An Empirical Study of Finding Approximate Equilibria in Bimatrix Games

Fearnley, John; Igwe, Tobenna Peter; Savani, Rahul

doi:10.1007/978-3-319-20086-6_26

Cited by 14 publications

(12 citation statements)

References 23 publications

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“…There is a natural extension to our tight instance generator: find a class of games rendering the performances of all existing polynomial-time algorithms unsatisfying. It is worth noting that the classic game generator GAMUT [17] is overwhelmed by the TS algorithm [9]. Therefore, a new benchmark is required, and finding such a benchmark is of great significance to understand the hardness of Nash equilibrium computing.…”

Section: Propose a Benchmark For Approximate Nash Equilibrium Computi...mentioning

confidence: 99%

See 1 more Smart Citation

On Tightness of Tsaknakis-Spirakis Descent Methods for Approximate Nash Equilibria

Chen¹,

Deng²,

Huang³

et al. 2021

Preprint

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Finding the minimum approximate ratio for Nash equilibrium of bi-matrix games has derived a series of studies, started with 3/4, followed by 1/2, 0.38 and 0.36, finally the best approximate ratio of 0.3393 by Tsaknakis and Spirakis (TS algorithm for short). Efforts to improve the results remain not successful in the past 14 years. This work makes the first progress to show that the bound of 0.3393 is indeed tight for the TS algorithm. Next, we characterize all possible tight game instances for the TS algorithm. It allows us to conduct extensive experiments to study the nature of the TS algorithm and to compare it with other algorithms. We find that this lower bound is not smoothed for the TS algorithm in that any perturbation on the initial point may deviate away from this tight bound approximate solution. Other approximate algorithms such as Fictitious Play and Regret Matching also find better approximate solutions. However, the new distributed algorithm for approximate Nash equilibrium by Czumaj et al. performs consistently at the same bound of 0.3393. This proves our lower bound instances generated against the TS algorithm can serve as a benchmark in design and analysis of approximate Nash equilibrium algorithms.

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Section: Propose a Benchmark For Approximate Nash Equilibrium Computi...mentioning

confidence: 99%

“…In literature, the experimental performance of the algorithm is far better than 0.3393 [20]. The worst ratio in an empirical trial by Fearnley et al shows that there is a game on which the TS algorithm gives a 0.3385-approximate Nash equilibrium [9].…”

Section: Introductionmentioning

confidence: 99%

On Tightness of Tsaknakis-Spirakis Descent Methods for Approximate Nash Equilibria

Chen¹,

Deng²,

Huang³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…This line of works originates with the celebrated Lemke-Howson algorithm (Lemke and Howson 1964), and for more recent works, see among others (Bhat and Leyton-Brown 2004;Thompson, Leung, and Leyton-Brown 2011) for the class of action-graph games and (Porter, Nudelman, and Shoham 2008) for the support enumeration method. More recently, there have also been experimental evaluations for methods that compute approximate equilibria, as reported in (Tsaknakis, Spirakis, and Kanoulas 2008;Kontogiannis and Spirakis 2011;Fearnley, Igwe, and Savani 2015), highlighting the need for creating new families of testbeds for such algorithms.…”

Section: Further Related Workmentioning

confidence: 99%

An Improved Quasi-Polynomial Algorithm for Approximate Well-Supported Nash Equilibria

Fasoulakis

Markakis

2019

AAAI

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We focus on the problem of computing approximate Nash equilibria in bimatrix games. In particular, we consider the notion of approximate well-supported equilibria, which is one of the standard approaches for approximating equilibria. It is already known that one can compute an ε-well-supported Nash equilibrium in time nO (log n/ε2), for any ε > 0, in games with n pure strategies per player. Such a running time is referred to as quasi-polynomial. Regarding faster algorithms, it has remained an open problem for many years if we can have better running times for small values of the approximation parameter, and it is only known that we can compute in polynomial-time a 0.6528-well-supported Nash equilibrium. In this paper, we investigate further this question and propose a much better quasi-polynomial time algorithm that computes a (1/2 + ε)-well-supported Nash equilibrium in time nO(log logn1/ε/ε2), for any ε > 0. Our algorithm is based on appropriately combining sampling arguments, support enumeration, and solutions to systems of linear inequalities.

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“…The column player then computes a best response j s against x * s , and uses log n communication rounds to transmit it to the row player. The row player then computes a best response r s against j s , then computes: A recent paper of Fearnley, Igwe, and Savani successfully applied genetic algorithms to find lower bounds against algorithms that compute approximate Nash equilibria [12]. Using the same approach, along with some hand tweaking of the output, we found the following game.…”

Section: Lemma 21 the Strategy Profilementioning

confidence: 99%

“…A line of work has studied the best guarantee that can be achieved in polynomial time [2,6,8]. The best algorithm known so far is the gradient descent method of Tsaknakis and Spirakis [20] that finds a 0.3393-NE in polynomial time, and examples upon which the algorithm finds no better than a 0.3385-NE have been found [12]. On the other hand, progress on computing approximate-well-supported Nash equilibria has been less forthcoming.…”

Section: Introductionmentioning

confidence: 99%

Distributed Methods for Computing Approximate Equilibria

Czumaj

Deligkas

Fasoulakis

et al. 2016

Web and Internet Economics

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We present a new, distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players' payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then computes an approximate Nash equilibrium using only limited communication between the players. Our method gives improved bounds on the complexity of computing approximate Nash equilibria in a number of different settings. Firstly, it gives a polynomial-time algorithm for computing approximate well supported Nash equilibria (WSNE) that always finds a 0.6528-WSNE, beating the previous best guarantee of 0.6608. Secondly, since our algorithm solves the two LPs separately, it can be applied to give an improved bound in the limited communication setting, giving a randomized expectedpolynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE, which beats the previous best known guarantee of 0.732. It can also be applied to the case of approximate Nash equilibria, where we obtain a randomized Artur Czumaj, Michail Fasoulakis and Marcin Jurdziński were partially supported by the Centre for Discrete Mathematics and its Applications (DIMAP) and EPSRC Award EP/D063191/1. Argyrios Deligkas, John Fearnley and Rahul Savani were supported by EPSRC Award EP/L011018/1. Argyrios Deligkas was also supported by ISF Award #2021296.

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An Empirical Study of Finding Approximate Equilibria in Bimatrix Games

Cited by 14 publications

References 23 publications

On Tightness of Tsaknakis-Spirakis Descent Methods for Approximate Nash Equilibria

On Tightness of Tsaknakis-Spirakis Descent Methods for Approximate Nash Equilibria

An Improved Quasi-Polynomial Algorithm for Approximate Well-Supported Nash Equilibria

Distributed Methods for Computing Approximate Equilibria

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