Constant Rank Bimatrix Games are PPAD-hard

The rank of a bimatrix game (A,B) is defined as rank(A+B). Computing a Nash equilibrium (NE) of a rank-$0$, i.e., zero-sum game is equivalent to linear programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an FPTAS for constant rank games, and asked if there exists a polynomial time algorithm to compute an exact NE. Adsul et al. (2011) answered this question affirmatively for rank-$1$ games, leaving rank-2 and beyond unresolved. In this paper we show that NE computation in games with rank $\ge 3$, is PPAD-hard, settling a decade long open problem. Interestingly, this is the first instance that a problem with an FPTAS turns out to be PPAD-hard. Our reduction bypasses graphical games and game gadgets, and provides a simpler proof of PPAD-hardness for NE computation in bimatrix games. In addition, we get: * An equivalence between 2D-Linear-FIXP and PPAD, improving a result by Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD. * NE computation in a bimatrix game with convex set of Nash equilibria is as hard as solving a simple stochastic game. * Computing a symmetric NE of a symmetric bimatrix game with rank $\ge 6$ is PPAD-hard. * Computing a (1/poly(n))-approximate fixed-point of a (Linear-FIXP) piecewise-linear function is PPAD-hard. The status of rank-$2$ games remains unresolved

A field deployable method for a rapid screening analysis of inorganic arsenic in seaweed

The authors thank the support for getting the seaweed samples from the projects funded under the Department of Agriculture, Food and the Marine’s Competitive research programmes in Ireland. Reference number 14 SF 860. The authors thank Corny Brombach for the graphical abstract.Peer reviewedPublisher PD

Nitric oxide modulates expression of extracellular matrix genes linked to fibrosis in kidney mesangial cells

Mesangial cells are thought to be important mediators of glomerular inflammation and fibrosis. Studies have established a direct role for nitric oxide (NO) in the regulation of gene expression in mesangial cells. Representational difference analysis was used to investigate changes in gene expression elicited by the treatment of S-nitroso-L-glutathione in rat mesangial cells. Seven upregulated and 11 downregulated genes were identified. Four out of 11 downregulated genes (connective tissue growth factor, thrombospondin-1, collagen type I all and collagen type I alpha 2) are known to be linked to inflammation and fibrosis. Results were verified across species in mesangial cells treated with a series of NO donors using Northern blot analysis, quantitative real-time PCR and protein analysis methods. Induction of endogenous NO production by cytokine stimulation also triggered regulation of the genes. One example gene, connective tissue growth factor, was studied at the promoter level. Promoter-reporter gene studies in mesangial cells demonstrated that NO acts at the transcriptional level to suppress gene expression. Our results reveal a complex role of NO in regulating gene expression in mesangial cells and suggest an antifibrotic potential for NO

Two-population replicator dynamics and number of Nash equilibria in random matrix games

We study the connection between the evolutionary replicator dynamics and the number of Nash equilibria in large random bi-matrix games. Using techniques of disordered systems theory we compute the statistical properties of both, the fixed points of the dynamics and the Nash equilibria. Except for the special case of zero-sum games one finds a transition as a function of the so-called co-operation pressure between a phase in which there is a unique stable fixed point of the dynamics coinciding with a unique Nash equilibrium, and an unstable phase in which there are exponentially many Nash equilibria with statistical properties different from the stationary state of the replicator equations. Our analytical results are confirmed by numerical simulations of the replicator dynamics, and by explicit enumeration of Nash equilibria.Comment: 9 pages, 2x2 figure

Settling Some Open Problems on 2-Player Symmetric Nash Equilibria

Over the years, researchers have studied the complexity of several decision versions of Nash equilibrium in (symmetric) two-player games (bimatrix games). To the best of our knowledge, the last remaining open problem of this sort is the following; it was stated by Papadimitriou in 2007: find a non-symmetric Nash equilibrium (NE) in a symmetric game. We show that this problem is NP-complete and the problem of counting the number of non-symmetric NE in a symmetric game is #P-complete. In 2005, Kannan and Theobald defined the "rank of a bimatrix game" represented by matrices (A, B) to be rank(A+B) and asked whether a NE can be computed in rank 1 games in polynomial time. Observe that the rank 0 case is precisely the zero sum case, for which a polynomial time algorithm follows from von Neumann's reduction of such games to linear programming. In 2011, Adsul et. al. obtained an algorithm for rank 1 games; however, it does not solve the case of symmetric rank 1 games. We resolve this problem

Entropy and typical properties of Nash equilibria in two-player games

We use techniques from the statistical mechanics of disordered systems to analyse the properties of Nash equilibria of bimatrix games with large random payoff matrices. By means of an annealed bound, we calculate their number and analyse the properties of typical Nash equilibria, which are exponentially dominant in number. We find that a randomly chosen equilibrium realizes almost always equal payoffs to either player. This value and the fraction of strategies played at an equilibrium point are calculated as a function of the correlation between the two payoff matrices. The picture is complemented by the calculation of the properties of Nash equilibria in pure strategies.Comment: 6 pages, was "Self averaging of Nash equilibria in two player games", main section rewritten, some new results, for additional information see http://itp.nat.uni-magdeburg.de/~jberg/games.htm

Statistical mechanics of random two-player games

Using methods from the statistical mechanics of disordered systems we analyze the properties of bimatrix games with random payoffs in the limit where the number of pure strategies of each player tends to infinity. We analytically calculate quantities such as the number of equilibrium points, the expected payoff, and the fraction of strategies played with non-zero probability as a function of the correlation between the payoff matrices of both players and compare the results with numerical simulations.Comment: 16 pages, 6 figures, for further information see http://itp.nat.uni-magdeburg.de/~jberg/games.htm

A global climatology of total columnar water vapour from SSM/I and MERIS

A global time series of total columnar water vapour from combined data of the Medium Resolution Imaging Spectrometer (MERIS) onboard ESA's Environmental Satellite (ENVISAT) and the Special Sensor Microwave/Imager (SSM/I) onboard the satellite series of the US Defense Meteorological Satellite Program (DMSP) is presented. The unique data set, generated in the framework of the ESA Data User Element (DUE) GlobVapour project, combines atmospheric water vapour observations over land and ocean, derived from measurements in the near- infrared and the microwave range, respectively. Daily composites and monthly means of total columnar water vapour are available as global maps on rectangular latitude–longitude grids with a spatial resolution of 0.05° × 0.05° over land and 0.5° × 0.5° over ocean for the years 2003 to 2008. The data are stored in NetCDF files and is fully compliant with the NetCDF Climate Forecast convention. Through the combination of high-quality microwave observations and near-infrared observations over ocean and land surfaces, respectively, the data set provides global coverage. The combination of both products is carried out such that the individual properties of the microwave and near-infrared products, in particular their uncertainties, are not modified by the merging process and are therefore well defined. Due to the global coverage and the provided uncertainty estimates this data set is potentially of high value for climate research. The SSM/I-MERIS TCWV data set is freely available via the GlobVapour project web page (www.globvapour.info) with associated doi:10.5676/DFE/WV_COMB/FP. In this paper, the details of the data set generation, i.e. the satellite data used, the retrieval techniques and merging approaches, are presented. The derived level 3 products are compared to global radiosonde data from the GCOS upper air network (GUAN), showing a high agreement with a root-mean-square deviation of roughly 4.4 kg m−2 and a small wet bias well below 1 kg m−2. Furthermore, the data set is shown to be free of seasonal biases. The consistency of the MERIS and SSM/I retrievals is demonstrated by applying the MERIS retrieval to sun glint areas over ocean

A global climatology of total columnar water vapour from SSM/I and MERIS

A global time series of total columnar water vapour from combined data of the Medium Resolution Imaging Spectrometer (MERIS) onboard ESA's Environmental Satellite (ENVISAT) and the Special Sensor Microwave/Imager (SSM/I) onboard the satellite series of the US Defense Meteorological Satellite Program (DMSP) is presented. The unique data set, generated in the framework of the ESA Data User Element (DUE) GlobVapour project, combines atmospheric water vapour observations over land and ocean, derived from measurements in the near- infrared and the microwave range, respectively. Daily composites and monthly means of total columnar water vapour are available as global maps on rectangular latitude–longitude grids with a spatial resolution of 0.05° × 0.05° over land and 0.5° × 0.5° over ocean for the years 2003 to 2008. The data are stored in NetCDF files and is fully compliant with the NetCDF Climate Forecast convention. Through the combination of high-quality microwave observations and near-infrared observations over ocean and land surfaces, respectively, the data set provides global coverage. The combination of both products is carried out such that the individual properties of the microwave and near-infrared products, in particular their uncertainties, are not modified by the merging process and are therefore well defined. Due to the global coverage and the provided uncertainty estimates this data set is potentially of high value for climate research. The SSM/I-MERIS TCWV data set is freely available via the GlobVapour project web page (www.globvapour.info) with associated doi:10.5676/DFE/WV_COMB/FP. In this paper, the details of the data set generation, i.e. the satellite data used, the retrieval techniques and merging approaches, are presented. The derived level 3 products are compared to global radiosonde data from the GCOS upper air network (GUAN), showing a high agreement with a root-mean-square deviation of roughly 4.4 kg m−2 and a small wet bias well below 1 kg m−2. Furthermore, the data set is shown to be free of seasonal biases. The consistency of the MERIS and SSM/I retrievals is demonstrated by applying the MERIS retrieval to sun glint areas over ocean