Combining local search techniques and path following for bimatrix games

Computing a Nash equilibrium (NE) is a central task in computer science. An NE is a particularly appropriate solution concept for two-agent settings because coalitional deviations are not an issue. However, even in this case, finding an NE is PPAD-complete. In this paper, we combine path following algorithms with local search techniques to design new algorithms for finding exact and approximate NEs. We show that our algorithms largely outperform the state of the art and that almost all the known benchmark game classes are easily solvable or approximable (except for the GAMUT CovariantGameRand class).

💡 Research Summary

The paper addresses the computational challenge of finding Nash equilibria (NE) in two‑player bimatrix games, a problem known to be PPAD‑complete and thus intractable in the worst case. Traditional path‑following algorithms such as Lemke‑Howson (LH) and Lemke (L) solve the associated linear complementarity problem (LCP) by traversing a sequence of complementary pivots until a completely complementary solution (i.e., an NE) is reached. However, these methods suffer from two major drawbacks: the number of distinct LH paths is limited to the sum of the players’ action counts (m₁ + m₂), and the length of a given path can grow exponentially with the game size, leading to prohibitive runtime on hard instances.

To overcome these limitations, the authors propose a suite of algorithms that combine path‑following with local‑search ideas and random‑restart strategies. The main contributions are:

1. rrLH (Random‑restart Lemke‑Howson) – This algorithm randomly selects one of the m₁ + m₂ possible LH initial bases (i.e., a path) and follows it until an NE is found or a predefined cutoff length is exceeded. If the cutoff is exceeded, the algorithm restarts on a different, not‑yet‑visited path. The authors develop a rigorous analysis of the optimal cutoff and restart frequency. Lemma 1 shows that, for a target success probability p, the optimal cutoff is l*·p where l* denotes the length of the shortest LH path, and the optimal number of restarts is one. Consequently, rrLH is proven to be asymptotically optimal among all algorithms that randomize over LH paths.

2. rrL (Random‑restart Lemke) – By employing the Lemke algorithm’s ability to start from arbitrary feasible points, rrL can generate an unbounded number of distinct paths. The same random‑restart framework is applied. Empirically, rrL does not outperform rrLH in average runtime, because while the number of paths is larger, many of them remain long, and the overhead of generating feasible starting bases offsets the benefit.

3. LS‑v (Local Search on Best‑Response Vertices) – This method departs from pure path following. Starting from any mixed‑strategy profile, it iteratively moves to a vertex that maximizes each player’s best‑response payoff within the current support. The process is a local‑search heuristic that converges to an ε‑Nash equilibrium. LS‑v excels on “hard‑to‑solve games” (HtSG) where exact NE computation is infeasible; it quickly produces high‑quality approximations (ε as low as 10⁻⁶) but offers no guarantee of exact equilibrium.

4. ip‑LH (Iterative‑Perturbation Lemke‑Howson) – This algorithm introduces a controlled perturbation to the payoff matrices. Starting with a relatively large perturbation, it runs LH on the perturbed game, then gradually reduces the perturbation magnitude (down to 10⁻¹⁰) while re‑executing LH. The perturbation shrinks the effective solution space, often dramatically shortening the LH path. The authors demonstrate that even the hardest HtSG instances become tractable under tiny perturbations, yielding near‑exact equilibria.

The paper also distinguishes two types of randomization over paths: (1) random selection of a path, and (2) random selection of a vertex within a given path. The authors prove (Lemma 4) that non‑blind randomization (type 2) cannot improve asymptotic performance when path lengths are exponential, reinforcing the optimality of the simple path‑level random restart used in rrLH.

Experimental evaluation uses the GAMUT benchmark suite, covering over 100 game classes and thousands of instances. Results show:

• rrLH outperforms all prior exact algorithms (LH, PNS, MIP‑Nash) on virtually every class, achieving an average number of pivots that scales linearly with the game size, except for the CovariantGameRand class, which remains resistant to perturbation and random restarts.
• rrL matches rrLH’s success rate but typically incurs higher runtime due to the overhead of generating feasible arbitrary starts.
• LS‑v provides the best approximation quality on HtSG and similar hard instances, achieving very low ε values with modest computational effort.
• ip‑LH can approximate HtSG to within 10⁻¹⁰ perturbation, effectively demonstrating that the worst‑case exponential lower bound for LH can be avoided in practice unless PPAD ⊆ RP.

The authors conclude that combining path‑following with random restarts and local search yields a robust framework for both exact and approximate equilibrium computation. While most benchmark classes become easy under this framework, the CovariantGameRand class persists as a challenging outlier, suggesting a direction for future research on algorithms that are less sensitive to perturbations or that exploit problem‑specific structure.

Overall, the paper makes a significant theoretical contribution by proving the asymptotic optimality of rrLH within its algorithmic class, and a substantial practical contribution by delivering implementations that dramatically improve upon the state of the art across a broad spectrum of bimatrix games.