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

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.

💡 Research Summary

The paper establishes that several widely used homotopy‑based algorithms for finding fixed points and Nash equilibria are computationally intractable in the strongest sense: implementing them is PSPACE‑complete. The authors begin by formalizing the classic homotopy method, which continuously deforms an easy‑to‑solve problem P0 into a target problem P1 by varying a parameter λ∈