The Complexity of Nash Equilibria in Stochastic Multiplayer Games

We analyse the computational complexity of finding Nash equilibria in stochastic multiplayer games with $\omega$-regular objectives. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game~$\mathcal{G}$, does there exist a pure-strategy Nash equilibrium of~$\mathcal{G}$ where player 0 wins with probability~$1$. Moreover, this problem remains undecidable if it is restricted to strategies with (unbounded) finite memory. However, if randomised strategies are allowed, decidability remains an open problem; we can only prove NP-hardness in this case. One way to obtain a provably decidable variant of the problem is to restrict the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSPACE respectively. Finally, we single out a special case of the general problem that, in many cases, admits an efficient solution. In particular, we prove that deciding the existence of an equilibrium in which each player either wins or loses with probability~$1$ can be done in polynomial time for games where, for instance, the objective of each player is given by a parity condition with a bounded number of priorities

Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata, and Pushdown Systems

We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems (pPDSs) and Recursive Markov Chains (RMCs). We then proceed to exploit these connections to obtain a number of new algorithmic upper and lower bounds for central computational problems about these models. Our main result is this: for an arbitrary QBD, we can approximate its termination probabilities (i.e., its $G$ matrix) to within $i$ bits of precision (i.e., within additive error $1/2^i$), in time polynomial in \underline{both} the encoding size of the QBD and in $i$, in the unit-cost rational arithmetic RAM model of computation. Specifically, we show that a decomposed Newton's method can be used to achieve this. We emphasize that this bound is very different from the well-known ``linear/quadratic convergence'' of numerical analysis, known for QBDs and TL-QBDs, which typically gives no constructive bound in terms of the encoding size of the system being solved. In fact, we observe (based on recent results) that for the more general TL-QBDs such a polynomial upper bound on Newton's method fails badly. Our upper bound proof for QBDs combines several ingredients: a detailed analysis of the structure of 1-counter automata, an iterative application of a classic condition number bound for errors in linear systems, and a very recent constructive bound on the performance of Newton's method for strongly connected monotone systems of polynomial equations. We show that the quantitative termination decision problem for QBDs (namely, ``is $G_{u,v} \geq 1/2$?'') is at least as hard as long standing open problems in the complexity of exact numerical computation, specifically the square-root sum problem. On the other hand, it follows from our earlier results for RMCs that any non-trivial approximation of termination probabilities for TL-QBDs is sqrt-root-sum-hard

Model-free reinforcement learning for stochastic parity games

This paper investigates the use of model-free reinforcement learning to compute the optimal value in two-player stochastic games with parity objectives. In this setting, two decision makers, player Min and player Max, compete on a finite game arena - a stochastic game graph with unknown but fixed probability distributions - to minimize and maximize, respectively, the probability of satisfying a parity objective. We give a reduction from stochastic parity games to a family of stochastic reachability games with a parameter ε, such that the value of a stochastic parity game equals the limit of the values of the corresponding simple stochastic games as the parameter ε tends to 0. Since this reduction does not require the knowledge of the probabilistic transition structure of the underlying game arena, model-free reinforcement learning algorithms, such as minimax Q-learning, can be used to approximate the value and mutual best-response strategies for both players in the underlying stochastic parity game. We also present a streamlined reduction from 112-player parity games to reachability games that avoids recourse to nondeterminism. Finally, we report on the experimental evaluations of both reductions

Calculations of giant magnetoresistance in Fe/Cr trilayers using layer potentials determined from {\it ab-initio} methods

The ab initio full-potential linearized augmented plane-wave method explicitly designed for the slab geometry was employed to elucidate the physical origin of the layer potentials for the trilayers nFe/3Cr/nFe(001), where n is the number of Fe monolayers. The thickness of the transition-metal ferromagnet has been ranged from $n=1$ up to n=8 while the spacer thickness was fixed to 3 monolayers. The calculated potentials were inserted in the Fuchs-Sondheimer formalism in order to calculate the giant magnetoresistance (GMR) ratio. The predicted GMR ratio was compared with the experiment and the oscillatory behavior of the GMR as a function of the ferromagnetic layer thickness was discussed in the context of the layer potentials. The reported results confirm that the interface monolayers play a dominant role in the intrinsic GMR.Comment: 17 pages, 7 figures, 3 tables. accepted in J. Phys.: Cond. Matte

Setting school-level outcome standards

To establish international standards for medical schools, an appropriate panel of experts must decide on performance standards. A pilot test of such standards was set in the context of a multidimensional (multiple-choice question examination, objective structured clinical examination, faculty observation) examination at 8 leading schools in China. Methods  A group of 16 medical education leaders from a broad array of countries met over a 3-day period. These individuals considered competency domains, examination items, and the percentage of students who could fall below a cut-off score if the school was still to be considered as meeting competencies. This 2-step process started with a discussion of the borderline school and the relative difficulty of a borderline school in achieving acceptable standards in a given competency domain. Committee members then estimated the percentage of students falling below the standard that is tolerable at a borderline school and were allowed to revise their ratings after viewing pilot data. Results  Tolerable failure rates ranged from 10% to 26% across competency domains and examination types. As with other standard-setting exercises, standard deviations from initial to final estimates of the tolerable failure rates fell, but the cut-off scores did not change significantly. Final, but not initial cut-off scores were correlated with student failure rates ( r =  0.59, P  = 0.03). Discussion  This paper describes a method to set school-level outcome standards at an international level based on prior established standard-setting methods. Further refinement of this process and validation using other examinations in other countries will be needed to achieve accurate international standards.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71572/1/j.1365-2929.2005.02374.x.pd

Transport of Cytoplasmically Synthesized Proteins into the Mitochondria in a Cell Free System from Neurospora crassa

Synthesis and transport of mitochondrial proteins were followed in a cell-free homogenate of Neurospora crassa in which mitochondrial translation was inhibited. Proteins synthesized on cytoplasmic ribosomes are transferred into the mitochondrial fraction. The relative amounts of proteins which are transferred in vitro are comparable to those transferred in whole cells. Cycloheximide and puromycin inhibit the synthesis of mitochondrial proteins but not their transfer into mitochondria. The transfer of immunoprecipitable mitochondrial proteins was demonstrated for matrix proteins, carboxyatractyloside-binding protein and cytochrome c. Import of proteins into mitochondria exhibits a degree of specificity. The transport mechanism differentiates between newly synthesized proteins and preexistent mitochondrial proteins, at least in the case of matrix proteins. In the cell-free homogenate membrane-bound ribosomes are more active in the synthesis of mitochondrial proteins than are free ribosomes. The finished translation products appear to be released from the membrane-bound ribosomes into the cytosol rather than into the membrane vesicles. The results suggest that the transport of cytoplasmically synthesized mitochondrial proteins is essentially independent of cytoplasmic translation; that cytoplasmically synthesized mitochondrial proteins exist in an extramitochondrial pool prior to import; that the site of this pool is the cytosol for at least some of the mitochondrial proteins; and that the precursors in the extramitochondrial pool differ in structure or conformation from the functional proteins in the mitochondria

Synthesis and structural analysis of the N-terminal domain of the thyroid hormone-binding protein transthyretin

Transthyretin (TTR) is a 55 kDa protein responsible for the transport of thyroid hormones and retinol in human serum. Misfolded forms of the protein are implicated in the amyloid diseases familial amyloidotic polyneuropathy and senile systemic amyloidosis. Its folding properties and stabilization by ligands are of current interest due to their importance in understanding and combating these diseases. To assist in such studies we developed a method for the solid phase synthesis of the monomeric unit of a TTR analogue and its folding to form a functional 55 kDa tetramer. The monomeric unit of the protein was chemically synthesized in three parts, comprising amino acid residues 151, 5499 and 102127, and ligated using chemoselective thioether ligation chemistry. The synthetic protein was folded and assembled to a tetrameric structure in the presence of the TTRs native ligand, thyroxine, as shown by gel filtration chromatography, native gel electrophoresis, TTR antibody recognition and thyroid hormone binding. In the current study the solution structure of the first of these fragment peptides, TTR(151) is examined to determine its intrinsic propensity to form beta-sheet structure, potentially involved in amyloid fibril formation by TTR. Despite the presence of extensive beta-structure in the native form of the protein, the Nterminal fragment adopts an essentially random coil conformation in solution

A phenomenological approach to the simulation of metabolism and proliferation dynamics of large tumour cell populations

A major goal of modern computational biology is to simulate the collective behaviour of large cell populations starting from the intricate web of molecular interactions occurring at the microscopic level. In this paper we describe a simplified model of cell metabolism, growth and proliferation, suitable for inclusion in a multicell simulator, now under development (Chignola R and Milotti E 2004 Physica A 338 261-6). Nutrients regulate the proliferation dynamics of tumor cells which adapt their behaviour to respond to changes in the biochemical composition of the environment. This modeling of nutrient metabolism and cell cycle at a mesoscopic scale level leads to a continuous flow of information between the two disparate spatiotemporal scales of molecular and cellular dynamics that can be simulated with modern computers and tested experimentally.Comment: 58 pages, 7 figures, 3 tables, pdf onl

Hormone affinity and fibril formation of piscine transthyretin: the role of the N-terminal

Transthyretin (TTR) transports thyroid hormones (THs), thyroxine (T4) and triiodothyronine (T3) in the blood of vertebrates. TH-binding sites are highly conserved in vertebrate TTR however, piscine TTR has a longer N-terminus which is thought to influence TH-binding affinity and may influence TTR stability. We produced recombinant wild-type sea bream TTR (sbTTRWT) plus two mutants in which six (sbTTRM6) and twelve (sbTTRM12) N-terminal residues were removed. Ligandbinding studies revealed similar affinities for T3 (Kd=10.6±1.7nM) and T4 (Kd=9.8±0.97nM) binding to sbTTRWT. Affinity for THs was unaltered in sbTTRM12 but sbTTRM6 had poorer affinity for T4 (Kd=252.3±15.8nM) implying that some residues in the N-terminus can influence T4 binding. sbTTRM6 inhibited acid-mediated fibril formation in vitro as shown by fluorometric measurements using thioflavine-T.In contrast, fibril formation by sbTTRM12 was significant, probably due to decreased stability of the tetramer. Such studies also suggested that sbTTRWT is more resistant to fibril formation than human TTR