Abstract: Multi-objective games (MOGs) have received much attention in recent years as a class of games with vector payoffs. Based on the semi-tensor product (STP), this paper discusses the MOG, including the existence, finite-step reachability, and finite-step controllability of Pareto equilibrium of this model, from both static and dynamic perspectives. First, the MOG concept is presented using multi-layer graphs, and STP is used to convert the payoff function into its algebraic form. Then, from the static perspective, two necessary and sufficient conditions are proposed to verify whether all players can meet their expectations and whether the strategy profile is a Pareto equilibrium, separately. Furthermore, from the dynamic perspective, a strategy updating rule is designed to investigate the finite-step reachability of the evolutionary MOG. Finally, the finite-step controllability of the evolutionary MOG is analyzed by adding pseudo-players, and a backward search algorithm is provided to find the shortest evolutionary process and control sequence.

Key words: Multi-objective game; Pareto equilibrium; Semi-tensor product; Finite-step reachability; Finite-step controllability

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