# American Institute of Mathematical Sciences

ISSN:
2164-6066

eISSN:
2164-6074

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## Journal of Dynamics and Games

April 2022 , Volume 9 , Issue 2

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2022, 9(2): 123-150 doi: 10.3934/jdg.2021031 +[Abstract](562) +[HTML](186) +[PDF](475.68KB)
Abstract:

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 \begin{document}$n$\end{document} 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.

2022, 9(2): 151-163 doi: 10.3934/jdg.2022001 +[Abstract](389) +[HTML](147) +[PDF](374.44KB)
Abstract:

We aim at characterizing the functions that could be explained (recoverable) as a best reply of payoff-maximizing players in contests for a fixed prize. We show that recoverability strongly differs between Decisive Contests, where the prize is allocated with certainty, and Possibly Indecisive Contests, where the prize might not be awarded. In the latter, any arbitrary set of best reply functions is recoverable, thus "anything goes." In the former, best reply functions have to satisfy strong conditions in some cases. We provide an outline of possible applications of our results to R & D and labor markets.

2022, 9(2): 165-189 doi: 10.3934/jdg.2022002 +[Abstract](290) +[HTML](150) +[PDF](901.44KB)
Abstract:

We study an optimal control problem arising from a generalization of rock-paper-scissors in which the number of strategies may be selected from any positive odd number greater than 1 and in which the payoff to the winner is controlled by a control variable \begin{document}$\gamma$\end{document}. Using the replicator dynamics as the equations of motion, we show that a quasi-linearization of the problem admits a special optimal control form in which explicit dynamics for the controller can be identified. We show that all optimal controls must satisfy a specific second order differential equation parameterized by the number of strategies in the game. We show that as the number of strategies increases, a limiting case admits a closed form for the open-loop optimal control. In performing our analysis we show necessary conditions on an optimal control problem that allow this analytic approach to function.

2022, 9(2): 191-202 doi: 10.3934/jdg.2022003 +[Abstract](286) +[HTML](117) +[PDF](315.15KB)
Abstract:

In this paper, stabilization problems for n-player noncooperative differential games of international pollution control (IPC) are analysed via the concept of the potential differential game (PDG) introduced by Fonseca-Morales and Hernández-Lerma (2018). By first identifying a game of IPC as a PDG, an associated optimal control problem (OCP) is obtained, whose optimal solution is a Nash equilibrium (NE) for the game of IPC. Thus, the problem of finding conditions for which the NE stabilizes the game of IPC reduces to finding conditions for which the optimal solution stabilizes the associated OCP. The concept not only yields mild conditions for saddle point stability analysed in the literature but also for the overtaking optimality of the NE of the game of IPC.

2022, 9(2): 203-217 doi: 10.3934/jdg.2022004 +[Abstract](265) +[HTML](127) +[PDF](383.68KB)
Abstract:

In this work the problem of optimal harvesting policy selection for natural resources management under model uncertainty is investigated. Under the framework of the neoclassical growth model dynamics, the associated optimal control problem is investigated by introducing the concept of model uncertainty on the initial conditions of the operational procedure. At this stage, the notion of convex risk measures, and in particular the class of Fréchet risk measures, is employed in order to quantify the total operational and marginal risk, whereas simultaneously obtaining robust to model uncertainty harvesting strategies.

2022, 9(2): 219-228 doi: 10.3934/jdg.2022005 +[Abstract](325) +[HTML](172) +[PDF](329.25KB)
Abstract:

We study the effects of bias on the gender gap by building a non-linear system of differential equations that model the evolution of the sex distribution in a closed market as a function of disaggregated bias and solve the equations explicitly. Thus, allowing us to make specific claims about the system's behavior that may shed some light on the development of policy geared towards a more equitable workplace.

2021 CiteScore: 3.3