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

WANG Yuanhua1, LI Haitao2

1. 1. Business School, Shandong Normal University, Ji'nan 250014, China;
2. School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250014, China
• Received:2021-03-09 Revised:2021-06-10 Online:2022-11-25 Published:2022-12-23
• Contact: WANG Yuanhua,Email:wyh_1005@163.com
• Supported by:
This research was supported by the National Natural Science Foundation of China under Grant Nos. 61903236, 62073202, and 61803240, Shandong Provincial National Science Foundation under Grant No. ZR2018BF021 and China Postdoctoral Science Foundation under Grant No. 2017M622262.

WANG Yuanhua, LI Haitao. Algebraic Verification of Finite Group-Based Potential Games with Vector Payoffs[J]. Journal of Systems Science and Complexity, 2022, 35(6): 2131-2144.

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.
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