On Tightness of the Tsaknakis-Spirakis Algorithm for Approximate Nash Equilibrium

Finding the minimum approximate ratio for Nash equilibrium of bi-matrix games has derived a series of studies, started with 3/4, followed by 1/2, 0.38 and 0.36, finally the best approximate ratio of 0.3393 by Tsaknakis and Spirakis (TS algorithm for short). Efforts to improve the results remain not successful in the past 14 years. This work makes the first progress to show that the bound of 0.3393 is indeed tight for the TS algorithm. Next, we characterize all possible tight game instances for the TS algorithm. It allows us to conduct extensive experiments to study the nature of the TS algorithm and to compare it with other algorithms. We find that this lower bound is not smoothed for the TS algorithm in that any perturbation on the initial point may deviate away from this tight bound approximate solution. Other approximate algorithms such as Fictitious Play and Regret Matching also find better approximate solutions. However, the new distributed algorithm for approximate Nash equilibrium by Czumaj et al. performs consistently at the same bound of 0.3393. This proves our lower bound instances generated against the TS algorithm can serve as a benchmark in design and analysis of approximate Nash equilibrium algorithms

Release of alkali metals during biomass thermal conversion

Biomass has great potential to become an economic source of renewable energy; however, its high chlorine and alkali metal content may cause series problems (e.g. slagging and corrosion) thus limiting its utilization. This paper reviews the release of potassium during biomass thermal conversion. Organic potassium is released first when the temperature is relatively low, starting at about 473 K and slowing down at about 773 K; the release of inorganic potassium occurs with the increase of processing temperature. The potassium vapors are mainly in the form of KCl, KOH and K 2 SO 4 . In addition to the temperature, the properties of biomass feedstock, fuel-air ratio, pressure and heating rate also significantly influence the release rate of alkali metals. Future studies are required to develop accurate kinetic models of potassium release to address the ash related challenges when firing and co-firing biomass with high inherent alkali content