Nonexchangeable logit dynamics with cost constraints: different viewpoints and numerical analysis

Abstract

The logit dynamic, a nonlinear dynamical system, subject to a const constraint with heterogeneous agents is formulated, and its well-posedness is studied from both agent-based (stochastic differential equation: SDE) and probabilistic (Fokker-Planck equation: FPE) viewpoints. The state space of agent actions is a compact domain in a finite-dimensional Euclidean space. The SDE is of the McKean-Vlasov type and is driven by jumps with finite variations, whereas the FPE is a nonlinear integro-differential equation. A key to our mathematical analysis of the FPE is adding a regularization factor into the cost constraint to mitigate the blow-up of the logit function. This property carries over to the McKean-Vlasov SDE. We also present a numerical method based on a naïve finite difference discretization for computing the FPE along with demonstrative computational examples, showing that the regularization method does not critically affect numerical solutions while preventing the breakdown of numerical computation. Finally, we conduct another demonstrative application study in which environmental, energy, and fishery resources intersect.