# American Institute of Mathematical Sciences

ISSN:
2164-6066

eISSN:
2164-6074

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## Journal of Dynamics & Games

January 2017 , Volume 4 , Issue 1

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2017, 4(1): 1-23 doi: 10.3934/jdg.2017001 +[Abstract](359) +[HTML](72) +[PDF](579.2KB)
Abstract:

Motivated by recent path-breaking contributions in the theory of repeated games in continuous time, this paper presents a family of discrete-time games which provides a consistent discrete-time approximation of the continuous-time limit game. Using probabilistic arguments, we prove that continuous-time games can be defined as the limit of a sequence of discrete-time games. Our convergence analysis reveals various intricacies of continuous-time games. First, we demonstrate the importance of correlated strategies in continuous-time. Second, we attach a precise meaning to the statement that a sequence of discrete-time games can be used to approximate a continuous-time game.

2017, 4(1): 25-39 doi: 10.3934/jdg.2017002 +[Abstract](371) +[HTML](69) +[PDF](318.9KB)
Abstract:

In this article, we study the Cournot-Theocharis duopoly with variable marginal cost. We present sufficient conditions such that, both firms enter the market at any stage, remain in the market, and maximize their profit at any stage. We suggest cost implementation strategies, under which the market might benefit from the variability of marginal cost. We exhibit strategies, for which, the variability of marginal cost might be hazardous for the duopoly competitors. We prove that there exist cases, in which the market forms cycles of length six. Within each cycle, there is an interchange between monopolies and duopolies. Finally, we present some new ideas to establish monopoly convergence under certain monopoly conditions.

2017, 4(1): 41-58 doi: 10.3934/jdg.2017003 +[Abstract](284) +[HTML](94) +[PDF](444.2KB)
Abstract:

We study the price of anarchy in graph coloring games (a subclass of polymatrix common-payoff games). Players are vertices of an undirected graph, and the strategies for each player are the colors \begin{document} $\left\{ {1, \ldots ,k} \right\}$ \end{document}. A tight bound of \begin{document} $\frac{k}{k-1}$ \end{document} is known (Hoefer 2007, Kun et al. 2013), if each player's payoff is the number of neighbors with different color than herself.

In our generalization, payoff is computed by determining the distance of the player's color to the color of each neighbor, applying a concave function \begin{document} $f$ \end{document} to each distance, and then summing up the resulting values. This is motivated, e. g., by spectrum sharing, and includes the payoff functions suggested by Kun et al. (2013) for future work as special cases.

Denote \begin{document} $f^*$ \end{document} the maximum value that \begin{document} $f$ \end{document} attains on \begin{document} $\left\{ {0, \ldots ,k - 1} \right\}$ \end{document}. We prove an upper bound of \begin{document} $2$ \end{document} on the price of anarchy if \begin{document} $f$ \end{document} is non-decreasing or assumes \begin{document} $f^*$ \end{document} somewhere in \begin{document} $\left\{ {0, \ldots ,{\frac{k}{2}}} \right\}$ \end{document}. Matching lower bounds are given for the monotone case and if \begin{document} $f^*$ \end{document} is assumed in \begin{document} $\frac{k}{2}$ \end{document} for even \begin{document} $k$ \end{document}. For general concave \begin{document} $f$ \end{document}, we prove an upper bound of \begin{document} $3$ \end{document}. We use a new technique that works by an appropriate splitting \begin{document} $\lambda = \lambda_1 + \ldots + \lambda_k$ \end{document} of the bound \begin{document} $\lambda$ \end{document} we are proving.

2017, 4(1): 59-74 doi: 10.3934/jdg.2017004 +[Abstract](256) +[HTML](74) +[PDF](426.1KB)
Abstract:

This paper deals with discrete-time stochastic systems composed of a large number of N interacting objects (a.k.a. agents or particles). There is a central controller whose decisions, at each stage, affect the system behavior. Each object evolves randomly among a finite set of classes, according to a transition law which depends on an unknown parameter. Such a parameter is possibly non observable and may change from stage to stage. Due to the lack of information and to the large number of agents, the control problem under study is rewritten as a game against nature according to the mean field theory; that is, we introduce a game model associated to the proportions of the objects in each class, whereas the values of the unknown parameter are now considered as "actions" selected by an opponent to the controller (the nature). Then, letting \begin{document} $N \to \infty$ \end{document} (the mean field limit) and considering a discounted optimality criterion, the objective for the controller is to minimize the maximum cost, where the maximum is taken over all possible strategies of the nature.

2017, 4(1): 75-86 doi: 10.3934/jdg.2017005 +[Abstract](523) +[HTML](84) +[PDF](289.3KB)
Abstract:

Discretizing a differential equation may change the qualitative behaviour drastically, even if the stepsize is small. We illustrate this by looking at the discretization of a piecewise continuous differential equation that models a population of agents playing the Rock-Paper-Scissors game. The globally asymptotically stable equilibrium of the differential equation turns, after discretization, into a repeller surrounded by an annulus shaped attracting region. In this region, more and more periodic orbits emerge as the discretization step approaches zero.