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Journal of Dynamics & Games

January 2021 , Volume 8 , Issue 1

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Generalized intransitive dice: Mimicking an arbitrary tournament
Ethan Akin
2021, 8(1): 1-20 doi: 10.3934/jdg.2020030 +[Abstract](369) +[HTML](211) +[PDF](390.14KB)

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} $\end{document}. Examples are known of intransitive \begin{document}$ 6 $\end{document}-sided dice, i.e. \begin{document}$ D_1 \to D_2 \to D_3 $\end{document} but \begin{document}$ D_3 \to D_1 $\end{document}. A tournament of size \begin{document}$ n $\end{document} is a choice of direction \begin{document}$ i \to j $\end{document} for each edge of the complete graph on \begin{document}$ n $\end{document} vertices. We show that if \begin{document}$ R $\end{document} is tournament on the set \begin{document}$ [n] = \{ 1, \dots, n \} $\end{document}, then for sufficiently large \begin{document}$ N $\end{document} there exist sets of independent \begin{document}$ N $\end{document}-sided dice \begin{document}$ \{ D_1, \dots, D_n \} $\end{document} such that \begin{document}$ D_i \to D_j $\end{document} if and only if \begin{document}$ i \to j $\end{document} in \begin{document}$ R $\end{document}.

Permanence in polymatrix replicators
Telmo Peixe
2021, 8(1): 21-34 doi: 10.3934/jdg.2020032 +[Abstract](296) +[HTML](122) +[PDF](2798.88KB)

Generally a biological system is said to be permanent if under small perturbations none of the species goes to extinction. In 1979 P. Schuster, K. Sigmund, and R. Wolff [15] introduced the concept of permanence as a stability notion for systems that models the self-organization of biological macromolecules. After, in 1987 W. Jansen [9], and J. Hofbauer and K. Sigmund [6] give sufficient conditions for permanence in the replicator equations. In this paper we extend these results for polymatrix replicators.

A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions
Laura Aquilanti, Simone Cacace, Fabio Camilli and Raul De Maio
2021, 8(1): 35-59 doi: 10.3934/jdg.2020033 +[Abstract](301) +[HTML](128) +[PDF](505.41KB)

Finite mixture models are an important tool in the statistical analysis of data, for example in data clustering. The optimal parameters of a mixture model are usually computed by maximizing the log-likelihood functional via the Expectation-Maximization algorithm. We propose an alternative approach based on the theory of Mean Field Games, a class of differential games with an infinite number of agents. We show that the solution of a finite state space multi-population Mean Field Games system characterizes the critical points of the log-likelihood functional for a Bernoulli mixture. The approach is then generalized to mixture models of categorical distributions. Hence, the Mean Field Games approach provides a method to compute the parameters of the mixture model, and we show its application to some standard examples in cluster analysis.

A note on the lattice structure for matching markets via linear programming
Pablo Neme and Jorge Oviedo
2021, 8(1): 61-67 doi: 10.3934/jdg.2021001 +[Abstract](257) +[HTML](105) +[PDF](305.34KB)

Given two stable matchings in a many-to-one matching market with \begin{document}$ q $\end{document}-responsive preferences, by manipulating the objective function of the linear program that characterizes the stable matching set, we compute the least upper bound and greatest lower bound between them.

A dynamic for production economies with multiple equilibria
Elvio Accinelli and Humberto Muñiz
2021, 8(1): 69-99 doi: 10.3934/jdg.2021002 +[Abstract](319) +[HTML](108) +[PDF](3714.72KB)

In this article, we extend to private ownership production economies, the results presented by Bergstrom, Shimomura, and Yamato (2009) on the multiplicity of equilibria for the special kind of pure-exchanges economies called Shapley-Shubik economies. Furthermore, a dynamic system that represents the changes in the distribution of the firms on the production branches is introduced. For the first purpose, we introduce a particular, but large enough, production sector to the Shapley-Shubik economies, for which a simple technique to build private-ownership economies with a multiplicity of equilibria is developed. In this context, we analyze the repercussions on the behavior of the economy when the number of possible equilibria changes due to rational decisions on the production side. For the second purpose, we assume that the rational decisions on the production side provoke a change in the distribution of the firms over the set of branches of production.



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