One-Step-Ahead Optimal Control vs Optimal Farsighted Control in a Dynamic Pollution Game

This paper studies how myopic and farsighted decision rules affect emissions, pollution dynamics, and payoffs in a discrete-time linear-quadratic pollution game with two players. We compare three behavioral regimes: a symmetric regime in which both players use one-step-ahead optimal control, a symmetric regime in which both players use farsighted feedback strategies, and an asymmetric regime in which a farsighted player interacts with a myopic opponent. The myopic regime is characterized by one-step-ahead payoff maximization, while the farsighted regimes are analyzed using dynamic optimization methods based on Bellman and Euler equations. The analysis characterizes the induced control rules and pollution trajectories and uses benchmark numerical experiments to compare the resulting discounted payoffs. The results suggest that farsighted behavior can improve payoffs in the symmetric benchmark case, whereas behavioral asymmetry may redistribute gains toward the farsighted player. At the same time, the comparison shows that conclusions about myopic behavior depend on the strategy class, the time horizon, and the parameter configuration. The paper therefore highlights the need to distinguish carefully between myopic decision rules and farsighted optimal control when evaluating dynamic pollution policies.