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Exponentially Many Steps for Finding a Nash Equilibrium in a Bimatrix Game
 IN PROCEEDINGS OF THE ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
, 2004
"... The LemkeHowson 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 PPA ..."
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Cited by 45 (1 self)
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The LemkeHowson 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 LemkeHowson algorithm takes, even in the best case, exponential time in the dimension d of the game, requiring #((# 3=4 ) d ) many steps, where # 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 dspace.
A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension
 JOURNAL OF THE ACM
, 1985
"... It has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplextype algorithms for linear programming. In this paper the average number of steps performed by a simplex algorithm, the socalled selfdual method, is analyzed. The algorithm is not started ..."
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Cited by 31 (2 self)
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It has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplextype algorithms for linear programming. In this paper the average number of steps performed by a simplex algorithm, the socalled selfdual method, is analyzed. The algorithm is not started at the traditional point (1,..., but points of the form (1, e, e2,...)T, with t sufficiently small, are used. The result is better, in two respects, than those of the previous analyses. First, it is shown that the expected number of steps is bounded between two quadratic functions cl(min(m, n))' and cz(min(m, n)) ' of the smaller dimension of the problem. This should be compared with the previous two major results in the field. Borgwardt proves an upper bound of 0(n4m1'(n1') under a model that implies that the zero vector satisfies all the constraints, and also the algorithm under his consideration solves only problems from that particular subclass. Smale analyzes the selfdual algorithm starting at (1,..., He shows that for any fixed m there is a constant c(m) such the expected number of steps is less than ~(m)(lnn)&quot;'(&quot;+~); Megiddo has shown that, under Smale's model, an upper bound C(m) exists. Thus, for the first time, a polynomial upper bound with no restrictions (except for nondegeneracy) on the problem is proved, and, for the first time, a nontrivial lower bound of precisely the same order of magnitude is established. Both Borgwardt and Smale require the input vectors to be drawn from
Hardtosolve bimatrix games
 ECONOMETRICA
, 2006
"... The Lemke–Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. This paper presents a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, th ..."
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Cited by 22 (1 self)
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The Lemke–Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. This paper presents a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, the games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in dspace. The construction is extended to nonsquare games where, in addition to exponentially long Lemke–Howson computations, finding an equilibrium by support enumeration takes on average exponential time.
Challenge instances for NASH
, 2004
"... Abstract. The problem NASH is that of finding one Nash equilibrium of a bimatrix game. The computational complexity of this problem is a longstanding open question. The Lemke–Howson algorithm is the classical algorithm for NASH. In an earlier paper, this algorithm was shown to be exponential, even ..."
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Cited by 3 (2 self)
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Abstract. The problem NASH is that of finding one Nash equilibrium of a bimatrix game. The computational complexity of this problem is a longstanding open question. The Lemke–Howson algorithm is the classical algorithm for NASH. In an earlier paper, this algorithm was shown to be exponential, even in the best case, for square bimatrix games using dual cyclic polytopes. However the games constructed there are easily solved by another standard method, support enumeration. In this paper we present “challenge instances ” for NASH. We extend the use of dual cyclic polytopes and construct a class of games which are shown to be hard to solve for both the Lemke–Howson algorithm and support enumeration. Other general methods for NASH are discussed. It is not obvious that they could solve these games efficiently. 1
Local Search Techniques for Nash Equilibrium Computation with Bimatrix
"... Ecco il momento dei ringraziamenti. Mi spoglio del gergo da Ingegnere per dedicarmi a questa pagina, come si deve. Un grazie en passant va a tutti coloro i quali ho incrociato tra i banchi in questi anni di studio, con cui si è passato un esame dopo l’altro, finchè si poteva, con cui si son trascor ..."
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Ecco il momento dei ringraziamenti. Mi spoglio del gergo da Ingegnere per dedicarmi a questa pagina, come si deve. Un grazie en passant va a tutti coloro i quali ho incrociato tra i banchi in questi anni di studio, con cui si è passato un esame dopo l’altro, finchè si poteva, con cui si son trascorsi i pochi momenti liberi o quella manciata di ore di studio (tra cui le tante ore nell’auletta della Nave, primo piano), a quelli fuggiti e rivisti di sfuggita, magari con la nuova guida al guinzaglio, o conosciuti a Madrid in 2 weekend, o scoperti cremaschi e mai visti prima. Perchè sapere che non sei l’unico a scalar la vetta, ti dà qualche forza in più. Sarà un piacere un giorno rivedersi, in altre vite. Ci conto. Ringrazio Riccardo che si è goduto una lettura in draft della tesi e con cui, negli ultimi mesi, ci siam caricati a vicenda per dare il massimo. Ma soprattutto un grazie a Marco, senza la cui passione, l’occhio critico e l’intensa amicizia, sarei arrivato a questo traguardo solo con uno sforzo ben maggiore. Voglio assolutamente che siano qui anche gli altri amici, quelli veri ma che col Poli han poco a che fare. Così quando riprenderò in mano questa pagina, ricorderò come se fosse oggi. Loro sono quelli che se capiterà di perdersi di vista, il giorno in cui ci rivedremo, non sarà cambiato nulla nei nostri sguardi. Forse alcuni non sanno quanto siano stati fondamentali per me. Ma anche lo sapessero, sono felice di abbracciarli ancora una volta con questa dedica. Loro sanno, nomi non ne servono. Un enorme Grazie lo devo e lo voglio rivolgere a Nicola. Sempre pronto ad invasioni di ufficio per i dubbi del giorno, a chiamate su skype da fusi orari di Taiwan o San Francisco e darmi un consiglio fidato quando capita che t’interroghi sul futuro. Ti ringrazio e mi spiace chiudere qui il lavoro insieme, certo che mi sarei goduto altri 3 anni di libera ricerca con te. Ma questo è un altro discorso. Un