Payoff design of network systems with multiple types of interactions via potential game approach

A potential network game with multiple fundamental game models is introduced to study the optimisation of a network system with multiple types of interactions. This paper aims to design individuals' payoff functions such that a network system can become a potential network game with the given system performance index function as its potential function. First, it is proved that a network game is a potential game if and only if all fundamental games are potential games, which is applied to simplify the game modeling. In addition, the algebraic characteristics of payoff functions in symmetric potential games are discussed to simplify the payoff design. Next, a necessary and sufficient condition about the designability of payoff functions is presented, as well as a formula to calculate payoff functions. Finally, the theoretical results are validated by the consensus problem of network systems.