How Hard Is It to Approximate the Best Nash Equilibrium?

SIAM Journal on Computing (Impact Factor: 0.74). 01/2011; 40(1):79-91. DOI: 10.1137/090766991
Source: DBLP


The quest for a PTAS for Nash equilibrium in a two-player game seeks to circumvent the PPAD-completeness of an (exact) Nash equilibrium by finding an approximate equilibrium, and has emerged as a major open question in Algorithmic Game Theory. A closely related problem is that of finding an equilibrium maximizing a certain objective, such as the social welfare. This optimization problem was shown to be NP-hard by Gilboa and Zemel [Games and Economic Behavior 1989]. However, this NP-hardness is unlikely to extend to finding an approximate equilibrium, since the latter admits a quasi-polynomial time algorithm, as proved by Lipton, Markakis and Mehta [Proc. of 4th EC, 2003]. We show that this optimization problem, namely, finding in a two-player game an approximate equilibrium achieving large social welfare is unlikely to have a polynomial time algorithm. One interpretation of our results is that the quest for a PTAS for Nash equilibrium should not extend to a PTAS for finding the best Nash equilibrium, which stands in contrast to certain algorithmic techniques used so far (e.g. sampling and enumeration). Technically, our result is a reduction from a notoriously difficult problem in modern Combinatorics, of finding a planted (but hidden) clique in a random graph G(n, 1/2). Our reduction starts from an instance with planted clique size k = O(log n). For comparison, the currently known algorithms due to Alon, Krivelevich and Sudakov [Random Struct. & Algorithms, 1998], and Krauthgamer and Feige [Random Struct. & Algorithms, 2000], are effective for a much larger clique size k = Ω(√n).

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Available from: Elad Hazan, Mar 09, 2015
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    • "Authors like Avis and Fukuda [2] and Audet et al. [3] [4] presented computational methods to enumerate all Nash extreme points for two-player games. Some other authors like Daskalakis et al. [5] and Hazan and Krauthgamer [6] have recently studied the Nash equilibrium computation complexity problem, also for two-player games. Etessami and Yannakakis [7] studied the complexity of computing approximated Nash equilibria for three or more players finite games. "
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    ABSTRACT: When confronted with multiple Nash equilibria, decision makers have to refine their choices. Among all known Nash equilibrium refinements, the perfectness concept is probably the most famous one. It is known that weakly dominated strategies of two-player games cannot be part of a perfect equilibrium. In general, this undominance property however does not extend to -player games (E. E. C. van Damme, 1983). In this paper we show that polymatrix games, which form a particular class of -player games, verify the undominance property. Consequently, we prove that every perfect equilibrium of a polymatrix game is undominated and that every undominated equilibrium of a polymatrix game is perfect. This result is used to set a new characterization of perfect Nash equilibria for polymatrix games. We also prove that the set of perfect Nash equilibria of a polymatrix game is a finite union of convex polytopes. In addition, we introduce a linear programming formulation to identify perfect equilibria for polymatrix games. These results are illustrated on two small game applications. Computational experiments on randomly generated polymatrix games with different size and density are provided.
    09/2014; 2014. DOI:10.1155/2014/937070
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    • "Researchers have used the hardness of the planted clique problem as an assumption to prove various impossibility results in other problems. Examples include cryptographic applications (Juels and Peinado, 2000; Applebaum, Barak and Wigderson, 2010), testing k-wise independence (Alon et al., 2007) and approximating Nash equilibria (Hazan and Krauthgamer, 2011). Recent works by Berthet and Rigollet (2013a,b) and Ma and Wu (2013) used similar hypotheses on the hardness of the Planted Clique problem to establish computational lower bounds in sparse principal component detection and sparse submatrix detection problems respectively. "
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    ABSTRACT: In recent years, Sparse Principal Component Analysis has emerged as an extremely popular dimension reduction technique for high-dimensional data. The theoretical challenge, in the simplest case, is to estimate the leading eigenvector of a population covariance matrix under the assumption that this eigenvector is sparse. An impressive range of estimators have been proposed; some of these are fast to compute, while others are known to achieve the minimax optimal rate over certain Gaussian or subgaussian classes. In this paper we show that, under a widely-believed assumption from computational complexity theory, there is a fundamental trade-off between statistical and computational performance in this problem. More precisely, working with new, larger classes satisfying a Restricted Covariance Concentration condition, we show that no randomised polynomial time algorithm can achieve the minimax optimal rate. On the other hand, we also study a (polynomial time) variant of the well-known semidefinite relaxation estimator, and show that it attains essentially the optimal rate among all randomised polynomial time algorithms.
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    • "Nash equilibria is known to be computationally hard [10] [12], and in light of these findings, a considerable effort has been directed towards understanding the complexity of approximate Nash equilibrium. Results in this direction include both upper bounds [22] [20] [13] [19] [14] [21] [16] [7] [29] [30] [3] [2] and lower bounds [18] [11] [8]. In particular, it is known that for a general bimatrix game an approximate Nash equilibrium can be computed in quasi-polynomial time [22]. "
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    ABSTRACT: We prove an approximate version of Carath\'{e}odory's theorem and present its algorithmic applications. In particular, given a set of vectors $X$ in $\mathbb{R}^d$, we show that for every vector in the convex hull of $X$ there exists a nearby (under the $p$-norm distance, for $2\leq p < \infty$) vector that can be expressed as a convex combination of at most $b$ vectors of $X$, where the bound $b$ is independent of the dimension $d$. Using this theorem we establish that in a bimatrix game with $ n \times n$ payoff matrices $A, B$, if the number of non-zero entries in any column of $A+B$ is at most $s$ then an approximate Nash equilibrium of the game can be computed in time $n^t$, where $t$ has a logarithmic dependence on $s$. Hence, our algorithm provides a novel understanding of the time required to compute an approximate Nash equilibrium in terms of the column sparsity $s$ of $A+B$. This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity $s$. Moreover, for arbitrary bimatrix games---since $s$ can be at most $n$---the running time of our algorithm matches the best-known upper bound, which is obtained in [Lipton, Markakis, and Mehta 2003]. The theorem also leads to an additive approximation algorithm for the densest $k$-bipartite subgraph problem. Given a graph with $n$ vertices and maximum degree $d$, the developed algorithm determines a $k \times k$ bipartite subgraph with near (in the additive sense) optimal density in time $n^t$, where $t$ has a logarithmic dependence on the degree $d$.
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