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

TL;DR

This paper introduces polynomial time algorithms for finding Tarski's fixed points and pure Nash equilibria in supermodular games, analyzing computational complexity and establishing bounds in different models.

Contribution

It presents the first polynomial time algorithms for Tarski's fixed points and explores computational complexity for equilibria in supermodular games.

Findings

First polynomial time algorithm for Tarski's fixed point

Matching oracle bound for uniqueness in oracle model

NP-hardness of determining uniqueness in polynomial model

Abstract

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.