  We consider two models of computation for Tarski's order preserving function
f related to fixed points in a complete lattice: the oracle function model and
the polynomial function model. In both models, we find the first polynomial
time algorithm for finding a Tarski's fixed point. In addition, we provide a
matching oracle bound for determining the uniqueness in the oracle function
model and prove it is Co-NP hard in the polynomial function model. The
existence of the pure Nash equilibrium in supermodular games is proved by
Tarski's fixed point theorem. Exploring the difference between supermodular
games and Tarski's fixed point, we also develop the computational results for
finding one pure Nash equilibrium and determining the uniqueness of the
equilibrium in supermodular games.
