Time-Delayed Game Strategy Analysis Among Japan, Other Nations, and the International Atomic Energy Agency in the Context of Fukushima Nuclear Wastewater Discharge Decision

The paper sets up the delayed replicator system (Eq. 14) and enumerates the eight pure equilibria γ1–γ8 . It then claims that γ4, γ6, and γ8 are the (only) asymptotically stable equilibria under parameter sign conditions listed in Table 3—namely: γ4 if CSJ < CMJ + CDJ; γ6 if CSJ > CMJ + CDJ and CLC < CSC; and γ8 if CLC > CSC and CSJ > CMJ + CDJ + TRJ + CHJ + CLC + IJ . In the appendix-style linearizations, the paper treats the delay characteristic equations of the form λ = a e^{−λτ} as if the “eigenvalue sign” were simply the sign of a (e.g., for γ4 it asserts stability “when CSJ < CMJ + CDJ and CII < CHJ” after writing dx/dt = (CSJ−CMJ−CDJ)x(t−τ), dy/dt = −CHJ y(t−τ), dz/dt = −CII z(t−τ)) and similarly for γ6 and γ8 . This reasoning omits the standard delay-dependent stability constraint for the scalar DDE u′(t) = a u(t−τ): asymptotic stability requires a < 0 and |a| τ < π/2. Ignoring the τ-threshold (and even introducing an extra, non-derivable “CII < CHJ” condition for γ4 in one place) leaves the paper’s argument incomplete and internally inconsistent with its own summary table. By contrast, the model’s solution correctly linearizes each corner into three decoupled scalar DDEs, applies the classical stability criterion, recovers exactly the paper’s sign conditions for γ4, γ6, and γ8, and adds the necessary small-delay bounds τ < min_i π/(2|a_i|). Hence the model’s solution is correct while the paper’s proof is incomplete.