# American Institute of Mathematical Sciences

ISSN:
2164-6066

eISSN:
2164-6074

All Issues

## Journal of Dynamics & Games

April 2020 , Volume 7 , Issue 2

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2020, 7(2): 97-104 doi: 10.3934/jdg.2020006 +[Abstract](864) +[HTML](195) +[PDF](319.11KB)
Abstract:

We consider a sufficient condition for the uniqueness of a Nash equilibrium in strategic-form games: for any two distinct strategy profiles, there is a player who can obtain a higher payoff by unilaterally changing the strategy from one strategy profile to the other strategy profile. An example of a game that satisfies this condition is the prisoner's dilemma. Viewed as a solution concept, the Nash equilibrium satisfying the condition is stronger than strict Nash Equilibrium and weaker than strict dominant strategy equilibrium.

2020, 7(2): 105-122 doi: 10.3934/jdg.2020007 +[Abstract](772) +[HTML](207) +[PDF](392.74KB)
Abstract:

In this paper, we present a systematic overview of different endogenous optimization-based characteristic functions and discuss their properties. Furthermore, we define and analyze in detail a new, \begin{document}$\eta$\end{document}-characteristic function. This characteristic function has a substantial advantage over other characteristic functions in that it can be obtained with a minimal computational effort and has a reasonable economic interpretation. In particular, the new characteristic function can be seen as a reduced version of the classical Neumann-Morgenstern characteristic function, where the players both from the coalition and from the complementary coalition use their previously computed strategies instead of solving respective optimization problems. Our finding are illustrated by a pollution control game with \begin{document}$n$\end{document} non-identical players. For the considered game, we compute all characteristic functions and compare their properties. Quite surprisingly, it turns out that both the characteristic functions and the resulting cooperative solutions satisfy some symmetry relations.

2020, 7(2): 123-140 doi: 10.3934/jdg.2020008 +[Abstract](811) +[HTML](205) +[PDF](818.38KB)
Abstract:

The possibility that a discrete process can be fruitfully approximated by a continuous one, with the latter involving a differential system, is fascinating. Important theoretical insights, as well as significant computational efficiency gains may lie in store. A great success story in this regard are the Navier-Stokes equations, which model many phenomena in fluid flow rather well. Recent years saw many attempts to formulate more such continuous limits, and thus harvest theoretical and practical advantages, in diverse areas including mathematical biology, economics, finance, computational optimization, image processing, game theory, and machine learning.

Caution must be applied as well, however. In fact, it is often the case that the given discrete process is richer in possibilities than its continuous differential system limit, and that a further study of the discrete process is practically rewarding. Furthermore, there are situations where the continuous limit process may provide important qualitative, but not quantitative, information about the actual discrete process. This paper considers several case studies of such continuous limits and demonstrates success as well as cause for caution. Consequences are discussed.

2020, 7(2): 141-162 doi: 10.3934/jdg.2020009 +[Abstract](691) +[HTML](193) +[PDF](408.87KB)
Abstract:

We consider difference and differential Stackelberg game theoretic models with several followers of opinion control in marketing networks. It is assumed that in the stage of analysis of the network its opinion leaders have already been found and are the only objects of control. The leading player determines the marketing budgets of the followers by resource allocation. In the basic version of the models both the leader and the followers maximize the summary opinions of the network agents. In the second version the leader has a target value of the summary opinion. In all four models we have found the Stackelberg equilibrium and the respective payoffs of the players analytically. It is shown that the hierarchical control system is ideally compatible in all cases.

2020, 7(2): 163-174 doi: 10.3934/jdg.2020010 +[Abstract](776) +[HTML](217) +[PDF](311.26KB)
Abstract:

A three-dimensional continuous-time stochastic model based on the classic Kermack-McKendrick model for the spread of epidemics is proposed for the propagation of a computer virus. Moreover, control variables are introduced into the model. We look for the controls that either minimize or maximize the expected time it takes to clean the infected computers, or to protect them from the virus. Using dynamic programming, the equations satisfied by the value functions are derived. Particular problems are solved explicitly.