Decision Theory and large deviations for dynamical hypotheses tests: The Neyman-Pearson Lemma, Min-Max and Bayesian tests

<p style='text-indent:20px;'>We analyze hypotheses tests using classical results on large deviations to compare two models, each one described by a different Hölder Gibbs probability measure. One main difference to the classical hypothesis tests in Decision Theory is that here the two measures are singular with respect to each other. Among other objectives, we are interested in the decay rate of the wrong decisions probability, when the sample size <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> goes to infinity. We show a dynamical version of the Neyman-Pearson Lemma displaying the ideal test within a certain class of similar tests. This test becomes exponentially better, compared to other alternative tests, when the sample size goes to infinity. We are able to present the explicit exponential decay rate. We also consider both, the Min-Max and a certain type of Bayesian hypotheses tests. We shall consider these tests in the log likelihood framework by using several tools of Thermodynamic Formalism. Versions of the Stein's Lemma and Chernoff's information are also presented.</p>

Stochastic dynamics and Edmonds' algorithm

<p style='text-indent:20px;'>Recently, there has been a revival of interest in cyclic decompositions of stochastic dynamics. These decompositions consider the behavior of dynamics over the short, medium and long run, aggregating cycles of behavior into progressively larger cycles, eventually encompassing the entire state space. We show that these decompositions are equivalent to the aggregative stage of Edmonds' algorithm and that this equivalence can be used to recover well-known results in the literature.</p>

Explaining the definition of wholesale access prices in the Portuguese telecommunications industry

<p style='text-indent:20px;'>The 2016–2018 triennium was a period marked by a fierce dispute between the European Commission and Autoridade Nacional de Comunicações, Portugal, on the need to regulate wholesale access prices. While the European Commission defended the imposition of Fiber-To-The-x regulation in non-competitive areas, the Portuguese sectoral regulator argued in favor of the persistence of Fiber-To-The-x deregulation. Following a Game Theory approach, the present study demonstrates that the transition from Fiber-To-The-x deregulation to Fiber-To-The-x regulation should only occur when a given territorial unit becomes a competitive area since the subgame perfect Nash equilibrium captures a regulatory framework optimally characterized by the imposition of active access price deregulation (regulation) in non-competitive (competitive) areas, that is, local administrative units characterized by a weak (strong) degree of vertical spillover, respectively. Meanwhile, ducts access regulation must be permanently imposed throughout the national territory, despite it can be relaxed in competitive areas if the regulator imposes intra-flexibility to establish a monopolistic bottleneck to ensure social welfare maximization. Previous conclusions require to introduce both facility-based and service-based competition at the entry stage as well as active and passive obligations at the regulation stage in a multi-stage game with complete information. The present analysis legitimizes the emergence of a new optimization theory in the telecommunications literature, whose modus operandi is contrary to (coincident with) the ladder of investment theory in non-competitive (competitive) areas, respectively. Differently from the view sustained by the ladder of investment theory, which defends that a short-term regulatory touch combined with long-term market deregulation is a socially optimal strategy, the new theory confirms that a regulatory intervention is socially desirable only in the long run. The conceptual refinement is meticulously explained and labeled as the <i>theory of creative creation</i> because, differently from the Schumpeterian gale of creative destruction, whose processes of industrial mutation are permanently market-driven by assumption, a period of regulatory holidays followed by successive regulatory interventions dependent on the degree of vertical spillover observed in the telecommunications industry can effectively promote investment realization that continuously revolutionizes the market structure from within, incessantly destroying the old technology. The theory of creative creation reflects the regulatory framework currently in force in the Portuguese Telecommunications Industry.</p>

Analysis of communities of countries with similar dynamics of the COVID-19 pandemic evolution

<p style='text-indent:20px;'>This work addresses the spread of the coronavirus through a non-parametric approach, with the aim of identifying communities of countries based on how similar their evolution of the disease is. The analysis focuses on the number of daily new COVID-19 cases per ten thousand people during a period covering at least 250 days after the confirmation of the tenth case. Dynamic analysis is performed by constructing Minimal Spanning Trees (MST) and identifying groups of similarity in contagions evolution in 95 time windows of a 150-day amplitude that moves one day at a time. The intensity measure considered was the number of times countries belonged to a similar performance group in constructed time windows. Groups' composition is not stable, indicating that the COVID-19 evolution needs to be treated as a dynamic problem in the context of complex systems. Three communities were identified by applying the Louvain algorithm. Identified communities analysis according to each country's socioeconomic characteristics and variables related to the disease sheds light on whether there is any suggested course of action. Even when strong testing and tracing cases policies may be related with a more stable dynamic of the disease, results indicate that communities are conformed by countries with diverse characteristics. The best option to counteract the harmful effects of a pandemic may be having strong health systems in place, with contingent capacity to deal with unforeseen events and available resources capable of a rapid expansion of its capacity.</p>

Using chemical reaction network theory to show stability of distributional dynamics in game theory

<p style='text-indent:20px;'>This article shows how to apply results of chemical reaction network theory (CRNT) to prove uniqueness and stability of a positive equilibrium for pairs/groups distributional dynamics that arise in game theoretic models. Evolutionary game theory assumes that individuals accrue their fitness through interactions with other individuals. When there are two or more different strategies in the population, this theory assumes that pairs (groups) are formed instantaneously and randomly so that the corresponding pairs (groups) distribution is described by the Hardy–Weinberg (binomial) distribution. If interactions times are phenotype dependent the Hardy-Weinberg distribution does not apply. Even if it becomes impossible to calculate the pairs/groups distribution analytically we show that CRNT is a general tool that is very useful to prove not only existence of the equilibrium, but also its stability. In this article, we apply CRNT to pair formation model that arises in two player games (e.g., Hawk-Dove, Prisoner's Dilemma game), to group formation that arises, e.g., in Public Goods Game, and to distribution of a single population in patchy environments. We also show by generalizing the Battle of the Sexes game that the methodology does not always apply.</p>

A note on the Nash equilibria of some multi-player reachability/safety games

<p style='text-indent:20px;'>In this short note we study a class of multi-player, turn-based games with deterministic state transitions and reachability / safety objectives (this class contains as special cases "classic" two-player reachability and safety games as well as multi-player and ""stay–in-a-set" and "reach-a-set" games). Quantitative and qualitative versions of the objectives are presented and for both cases we prove the existence of a deterministic and memoryless Nash equilibrium; the proof is short and simple, using only Fink's classic result about the existence of Nash equilibria for <i>multi-player discounted stochastic games</i></p>

Linear-quadratic zero-sum mean-field type games: Optimality conditions and policy optimization

<p style='text-indent:20px;'>In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics and quadratic cost are studied under infinite-horizon discounted utility function. ZSMFTG are a class of games in which two decision makers whose utilities sum to zero, compete to influence a large population of indistinguishable agents. In particular, the case in which the transition and utility functions depend on the state, the action of the controllers, and the mean of the state and the actions, is investigated. The optimality conditions of the game are analysed for both open-loop and closed-loop controls, and explicit expressions for the Nash equilibrium strategies are derived. Moreover, two policy optimization methods that rely on policy gradient are proposed for both model-based and sample-based frameworks. In the model-based case, the gradients are computed exactly using the model, whereas they are estimated using Monte-Carlo simulations in the sample-based case. Numerical experiments are conducted to show the convergence of the utility function as well as the two players' controls.</p>

Crisis risk prediction with concavity from Polymodel

<p style='text-indent:20px;'>Financial crises are an important research topic because of their impact on the economy, businesses, and populations. However, prior research tends to generate reactive systemic risk measures, in the sense that the measure surges after the crisis starts. Few of them succeed in warning of financial crises in advance. In this paper, we first sketch a toy model that produces normal mixture distributions based on a dynamic regime switching model. We derive that the relative concavity among various indices tends to increase before a crisis. Using Polymodel theory, we introduce a measure of concavity as a crisis risk indicator, and test it against known crises observed in the past. We validate this indicator by a trading strategy holding long or short positions on the S &amp; P 500 Index, depending on the indicator value.</p>