Dang C., Ye Y. (2010) The Complexity of Determining the Uniqueness of Tarski’s Fixed Point under the Lexicographic Ordering. In: Saberi A. (eds) Internet and Network Economics. WINE 2010. Lecture Notes in Computer Science, vol 6484. Springer, Berlin, Heidelberg
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.