Approximate well-supported Nash equilibria in symmetric bimatrix games

The $\varepsilon$-well-supported Nash equilibrium is a strong notion of approximation of a Nash equilibrium, where no player has an incentive greater than $\varepsilon$ to deviate from any of the pure strategies that she uses in her mixed strategy. The smallest constant $\varepsilon$ currently known for which there is a polynomial-time algorithm that computes an $\varepsilon$-well-supported Nash equilibrium in bimatrix games is slightly below $2/3$. In this paper we study this problem for symmetric bimatrix games and we provide a polynomial-time algorithm that gives a $(1/2+\delta)$-well-supported Nash equilibrium, for an arbitrarily small positive constant $\delta$

An Empirical Study of Finding Approximate Equilibria in Bimatrix Games

While there have been a number of studies about the efficacy of methods to find exact Nash equilibria in bimatrix games, there has been little empirical work on finding approximate Nash equilibria. Here we provide such a study that compares a number of approximation methods and exact methods. In particular, we explore the trade-off between the quality of approximate equilibrium and the required running time to find one. We found that the existing library GAMUT, which has been the de facto standard that has been used to test exact methods, is insufficient as a test bed for approximation methods since many of its games have pure equilibria or other easy-to-find good approximate equilibria. We extend the breadth and depth of our study by including new interesting families of bimatrix games, and studying bimatrix games upto size $2000 \times 2000$. Finally, we provide new close-to-worst-case examples for the best-performing algorithms for finding approximate Nash equilibria

Polylogarithmic Supports are required for Approximate Well-Supported Nash Equilibria below 2/3

In an epsilon-approximate Nash equilibrium, a player can gain at most epsilon in expectation by unilateral deviation. An epsilon well-supported approximate Nash equilibrium has the stronger requirement that every pure strategy used with positive probability must have payoff within epsilon of the best response payoff. Daskalakis, Mehta and Papadimitriou conjectured that every win-lose bimatrix game has a 2/3-well-supported Nash equilibrium that uses supports of cardinality at most three. Indeed, they showed that such an equilibrium will exist subject to the correctness of a graph-theoretic conjecture. Regardless of the correctness of this conjecture, we show that the barrier of a 2/3 payoff guarantee cannot be broken with constant size supports; we construct win-lose games that require supports of cardinality at least Omega((log n)^(1/3)) in any epsilon-well supported equilibrium with epsilon < 2/3. The key tool in showing the validity of the construction is a proof of a bipartite digraph variant of the well-known Caccetta-Haggkvist conjecture. A probabilistic argument shows that there exist epsilon-well-supported equilibria with supports of cardinality O(log n/(epsilon^2)), for any epsilon> 0; thus, the polylogarithmic cardinality bound presented cannot be greatly improved. We also show that for any delta > 0, there exist win-lose games for which no pair of strategies with support sizes at most two is a (1-delta)-well-supported Nash equilibrium. In contrast, every bimatrix game with payoffs in [0,1] has a 1/2-approximate Nash equilibrium where the supports of the players have cardinality at most two.Comment: Added details on related work (footnote 7 expanded

Approximate Well-supported Nash Equilibria below Two-thirds

In an epsilon-Nash equilibrium, a player can gain at most epsilon by changing his behaviour. Recent work has addressed the question of how best to compute epsilon-Nash equilibria, and for what values of epsilon a polynomial-time algorithm exists. An epsilon-well-supported Nash equilibrium (epsilon-WSNE) has the additional requirement that any strategy that is used with non-zero probability by a player must have payoff at most epsilon less than the best response. A recent algorithm of Kontogiannis and Spirakis shows how to compute a 2/3-WSNE in polynomial time, for bimatrix games. Here we introduce a new technique that leads to an improvement to the worst-case approximation guarantee

On the Approximation Performance of Fictitious Play in Finite Games

We study the performance of Fictitious Play, when used as a heuristic for finding an approximate Nash equilibrium of a 2-player game. We exhibit a class of 2-player games having payoffs in the range [0,1] that show that Fictitious Play fails to find a solution having an additive approximation guarantee significantly better than 1/2. Our construction shows that for n times n games, in the worst case both players may perpetually have mixed strategies whose payoffs fall short of the best response by an additive quantity 1/2 - O(1/n^(1-delta)) for arbitrarily small delta. We also show an essentially matching upper bound of 1/2 - O(1/n)

A Direct Reduction from k-Player to 2-Player Approximate Nash Equilibrium

We present a direct reduction from k-player games to 2-player games that preserves approximate Nash equilibrium. Previously, the computational equivalence of computing approximate Nash equilibrium in k-player and 2-player games was established via an indirect reduction. This included a sequence of works defining the complexity class PPAD, identifying complete problems for this class, showing that computing approximate Nash equilibrium for k-player games is in PPAD, and reducing a PPAD-complete problem to computing approximate Nash equilibrium for 2-player games. Our direct reduction makes no use of the concept of PPAD, thus eliminating some of the difficulties involved in following the known indirect reduction.Comment: 21 page

Approximating Nash Equilibria and Dense Bipartite Subgraphs via an Approximate Version of Carathéodory's Theorem

We present algorithmic applications of an approximate version of Caratheodory's theorem. The theorem states that given a set of vectors X in R^d, for every vector in the convex hull of X there exists an ε-close (under the p-norm distance, for 2 ≤ p < ∞) vector that can be expressed as a convex combination of at most b vectors of X, where the bound b depends on ε and the norm p and is independent of the dimension d. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pisier (1981) and Carl (1985). However, in this paper we present a self-contained proof of this result. Using this theorem we establish that in a bimatrix game with n x n payoff matrices A, B, if the number of non-zero entries in any column of A+B is at most s then an ε-Nash equilibrium of the game can be computed in time n^O(log s/ε^2}). 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 was obtained by Lipton, Markakis, and Mehta (2003). The approximate Carathéodory's 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 x k bipartite subgraph with density within ε (in the additive sense) of the optimal density in time n^O(log d/ε^2)

Spectral Inversion of Multi-Line Full-Disk Observations of Quiet Sun Magnetic Fields

Spectral inversion codes are powerful tools to analyze spectropolarimetric observations, and they provide important diagnostics of solar magnetic fields. Inversion codes differ by numerical procedures, approximations of the atmospheric model, and description of radiative transfer. Stokes Inversion based on Response functions (SIR) is an implementation widely used by the solar physics community. It allows to work with different atmospheric components, where gradients of different physical parameters are possible, e.g., magnetic field strength and velocities. The spectropolarimetric full-disk observations were carried out with the Stokesmeter of the Solar Telescope for Operative Predictions (STOP) at the Sayan Observatory on 3 February 2009, when neither an active region nor any other extended flux concentration was present on the Sun. In this study of quiet Sun magnetic fields, we apply the SIR code simultaneously to 15 spectral lines. A tendency is found that weaker magnetic field strengths occur closer to the limb. We explain this finding by the fact that close to the limb, we are more sensitive to higher altitudes in an expanding flux tube, where the field strength should be smaller since the magnetic flux is conserved with height. Typically, the inversions deliver two populations of magnetic elements: (1) high magnetic field strengths (1500-2000 G) and high temperatures (5500-6500 K) and (2) weak magnetic fields (50-150 G) and low temperatures (5000-5300 K).Comment: 10 pages, 6 figures, accepted for Solar Physic