Quantitative comparison of closed- and open-loop linear-quadratic $N$-player differential games

RESUMO

We compare closed-loop and open-loop Nash equilibria in a finite-horizon stochastic linear-quadratic $N$-player game with decoupled state dynamics and interaction through the state costs. We introduce a block-diagonal reference game and a nearby perturbed game, and study each under both information structures. The closed-loop and open-loop problems lead to different Riccati systems, but for the reference game their induced equilibrium state and control processes coincide exactly. We then prove solvability and stability under weak perturbations of the state costs, yielding quantitative bounds between the corresponding equilibria. In particular, when the perturbation decreases sufficiently fast with the population size, the closed-loop and open-loop equilibria of the perturbed game become asymptotically equivalent. The comparison is carried out directly at the finite-player level, without requiring exchangeability or a mean-field limit.