When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the $n$-stage repeated version of the game exist if and only if both players have $\Omega(n)$ random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the $n$-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on "random-like" sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators \emph{do not} exist, then $\Omega(n)$ random bits remain necessary for equilibria to exist

Data-driven efficient score tests for deconvolution problems

We consider testing statistical hypotheses about densities of signals in deconvolution models. A new approach to this problem is proposed. We constructed score tests for the deconvolution with the known noise density and efficient score tests for the case of unknown density. The tests are incorporated with model selection rules to choose reasonable model dimensions automatically by the data. Consistency of the tests is proved

Unambiguous quantum state filtering

In this paper, we consider the generalized measurement where one particular quantum signal is unambiguously extracted from a set of non-commutative quantum signals and the other signals are filtered out. Simple expressions for the maximum detection probability and its POVM are derived. We applyl such unambiguous quantum state filtering to evaluation of the sensing of decoherence channels. The bounds of the precision limit for a given quantum state of probes and possible device implementations are discussed.Comment: 7 pages, 5 figure

The singular continuous diffraction measure of the Thue-Morse chain

The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the Thue-Morse chain, in its realisation as a binary sequence with values in $\{\pm 1\}$. We revisit this example and derive a functional equation together with an explicit form of the corresponding singular continuous diffraction measure, which is related to the known representation as a Riesz product.Comment: 6 pages, 1 figure; revised and improved versio

A galaxy-halo model of large-scale structure

We present a new, galaxy-halo model of large-scale structure, in which the galaxies entering a given sample are the fundamental objects. Haloes attach to galaxies, in contrast to the standard halo model, in which galaxies attach to haloes. The galaxy-halo model pertains mainly to the relationships between the power spectra of galaxies and mass, and their cross-power spectrum. With surprisingly little input, an intuition-aiding approximation to the galaxy-matter cross-correlation coefficient R(k) emerges, in terms of the halo mass dispersion. This approximation seems valid to mildly non-linear scales (k < ~3 h/Mpc), allowing measurement of the bias and the matter power spectrum from measurements of the galaxy and galaxy-matter power spectra (or correlation functions). This is especially relevant given the recent advances in precision in measurements of the galaxy-matter correlation function from weak gravitational lensing. The galaxy-halo model also addresses the issue of interpreting the galaxy-matter correlation function as an average halo density profile, and provides a simple description of galaxy bias as a function of scale.Comment: 13 pages, 9 figures, submitted to MNRAS. Minor changes, suggested by refere

A global descriptor of spatial pattern interaction in the galaxy distribution

We present the function J as a morphological descriptor for point patterns formed by the distribution of galaxies in the Universe. This function was recently introduced in the field of spatial statistics, and is based on the nearest neighbor distribution and the void probability function. The J descriptor allows to distinguish clustered (i.e. correlated) from ``regular'' (i.e. anti-correlated) point distributions. We outline the theoretical foundations of the method, perform tests with a Matern cluster process as an idealised model of galaxy clustering, and apply the descriptor to galaxies and loose groups in the Perseus-Pisces Survey. A comparison with mock-samples extracted from a mixed dark matter simulation shows that the J descriptor can be profitably used to constrain (in this case reject) viable models of cosmic structure formation.Comment: Significantly enhanced version, 14 pages, LaTeX using epsf, aaspp4, 7 eps-figures, accepted for publication in the Astrophysical Journa

Pt/CeO2 and Pt/CeSnOx catalysts for low-temperature CO oxidation prepared by plasma-arc technique

We applied a method of plasma arc synthesis to study effects of modification of the fluorite phase of ceria by tin ions. By sputtering active components (Pt, Ce, Sn) together with carbon from a graphite electrode in a helium ambient we prepared samples of complex highly defective composite PtCeC and PtCeSnC oxide particles stabilized in a matrix of carbon. Subsequent high-temperature annealing of the samples in oxygen removes the carbon matrix and causes the formation of active catalysts Pt/CeOx and Pt/CeSnOx for CO oxidation. In the presence of Sn, X-Ray Diffraction (XRD) and High-Resolution Transmission Electron Microscopy (HRTEM) show formation of a mixed phase CeSnOx and stabilization of more dispersed species with a fluorite-type structure. These factors are essential for the observed high activity and thermic stability of the catalyst modified by Sn. X-Ray Photoelectron Spectroscopy (XPS) reveals the presence of both Pt2+ and Pt4+ ions in the catalyst Pt/CeOx, whereas only the state Pt2+ of platinum could be detected in the Sn-modified catalyst Pt/CeSnOx. Insertion of Sn ions into the Pt/CeOx lattice destabilizes/reduces Pt4+ cations in the Pt/CeSnOx catalyst and induces formation of strikingly high concentration (up to 50% at.) of lattice Ce3+ ions. Our DFT calculations corroborate destabilization of Pt4+ ions by incorporation of cationic Sn in Pt/CeOx. The presented results show that modification of the fluorite lattice of ceria by tin induces substantial amount of mobile reactive oxygen partly due to affecting geometric parameters of ceria by tin ions

Algorithms for Game Metrics

Simulation and bisimulation metrics for stochastic systems provide a quantitative generalization of the classical simulation and bisimulation relations. These metrics capture the similarity of states with respect to quantitative specifications written in the quantitative {\mu}-calculus and related probabilistic logics. We first show that the metrics provide a bound for the difference in long-run average and discounted average behavior across states, indicating that the metrics can be used both in system verification, and in performance evaluation. For turn-based games and MDPs, we provide a polynomial-time algorithm for the computation of the one-step metric distance between states. The algorithm is based on linear programming; it improves on the previous known exponential-time algorithm based on a reduction to the theory of reals. We then present PSPACE algorithms for both the decision problem and the problem of approximating the metric distance between two states, matching the best known algorithms for Markov chains. For the bisimulation kernel of the metric our algorithm works in time O(n^4) for both turn-based games and MDPs; improving the previously best known O(n^9\cdot log(n)) time algorithm for MDPs. For a concurrent game G, we show that computing the exact distance between states is at least as hard as computing the value of concurrent reachability games and the square-root-sum problem in computational geometry. We show that checking whether the metric distance is bounded by a rational r, can be done via a reduction to the theory of real closed fields, involving a formula with three quantifier alternations, yielding O(|G|^O(|G|^5)) time complexity, improving the previously known reduction, which yielded O(|G|^O(|G|^7)) time complexity. These algorithms can be iterated to approximate the metrics using binary search.Comment: 27 pages. Full version of the paper accepted at FSTTCS 200

Pt/CeO2 and Pt/CeSnOx Catalysts for Low-Temperature CO Oxidation Prepared by Plasma-Arc Technique

We applied a method of plasma arc synthesis to study effects of modification of the fluorite phase of ceria by tin ions. By sputtering active components (Pt, Ce, Sn) together with carbon from a graphite electrode in a helium ambient we prepared samples of complex highly defective composite PtCeC and PtCeSnC oxide particles stabilized in a matrix of carbon. Subsequent high-temperature annealing of the samples in oxygen removes the carbon matrix and causes the formation of active catalysts Pt/CeOx and Pt/CeSnOx for CO oxidation. In the presence of Sn, X-Ray Diffraction (XRD) and High-Resolution Transmission Electron Microscopy (HRTEM) show formation of a mixed phase CeSnOx and stabilization of more dispersed species with a fluorite-type structure. These factors are essential for the observed high activity and thermic stability of the catalyst modified by Sn. X-Ray Photoelectron Spectroscopy (XPS) reveals the presence of both Pt2+ and Pt4+ ions in the catalyst Pt/CeOx, whereas only the state Pt2+ of platinum could be detected in the Sn-modified catalyst Pt/CeSnOx. Insertion of Sn ions into the Pt/CeOx lattice destabilizes/reduces Pt4+ cations in the Pt/CeSnOx catalyst and induces formation of strikingly high concentration (up to 50% at.) of lattice Ce3+ ions. Our DFT calculations corroborate destabilization of Pt4+ ions by incorporation of cationic Sn in Pt/CeOx. The presented results show that modification of the fluorite lattice of ceria by tin induces substantial amount of mobile reactive oxygen partly due to affecting geometric parameters of ceria by tin ions