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

Computations and Complexities of Tarski's Fixed Points and Supermodular Games