenique, F.: Finding all equilibria in games of strategic complements. Journal of Economic Theory 135, 514–532 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fudenberg, D., Tirole, J.: Game Theory. MIT Press, Cambridge (1991)MATHGoogle Scholar
  8. 8.
    Gilboa, I., Zemmel, E.: Nash and correlated equilibria: some complexity considerations. Games and Economic Behavior 1, 80–93 (1989)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lippman, S.A., McCardle, K.F.: The competitive newsboy. Operations Research 45, 54–65 (1997)CrossRefMATHGoogle Scholar
  10. 10.
    Milgrom, P., Roberts, J.: Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58, 155–1277 (1990)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Milgrom, P., Roberts, J.: Comparing equilibria. American Economic Review 84, 441–459 (1994)Google Scholar
  12. 12.
    Milgrom, P., Shannon, C.: Monotone comparative statics. Econometrica 62, 157–180 (1994)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics 5, 285–308 (1955)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Topkis, D.M.: Equilibrium points in nonzero-sum n-person submodular games. SIAM Journal on Control and Optimization 17, 773–787 (1979)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Topkis, D.M.: Supermodularity and Complementarity. Princeton University Press, Princeton (1998)Google Scholar
  16. 16.
    Vives, X.: Nash equilibrium with strategic complementarities. Journal of Mathematical Economics 19, 305–321 (1990)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Vives, X.: Oligopoly Pricing. MIT Press, Cambridge (1999)Google Scholar
  18. 18.
    Vives, X.: Complemetarities and games: new developments. Journal of Economic Literature XLIII, 437–479 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Chuangyin Dang
    • 1
  • Yinyu Ye
    • 2
  1. 1.Dept. of Manufacturing Engineering & Engineering ManagementCity University of Hong KongKowloonHong Kong SAR, China
  2. 2.Dept. of Management Science & EngineeringStanford UniversityStanford

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