Linear Complementarity Algorithms for Infinite Games

TL;DR

This paper investigates the performance of two pivoting algorithms, Lemke and Cottle-Dantzig, on linear complementarity problems derived from infinite games, revealing their exponential worst-case complexity for certain game families.

Contribution

It introduces the application of these pivoting algorithms to infinite games and analyzes their performance, showing they can be exponential in some cases, unlike classical algorithms.

Findings

Both algorithms run in exponential time on certain parity games.

They perform similarly on infinite games as on general P-matrix LCPs.

The algorithms offer alternative approaches to classical strategy-improvement methods.

Abstract

The performance of two pivoting algorithms, due to Lemke and Cottle and Dantzig, is studied on linear complementarity problems (LCPs) that arise from infinite games, such as parity, average-reward, and discounted games. The algorithms have not been previously studied in the context of infinite games, and they offer alternatives to the classical strategy-improvement algorithms. The two algorithms are described purely in terms of discounted games, thus bypassing the reduction from the games to LCPs, and hence facilitating a better understanding of the algorithms when applied to games. A family of parity games is given, on which both algorithms run in exponential time, indicating that in the worst case they perform no better for parity, average-reward, or discounted games than they do for general P-matrix LCPs.