Optimizing high-speed railway (HSR) timetables requires coordinated decisions on train stopping patterns and local feasibility constraints, such as headway separation and service ordering, particularly at stations where consecutive trains share infrastructure. As network size and service density increase, station-level interactions can induce strategic behavior among trains competing for overlapping passenger markets. This study introduces a delayed-feedback evolutionary game framework to model interactions between two trains under incomplete information. Here, “delayed feedback” represents the lag in information transmission and periodic strategy updates, rather than operational train delays. We first formulate replicator dynamics for the non-delayed scenario, deriving equilibrium points and local stability conditions. The model is then extended to include delayed feedback in payoff evaluation, and the impact of delay parameters on stability and convergence is analyzed. To account for operational heterogeneity, two station layouts are considered: (i) two tracks per direction, representing small stations with limited overtaking and stop–pass combinations, and (ii) four tracks per direction, representing large stations that allow simultaneous stopping and richer operational patterns. Numerical simulations examine convergence, oscillatory behavior, and parameter sensitivity. Results indicate that delayed feedback significantly influences system dynamics: small delays maintain convergence to evolutionarily stable strategies, whereas larger delays induce persistent oscillations and complex transient trajectories. Station layout further affects stability regions and long-term strategy profiles. This study is based on analytical derivation and numerical simulation rather than sample-based statistical inference; therefore, non-parametric hypothesis testing is not applicable in the present framework. This framework provides a game-theoretic and stability-oriented tool for station-level timetable analysis, offering methodological guidance for timetable design under delayed decision feedback in HSR operations.
Lei et al. (2026) studied this question.