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This study demonstrates that classical algorithms for solving game theory problems—specifically computing Nash equilibria, fixed points, and variational inequalities—perform significantly better than worst-case predictions when analyzed under realistic conditions with small random perturbations. The researchers prove that cover-based search and homotopy path-following methods can solve these PPAD-complete problems exponentially faster in smoothed analysis scenarios compared to worst-case theoretical bounds. For games with constant players and n strategies, their recursive algorithm achieves 2^O(n) expected runtime versus previous n^O(n) or worse complexities.
Why it matters
These findings suggest that computational problems in game theory and economic equilibrium analysis, while theoretically hard in worst-case scenarios, may be tractable in practice when real-world noise and perturbations are considered. This has implications for applications in economics, multi-agent systems, and machine learning where computing equilibria is fundamental.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: We study, under a smoothed analysis perspective, the complexity of three PPAD-complete problems: computing Nash equilibria, computing fixed points, and solving variational inequalities. We consider a general perturbation model that bounds the probability density of the quantities evaluated by our algorithms. As special cases, it captures worst-case games perturbed by small random per-action costs, and worst-case maps perturbed by a small random affine function.
Under these models, we show that classical algorithmic paradigms, cover-based search and homotopy path-following provably outperform the worst-case state-of-the-art. For games with a constant number of players and $n$ strategies per player, a recursive cover-based algorithm computes an $varepsilon$-Nash equilibrium using $log(1/varepsilon)cdot 2^{O(n)}$ expected payoff-vector computations, compared to the worst-case runtime tradeoffs of $log(1/varepsilon)cdot n^{O(n)}$, $(1/varepsilon)^{O(n)}$, $2^{O(log^2 n/varepsilon^2)}$, and $2^{O(n)}mathrm{poly}(log(1/varepsilon))$. For fixed points and variational inequalities of bounded and Lipschitz maps on the $n$-dimensional $ell_2$ unit ball, the same approach runs in expected time $log(1/varepsilon)cdot 2^{O(n)}$, compared to the worst-case $(1/varepsilon)^{O(n)}$; the $2^{Omega(n)}$ factor is necessary by prior work [Rubinstein, 2017]. For general smooth maps, we further analyze the classical predictor–corrector homotopy method, for which only qualitative convergence guarantees were previously known, showing that under perturbations the homotopy path is well conditioned and has length $2^{O(n)}$, so that the method finds an $varepsilon$-approximate solution in time $2^{O(n)}cdotmathrm{poly}(1/p)/varepsilon$ with probability at least $1-p$.