Exploring multi-potential games in strategic form: A graph theoretic approach

Multi-potential games (MPGs), wherein the facility cost functions of players are distinct, significantly expand the scope of traditional potential games. This paper explores the payoff-related structure properties within MPGs, with a focus on the categorization of players according to their conflicting interests. Initially, the study establishes a necessary and sufficient condition to determine if a finite non-cooperative game qualifies as a potential game. Subsequently, it demonstrates that players with conflicting interests cannot share the same potential function. Leveraging the principles of graph theory in alignment with players' conflicting interests, the study identifies the minimal potential index in MPGs, which provides the construction of all possible player partitions. Finally, the study examines a network game affected by external payoff matrix perturbations. It demonstrates how MPGs can effectively illuminate the intricate connection of common and conflicting interests among players.