Delayed information and stability fragility in a quantum Bertrand duopoly

We study a quantum Bertrand duopoly in which a boundedly rational firm adjusts its strategy using delayed marginal-profit information, while its rival follows a naive best response. Although the information delay leaves the stationary strategies unchanged, it modifies the dimension and spectral properties of the adjustment dynamics. We derive the characteristic equation governing local stability for a general integer delay and establish explicit necessary and sufficient stability conditions, the critical adjustment threshold, and the critical multipliers for contemporaneous information and for a one-period delay. In the absence of delay, instability occurs through a real multiplier at −1, whereas a one-period delay produces a complex-conjugate crossing of the unit circle. We prove that a one-period information delay strictly reduces the range of adjustment speeds compatible with local stability. For longer integer delays, for which compact closed-form Jury conditions become cumbersome, we provide numerical root-continuation evidence rather than making a general analytical claim. Numerical bifurcation diagrams, Lyapunov exponents, root trajectories, stability maps, time series, projected trajectories, and orbit-separation measures support the analytical results. Finally, an extended entanglement range and an alternative calibration show that the effect of entanglement is not globally monotone, whereas the one-period delay penalty remains pointwise valid.