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Journal of Dynamics & Games

January 2015 , Volume 2 , Issue 1

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Nash Equilibrium in evolutionary competitive models of firms and workers under external regulation
Elvio Accinelli, Bruno Bazzano, Franco Robledo and Pablo Romero
2015, 2(1): 1-32 doi: 10.3934/jdg.2015.2.1 +[Abstract](630) +[PDF](1289.6KB)
Abstract:
The object of this paper is to study the labor market using evolutionary game theory as a framework. The entities of this competitive model are firms and workers, with and without external regulation. Firms can either innovate or not, while workers can either be skilled or not. Under the most simple model, called normal model, the economy rests in a poverty trap, where workers are not skilled and firms are not innovative. This Nash equilibria is stable even when both entities follow the optimum strategy in an on-off fashion. This fact suggests the need of an external agent that promotes the economy in order not to fall in a poverty trap.
    Therefore, an evolutionary competitive model is introduced, where an external regulator provides loans to encourage workers to be skilled and firms to be innovative. This model includes poverty traps but also other Nash equilibria, where firms and workers are jointly innovative and skilled.
    The external regulator, in a three-phase process (loans, taxes and inactivity) achieves a common wealth, with a prosperous economy, with innovative firms and skilled workers.
Hamiltonian evolutionary games
Hassan Najafi Alishah and Pedro Duarte
2015, 2(1): 33-49 doi: 10.3934/jdg.2015.2.33 +[Abstract](856) +[PDF](690.7KB)
Abstract:
We introduce a class of o.d.e.'s that generalizes to polymatrix games the replicator equations on symmetric and asymmetric games. We also introduce a new class of Poisson structures on the phase space of these systems, and characterize the corresponding subclass of Hamiltonian polymatrix replicator systems. This extends known results for symmetric and asymmetric replicator systems.
On noncooperative $n$-player principal eigenvalue games
Getachew K. Befekadu and Panos J. Antsaklis
2015, 2(1): 51-63 doi: 10.3934/jdg.2015.2.51 +[Abstract](610) +[PDF](447.8KB)
Abstract:
We consider a noncooperative $n$-player principal eigenvalue game which is associated with an infinitesimal generator of a stochastically perturbed multi-channel dynamical system -- where, in the course of such a game, each player attempts to minimize the asymptotic rate with which the controlled state trajectory of the system exits from a given bounded open domain. In particular, we show the existence of a Nash-equilibrium point (i.e., an $n$-tuple of equilibrium linear feedback operators) in a game-theoretic setting that is connected to a maximum closed invariant set of the corresponding deterministic multi-channel dynamical system, when the latter is composed with this $n$-tuple of equilibrium linear feedback operators.
Discrete time dynamic oligopolies with adjustment constraints
Chrystie Burr, Laura Gardini and Ferenc Szidarovszky
2015, 2(1): 65-87 doi: 10.3934/jdg.2015.2.65 +[Abstract](798) +[PDF](535.9KB)
Abstract:
A classical $n$-firm oligopoly is considered first with linear demand and cost functions which has a unique equilibrium. We then assume that the output levels of the firms are bounded in a sense that they are unwilling to make small changes, the output levels are bounded from above, and if the optimal output level is very small then the firms quit producing, which are realistic assumptions in real economies. In the first part of the paper, the best responses of the firms are determined and the existence of infinitely many equilibria is verified. The second part of the paper examines the global dynamics of the duopoly version of the game. In particular we study the stability of the system, the bifurcations which can occur and the basins of attraction of the existing attracting sets, as a function of the speed of adjustment parameter.
Discrete time mean field games: The short-stage limit
Juan Pablo Maldonado López
2015, 2(1): 89-101 doi: 10.3934/jdg.2015.2.89 +[Abstract](760) +[PDF](370.0KB)
Abstract:
In this note we provide a model for discrete time mean field games. Our main contributions are an explicit approximation in the discounted case and an approximation result for a mean field game with short-stage duration.
Reversibility and oscillations in zero-sum discounted stochastic games
Sylvain Sorin and Guillaume Vigeral
2015, 2(1): 103-115 doi: 10.3934/jdg.2015.2.103 +[Abstract](818) +[PDF](364.9KB)
Abstract:
We show that by coupling two well-behaved exit-time problems one can construct two-person zero-sum dynamic games having oscillating discounted values. This unifies and generalizes recent examples of stochastic games with finite state space, due to Vigeral (2013) and Ziliotto (2013).

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