# American Institute of Mathematical Sciences

ISSN:
2164-6066

eISSN:
2164-6074

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## Journal of Dynamics and Games

January 2022 , Volume 9 , Issue 1

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2022, 9(1): 1-12 doi: 10.3934/jdg.2021019 +[Abstract](997) +[HTML](447) +[PDF](322.29KB)
Abstract:

We consider a pursuit-evasion differential game problem with countable number pursuers and one evader in the Hilbert space \begin{document}$l_{2}.$\end{document} Players' dynamic equations described by certain \begin{document}$n^{th}$\end{document} order ordinary differential equations. Control functions of the players subject to integral constraints. The goal of the pursuers is to minimize the distance to the evader and that of the evader is the opposite. The stoppage time of the game is fixed and the game payoff is the distance between evader and closest pursuer when the game is stopped. We study this game problem and find the value of the game. In addition to this, we construct players' optimal strategies.

2022, 9(1): 13-25 doi: 10.3934/jdg.2021020 +[Abstract](802) +[HTML](404) +[PDF](359.52KB)
Abstract:

In this paper we study zero-sum stochastic games for pure jump processes on a general state space with risk sensitive discounted criteria. We establish a saddle point equilibrium in Markov strategies for bounded cost function. We achieve our results by studying relevant Hamilton-Jacobi-Isaacs equations.

2022, 9(1): 27-32 doi: 10.3934/jdg.2021022 +[Abstract](667) +[HTML](368) +[PDF](247.39KB)
Abstract:

Based on a work by M. Dur and J.-B. Hiriart-Urruty[3], we consider the problem of whether a symmetric matrix is copositive formulated as a difference of convex functions problem. The convex nondifferentiable function in this d.c. decomposition being proximable, we then apply a proximal-gradient method to approximate the related stationary points. Whereas, in [3], the DCA algorithm was used.

2022, 9(1): 33-74 doi: 10.3934/jdg.2021024 +[Abstract](741) +[HTML](318) +[PDF](617.88KB)
Abstract:

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 theory of creative creation 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.

2022, 9(1): 75-96 doi: 10.3934/jdg.2021026 +[Abstract](679) +[HTML](272) +[PDF](2372.24KB)
Abstract:

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.

2022, 9(1): 97-115 doi: 10.3934/jdg.2021027 +[Abstract](546) +[HTML](269) +[PDF](898.63KB)
Abstract:

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 & P 500 Index, depending on the indicator value.

2022, 9(1): 117-122 doi: 10.3934/jdg.2021028 +[Abstract](397) +[HTML](194) +[PDF](310.65KB)
Abstract:

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 multi-player discounted stochastic games

2020 CiteScore: 0.6