978-3-540-68138-0 Internet and Network Economics 978-3-540-68141-0 https://doi.org/10.1007/11944874_27 2006 algorithm limits of approximation linear-time algorithm strategies logn strategies conjecture chapter limit intractability context cases players view approximation chapters game approximate equilibrium 297-306 approximation scheme 2022-01-01T19:17 demanding notion approximate Nash equilibria equilibrium graph-theoretic conjecture note en A Note on Approximate Nash Equilibria contingent 2006-01-01 win-lose games https://scigraph.springernature.com/explorer/license/ Nash equilibrium false possible contingent subexponential approximation scheme notion In view of the intractability of finding a Nash equilibrium, it is important to understand the limits of approximation in this context. A subexponential approximation scheme is known [LMM03], and no approximation better than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\over 4$\end{document} is possible by any algorithm that examines equilibria involving fewer than logn strategies [Alt94]. We give a simple, linear-time algorithm examining just two strategies per player and resulting in a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\over 2$\end{document}-approximate Nash equilibrium in any 2-player game. For the more demanding notion of well-supported approximate equilibrium due to [DGP06] no nontrivial bound is known; we show that the problem can be reduced to the case of win-lose games (games with all utilities 0–1), and that an approximation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$5\over 6$\end{document} is possible contingent upon a graph-theoretic conjecture. problem scheme Papadimitriou Christos Springer Nature - SN SciGraph project University of California, Berkeley, USA University of California, Berkeley, USA Kontogiannis Spyros Mavronicolas Marios Computation Theory and Mathematics pub.1016417682 dimensions_id Information and Computing Sciences doi 10.1007/11944874_27 Aranyak Mehta Springer Nature IBM Almaden Research Center, San Jose, USA IBM Almaden Research Center, San Jose, USA Daskalakis Constantinos Spirakis Paul