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Exponentially many steps for finding a Nash equilibrium in a bimatrix game

Savani, Rahul and von Stengel, Bernhard (2004) Exponentially many steps for finding a Nash equilibrium in a bimatrix game. CDAM research report series, LSE-CDAM-2004-03. Centre for Discrete and Applicable Mathematics, London School of Economics and Political Science, London, UK.

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Abstract

The Lemke–Howson algorithm is the classical algorithm for the problem NASH of finding one Nash equilibrium of a bimatrix game. It provides a constructive, elementary proof of existence of an equilibrium, by a typical “directed parity argument”, which puts NASH into the complexity class PPAD. This paper presents a class of bimatrix games for which the Lemke–Howson algorithm takes, even in the best case, exponential time in the dimension d of the game, requiring ­((µ3=4)d) many steps, where µ is the Golden Ratio. The “parity argument” for NASH is thus explicitly shown to be inefficient. The games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in d-space.

Item Type: Monograph (Report)
Official URL: http://www.cdam.lse.ac.uk
Additional Information: © 2004 the authors
Library of Congress subject classification: Q Science > QA Mathematics
Sets: Departments > Mathematics
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Identification Number: LSE-CDAM-2004-03
Date Deposited: 03 Dec 2008 12:30
URL: http://eprints.lse.ac.uk/13339/

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