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 $\delta$

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

Revealing the nature of magnetic shadows with numerical 3D-MHD simulations

We investigate the interaction of magneto-acoustic waves with magnetic network elements with the aim of finding possible signatures of the magnetic shadow phenomenon in the vicinity of network elements. We carried out three-dimensional numerical simulations of magneto-acoustic wave propagation in a model solar atmosphere that is threaded by a complexly structured magnetic field, resembling that of a typical magnetic network element and of internetwork regions. High-frequency waves of 10 mHz are excited at the bottom of the simulation domain. On their way through the upper convection zone and through the photosphere and the chromosphere they become perturbed, refracted, and converted into different mode types. We applied a standard Fourier analysis to produce oscillatory power-maps of the line-of-sight velocity. In the power maps of the upper photosphere and the lower chromosphere, we clearly see the magnetic shadow: a seam of suppressed power surrounding the magnetic network elements. We demonstrate that this shadow is linked to the mode conversion process and that power maps at these height levels show the signature of three different magneto-acoustic wave modes.Comment: Astronomy & Astrophysics Letters, in print 4 pages, 4 figure

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

Emergence of small-scale magnetic flux in the quiet Sun

We study the evolution of a small-scale emerging flux region (EFR) in the quiet Sun, from its emergence to its decay. We track processes and phenomena across all atmospheric layers, explore their interrelations and compare our findings with recent numerical modelling studies. We used imaging, spectral and spectropolarimetric observations from space-borne and ground-based instruments. The EFR appears next to the chromospheric network and shows all characteristics predicted by numerical simulations. The total magnetic flux of the EFR exhibits distinct evolutionary phases, namely an initial subtle increase, a fast increase and expansion of the region area, a more gradual increase, and a slow decay. During the initial stages, bright points coalesce, forming clusters of positive- and negative-polarity in a largely bipolar configuration. During the fast expansion, flux tubes make their way to the chromosphere, producing pressure-driven absorption fronts, visible as blueshifted chromospheric features. The connectivity of the quiet-Sun network gradually changes and part of the existing network forms new connections with the EFR. A few minutes after the bipole has reached its maximum magnetic flux, it brightens in soft X-rays forming a coronal bright point, exhibiting episodic brightenings on top of a long smooth increase. These coronal brightenings are also associated with surge-like chromospheric features, which can be attributed to reconnection with adjacent small-scale magnetic fields and the ambient magnetic field. The emergence of magnetic flux even at the smallest scales can be the driver of a series of energetic phenomena visible at various atmospheric heights and temperature regimes. Multi-wavelength observations reveal a wealth of mechanisms which produce diverse observable effects during the different evolutionary stages of these small-scale structures.Comment: Accepted for publication in Astronomy & Astrophysics 14 pages, 14 figure

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

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)

Observational study of chromospheric heating by acoustic waves

Aims. To investigate the role of acoustic and magneto-acoustic waves in heating the solar chromosphere, observations in strong chromospheric lines are analyzed by comparing the deposited acoustic-energy flux with the total integrated radiative losses. Methods. Quiet-Sun and weak-plage regions were observed in the Ca II 854.2 nm and H-alpha lines with the Fast Imaging Solar Spectrograph (FISS) at the 1.6-m Goode Solar Telescope (GST) on 2019 October 3 and in the H-alpha and H-beta lines with the echelle spectrograph attached to the Vacuum Tower Telescope (VTT) on 2018 December 11 and 2019 June 6. The deposited acoustic energy flux at frequencies up to 20 mHz was derived from Doppler velocities observed in line centers and wings. Radiative losses were computed by means of a set of scaled non-LTE 1D hydrostatic semi-empirical models obtained by fitting synthetic to observed line profiles. Results. In the middle chromosphere (h = 1000-1400 km), the radiative losses can be fully balanced by the deposited acoustic energy flux in a quiet-Sun region. In the upper chromosphere (h > 1400 km), the deposited acoustic flux is small compared to the radiative losses in quiet as well as in plage regions. The crucial parameter determining the amount of deposited acoustic flux is the gas density at a given height. Conclusions. The acoustic energy flux is efficiently deposited in the middle chromosphere, where the density of gas is sufficiently high. About 90% of the available acoustic energy flux in the quiet-Sun region is deposited in these layers, and thus it is a major contributor to the radiative losses of the middle chromosphere. In the upper chromosphere, the deposited acoustic flux is too low, so that other heating mechanisms have to act to balance the radiative cooling.Comment: 11 pages, 10 figures, 3 table