The complexity of determining the uniqueness of Tarski's fixed point under the lexicographic ordering

  • Authors:

  • Chuangyin Dang;Yinyu Ye

  • Affiliations:

  • Dept. of Manufacturing Engineering & Engineering Management, City University of Hong Kong, Kowloon, Hong Kong SAR, China;Dept. of Management Science & Engineering, Stanford University, Stanford, CA

  • Venue:
  • WINE'10 Proceedings of the 6th international conference on Internet and network economics
  • Year:
  • 2010

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Abstract

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 Cinto 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.