# American Institute of Mathematical Sciences

ISSN:
2164-6066

eISSN:
2164-6074

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## Journal of Dynamics and Games

July 2021 , Volume 8 , Issue 3

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2021, 8(3): 167-186 doi: 10.3934/jdg.2021004 +[Abstract](1450) +[HTML](514) +[PDF](985.86KB)
Abstract:

We utilize systems theory in the study of the implementation of non pharmaceutical strategies for the mitigation of the COVID-19 pandemic. We present two models. The first one is a model of predictive control with receding horizon and discontinuous actions of unknown costs for the implementation of adaptive triggering policies during the disease. This model is based on a periodic assessment of the peak of the pandemic (and, thus, of the health care demand) utilizing the latest data about the transmission and recovery rate of the disease. Consequently, the model seems to be suitable for discontinuous, non-mechanical (i.e. human) actions with unknown effectiveness, like those applied in the case of COVID-19. Secondly, we consider a feedback control problem in order to contain the pandemic at the capacity of the NHS (National Health System). As input parameter we consider the value \begin{document}$p$\end{document} that reflects the intensity-effectiveness of the measures applied and as output the predicted maximum of infected people to be treated by NHS. The feedback control regulates \begin{document}$p$\end{document} so that the number of infected people is manageable. Based on this approach, we address the following questions: (a) the limits of improvement of this approach; (b) the effectiveness of this approach; (c) the time horizon and timing of the application.

2021, 8(3): 187-202 doi: 10.3934/jdg.2021005 +[Abstract](893) +[HTML](389) +[PDF](400.1KB)
Abstract:

A generalized \begin{document}$N$\end{document}-sided die is a random variable \begin{document}$D$\end{document} on a sample space of \begin{document}$N$\end{document} equally likely outcomes taking values in the set of positive integers. We say of independent \begin{document}$N$\end{document}-sided dice \begin{document}$D_i, D_j$\end{document} that \begin{document}$D_i$\end{document} beats \begin{document}$D_j$\end{document}, written \begin{document}$D_i \to D_j$\end{document}, if \begin{document}$Prob(D_i > D_j) > \frac{1}{2}$\end{document}. A collection of dice \begin{document}$\{ D_i : i = 1, \dots, n \}$\end{document} models a tournament on the set \begin{document}$[n] = \{ 1, 2, \dots, n \}$\end{document}, i.e. a complete digraph with \begin{document}$n$\end{document} vertices, when \begin{document}$D_i \to D_j$\end{document} if and only if \begin{document}$i \to j$\end{document} in the tournament. By using regular \begin{document}$n$\end{document}-fold partitions of the set \begin{document}$[Nn]$\end{document} to label the \begin{document}$N$\end{document}-sided dice we can model an arbitrary tournament on \begin{document}$[n]$\end{document} and \begin{document}$N$\end{document} can be chosen to be less than or equal to \begin{document}$N = 3^{n-2}$\end{document}.

2021, 8(3): 203-231 doi: 10.3934/jdg.2021008 +[Abstract](4698) +[HTML](1137) +[PDF](772.97KB)
Abstract:

Determining the cause of a particular event has been a case of study for several researchers over the years. Finding out why an event happens (its cause) means that, for example, if we remove the cause from the equation, we can stop the effect from happening or if we replicate it, we can create the subsequent effect. Causality can be seen as a mean of predicting the future, based on information about past events, and with that, prevent or alter future outcomes. This temporal notion of past and future is often one of the critical points in discovering the causes of a given event. The purpose of this survey is to present a cross-sectional view of causal discovery domain, with an emphasis in the machine learning/data mining area.

2021, 8(3): 233-266 doi: 10.3934/jdg.2021009 +[Abstract](1008) +[HTML](390) +[PDF](589.92KB)
Abstract:

We study a two player zero sum game where the initial position \begin{document}$z_0$\end{document} is not communicated to any player. The initial position is a function of a couple \begin{document}$(x_0,y_0)$\end{document} where \begin{document}$x_0$\end{document} is communicated to player Ⅰ while \begin{document}$y_0$\end{document} is communicated to player Ⅱ. The couple \begin{document}$(x_0,y_0)$\end{document} is chosen according to a probability measure \begin{document}$dm(x,y) = h(x,y) d\mu(x) d\nu(y)$\end{document}. We show that the game has a value and, under additional regularity assumptions, that the value is a solution of Hamilton Jacobi Isaacs equation in a dual sense.

2021, 8(3): 267-275 doi: 10.3934/jdg.2021010 +[Abstract](1049) +[HTML](384) +[PDF](336.93KB)
Abstract:

The allocation problem of rewards/costs is a basic question for players namely individuals and companies that planning cooperation under uncertainty. The involvement of uncertainty in cooperative game theory is motivated by the real world where noise in observation and experimental design, incomplete information and further vagueness in preference structures and decision-making play an important role. In this paper we extend cooperative bubbly games to cooperative fuzzy bubbly games, where the worth of each coalition is a fuzzy bubble instead of an interval. Further, we introduce a set-valued concept called the fuzzy bubbly core. Finally, some results on fuzzy bubbly core are given.

2021, 8(3): 277-297 doi: 10.3934/jdg.2021011 +[Abstract](1333) +[HTML](633) +[PDF](1125.46KB)
Abstract:

A novel approach depicting the dynamics of marijuana usage to gauge the effects of peer influence in a school population, is the site of investigation. Consumption of drug is considered as a contagious social epidemic which is spread mainly by peer influences. A relation-based graph-CA (r-GCA) model consisting of 4 states namely, Nonusers (N), Experimental users (E), Recreational users (R) and Addicts (A), is formulated in order to represent the prevalence of the epidemic on a campus. The r-GCA model is set up by local transition rules which delineates the proliferation of marijuana use. Data available in [4] is opted to verify and validate the r-GCA. Simulations of the r-GCA system are presented and discussed. The numerical results agree quite accurately with the observed data. Using the model, the enactment of campaigns of prevention targeting N, E and R states respectively were conducted and analysed. The results indicate a significant decline in marijuana consumption on the campus when a campaign of prevention targeting the latter three states simultaneously, is enacted.

2021, 8(3): 299-329 doi: 10.3934/jdg.2021017 +[Abstract](979) +[HTML](402) +[PDF](5677.59KB)
Abstract:

We propose a (toy) MFG model for the evolution of residents and firms densities, coupled both by labour market equilibrium conditions and competition for land use (congestion). This results in a system of two Hamilton-Jacobi-Bellman and two Fokker-Planck equations with a new form of coupling related to optimal transport. This MFG has a convex potential which enables us to find weak solutions by a variational approach. In the case of quadratic Hamiltonians, the problem can be reformulated in Lagrangian terms and solved numerically by an IPFP/Sinkhorn-like scheme as in [4]. We present numerical results based on this approach, these simulations exhibit different behaviours with either agglomeration or segregation dominating depending on the initial conditions and parameters.

2021 CiteScore: 3.3