• STP-based algebraic formulation converts vector payoff functions of multi-objective games into a tractable algebraic form, enabling rigorous equilibrium analysis.
• Two necessary and sufficient conditions are derived to verify whether all players meet their expectations and whether a strategy profile is a Pareto equilibrium.
• A strategy updating rule is designed to analyze finite-step reachability of evolutionary multi-objective games, providing a dynamic perspective on equilibrium attainment.
• Finite-step controllability is achieved by introducing pseudo-players, and a backward search algorithm identifies the shortest evolutionary process and control sequence.