An Exponential Lower Bound for the Latest Deterministic Strategy Iteration Algorithms

TL;DR

This paper establishes an exponential lower bound for two main deterministic strategy iteration algorithms used in solving various types of games, demonstrating their limitations in worst-case scenarios.

Contribution

It introduces new exponential lower bounds for both local and global improvement variants of strategy iteration algorithms in game solving.

Findings

Both variants require exponentially many iterations on certain game families.

The results highlight fundamental limitations of these deterministic algorithms.

The bounds apply to parity, mean payoff, discounted payoff, and stochastic games.

Abstract

This paper presents a new exponential lower bound for the two most popular deterministic variants of the strategy improvement algorithms for solving parity, mean payoff, discounted payoff and simple stochastic games. The first variant improves every node in each step maximizing the current valuation locally, whereas the second variant computes the globally optimal improvement in each step. We outline families of games on which both variants require exponentially many strategy iterations.