A Fractional Calculus Approach to Modeling Fractal Dynamic Games

Motivated by the complexity of spatio-temporal patterns of interconnected human processes (e.g., crowds, car traffic, social networks), this paper sets forth the fractal dynamic games as an analytical tool for modeling and predicting human dynamics. Starting from a statistical physics description of interactions between agents and from the observed statistical properties of economic measures, we construct a master equation characterizing the dynamics of cost functionals as stochastic variables affected by additive and multiplicative noise forces. Given the significance of human behavior, we allow the cost distribution to depend on the evolution of agents density. By coupling the description of agent dynamics through a fractal structure with a generic stochastic utility function, we formulate a new dynamic game. Employing optimal control theory concepts, we derive a continuum formulation of the car traffic dynamics optimization resulting in a nonlinear fractional partial differential equation.