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

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

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

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

The Path to Eliminating Raccoon Rabies in the Eastern US-Obstacles and Opportunities in Urban-Suburban Landscapes

Rabies in terrestrial wildlife poses a significant public and animal health threat. Oral rabies vaccination (ORV) targeting specific vector species has proven effective in eliminating certain rabies variants in Europe and Canada. The goal of eliminating the raccoon rabies variant (RRV) in the US is achievable through an integrated ORV program at the landscape scale. Current wildlife rabies management in the US includes extensive air and ground ORV programs in 16 eastern states coordinated by Wildlife Services (WS)’ National Rabies Management Program. More than 10 million vaccine-baits are distributed annually targeting raccoons (Procyon lotor) and striped skunks (Mephitis mephitis) with the long-term goal of eliminating RRV. Achieving vaccine-induced herd immunity in target species in developed landscapes has proven challenging due to abundant anthropogenic food sources, higher wildlife densities, decreased home ranges, habitat fragmentation, and non-target bait competition. Effectively managing RRV in the urban-suburban landscape requires greater understanding of meso-carnivore ecology in these landscapes and critical analyses of current baiting strategies. Preliminary results from urban-suburban studies demonstrate fewer potential ORV bait encounters for target species than expected, lower seroconversion rates compared to rural habitats and patchy bait distribution patterns. New technologies including the use of Point of Interest GPS units to document ground bait distribution in combination with research conducted by WS including ORV field trials, urban density studies, and raccoon, skunk, and opossum (Didelphis virginiana) ecology have provided valuable insight to overcome the obstacles of urban rabies management and make eliminating RRV a reality