Algebraic Verification of Finite Group-Based Potential Games with Vector Payoffs

This paper studies a class of strategic games,where players often collaborate with other players to form a group when making decisions,and the payoff functions of players in such games are presented as vector functions.First,using the semi-tensor product(STP)method,it is proved that a finite game with vector payoffs is potential if and only if its potential equation has solution.By adding a suitable weight vector to the vector payoffs of each player,a finite game with vector payoffs that is not potential can be converted into a potential game.Second,as a natural generalization,the authors consider the verification problem of the group-based potential games with vector payoffs.By solving a linear potential equation,a simple formula is obtained to calculate the corresponding potential function.Finally,some examples are presented and discussed in detail to illustrate the theoretical results.