Computing fixed points of an increasing mapping from a finite lattice formed by integer points of a box into itself

For a finite lattice (D_I, ≤) consisting of integer points in an n-dimensional box D of Rⁿ and an increasing mapping ψ from D_I into itself, the well-known Tarski´s fixed point theorem asserts that there exists a point x* ∈ D_I such that ψ(x*) = x*, which is a fixed point of ψ in D_I and has applications in supermodular games from economic analysis. To compute such a fixed point, an arbitrary starting simplicial algorithm is developed in this paper through an introduction of an integer labeling rule and an application of a triangulation of Rⁿ. The algorithm consists of two phases, one of which forms a variable dimension pivoting procedure and the other a full-dimensional pivoting procedure. Starting from an arbitrary integer point in D, the algorithm interchanges from one phase to the other, if necessary, and follows a finite simplicial path that leads to a fixed point of ψ in D_I. The interchanging step is essential to ensure that the algorithm converges to a fixed point. A significant advantage of the algorithm is its arbitrary starting feature, which can certainly benefit from any information on where a fixed point may be found.