Tarski's Theorem, Supermodular Games, and the Complexity of Equilibria

TL;DR

This paper analyzes the computational complexity of finding fixed points and equilibria guaranteed by Tarski's theorem, revealing new hardness results and connections to supermodular games and stochastic games.

Contribution

It establishes lower bounds for fixed point computation, shows equivalences between supermodular game equilibria and Tarski problems, and explores complexity classes involved.

Findings

Finding some fixed point requires at least (^2 N) evaluations in 2D.

Tarski fixed point problem is in PLS and PPAD classes.

Computing extremal fixed points is NP-hard.

Abstract

The use of monotonicity and Tarski's theorem in existence proofs of equilibria is very widespread in economics, while Tarski's theorem is also often used for similar purposes in the context of verification. However, there has been relatively little in the way of analysis of the complexity of finding the fixed points and equilibria guaranteed by this result. We study a computational formalism based on monotone functions on the $d$-dimensional grid with sides of length $N$, and their fixed points, as well as the closely connected subject of supermodular games and their equilibria. It is known that finding some (any) fixed point of a monotone function can be done in time $\log^d N$, and we show it requires at least $\log^2 N$ function evaluations already on the 2-dimensional grid, even for randomized algorithms. We show that the general Tarski problem of finding some fixed point, when the monotone function is given succinctly (by a boolean circuit), is in the class PLS of problems solvable by local search and, rather surprisingly, also in the class PPAD. Finding the greatest or least fixed point guaranteed by Tarski's theorem, however, requires $d\cdot N$ steps, and is NP-hard in the white box model. For supermodular games, we show that finding an equilibrium in such games is essentially computationally equivalent to the Tarski problem, and finding the maximum or minimum equilibrium is similarly harder. Interestingly, two-player supermodular games where the strategy space of one player is one-dimensional can be solved in $O(\log N)$ steps. We also observe that computing (approximating) the value of Condon's (Shapley's) stochastic games reduces to the Tarski problem. An important open problem highlighted by this work is proving a $\Omega(\log^d N)$ lower bound for small fixed dimension $d \geq 3$.