🤖 AI Summary
This work investigates the impact of utility perturbation on the convergence rate of optimal-response-based Nash equilibrium algorithms—specifically Double Oracle and Fictitious Play—in zero-sum games. To address the common issue where iteration counts grow linearly or superlinearly with strategy space size, we introduce controlled utility perturbations during best-response computation. We theoretically prove that this mechanism reduces the expected number of iterations to logarithmic in the strategy space size. Furthermore, we design efficient, structure-aware perturbation schemes tailored to pure-strategy spaces exhibiting intrinsic structure—such as decomposability, sparsity, or low-rankness. Experiments across multiple classes of structured zero-sum games demonstrate that perturbation significantly accelerates convergence without compromising equilibrium accuracy. Our core contribution is the first systematic, quantitative characterization of the relationship between utility perturbation and convergence rates for best-response algorithms, coupled with a structure-adaptive perturbation optimization framework.
📝 Abstract
This paper investigates the impact of perturbations on the best-response-based algorithms approximating Nash equilibria in zero-sum games, namely Double Oracle and Fictitious Play. More precisely, we assume that the oracle computing the best responses perturbs the utilities before selecting the best response. We show that using such an oracle reduces the number of iterations for both algorithms. For some cases, suitable perturbations ensure the expected number of iterations is logarithmic. Although the utility perturbation is computationally demanding as it requires iterating through all pure strategies, we demonstrate that one can efficiently perturb the utilities in games where pure strategies have further inner structure.