Delayed information and quantum entanglement in Cournot duopoly: a protocol-invariant boundary for oscillatory instability

We study a continuous-time quantum Cournot duopoly with isoelastic demand and delayed marginal-profit adjustment. A unified representation covers the Li–Du–Massar and Frąckiewicz protocols and separates protocol effects from cost asymmetry. We derive the interior quantum Nash equilibrium, establish positivity, global existence, and boundedness, and obtain the exact characteristic equation. Writing m=c2/c1, the equilibrium is stable for every finite delay whenever 3-22≤m≤3+22, whereas a finite critical delay exists when m<3-22 or m>3+22. In the latter regime, a simple conjugate pair crosses the imaginary axis transversally, identifying the onset of an oscillatory instability. Numerical simulations are consistent with the resulting oscillatory dynamics beyond the stability threshold. The existence of a finite stability threshold depends only on relative marginal costs; entanglement and protocol choice determine its numerical location. Root continuation further supports the analytical results.