The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions

TL;DR

This paper proves that several widely used homotopy and equilibrium selection algorithms in game theory are PSPACE-complete to implement, indicating their computational intractability in the worst case.

Contribution

It establishes the PSPACE-completeness of implementing homotopy methods, equilibrium selection, and Lemke-Howson solutions, extending previous complexity results.

Findings

Homotopy method implementation is PSPACE-complete.

Equilibrium selection via Harsanyi-Selten is PSPACE-complete.

Computing Lemke-Howson equilibria is PSPACE-complete.

Abstract

We show that the widely used homotopy method for solving fixpoint problems, as well as the Harsanyi-Selten equilibrium selection process for games, are PSPACE-complete to implement. Extending our result for the Harsanyi-Selten process, we show that several other homotopy-based algorithms for finding equilibria of games are also PSPACE-complete to implement. A further application of our techniques yields the result that it is PSPACE-complete to compute any of the equilibria that could be found via the classical Lemke-Howson algorithm, a complexity-theoretic strengthening of the result in [Savani and von Stengel]. These results show that our techniques can be widely applied and suggest that the PSPACE-completeness of implementing homotopy methods is a general principle.