Equilibrium and Stability Analysis of Discrete Decision Models with Heterogeneous Players

Description

We investigate Nash equilibrium strategies in a discrete decision model with heterogeneous players whose utilities incorporate both internal and external effects. We first characterize all pure Nash equilibria and classify them into coherent and split equilibria. Subsequently, we derive necessary and sufficient conditions for the existence of mixed Nash equilibria and establish their relationship with the fixed points of a smooth best-response dynamical system. We study the evolution of players' decisions by formulating a nonlinear discrete-time dynamical system based on the logit best-response rule. We derive conditions for local asymptotic stability of the. Moreover, by employing the Banach contraction principle, we establish the existence and uniqueness of a globally asymptotically stable equilibrium and provide an explicit geometric estimate for the rate of convergence of the trajectories toward equilibrium. Finally, we identify critical bifurcation thresholds and investigate the resulting qualitative behavior of the dynamics through numerical simulations, including bifurcation diagrams and Lyapunov exponent analysis, revealing transitions between stable and more complex dynamical regimes. Several examples are presented to illustrate the theoretical results and to demonstrate the rich dynamical behavior generated by the proposed model.