Group-Based State Potential Games

Facing different environments, i.e., states, there will be different game relationships among players, but there will also exist interactions among players to form a group. This article focuses on presenting the group-based state potential game (GSPG) and the strongly GSPG (SGSPG) to better describe such situations. Players who interact with each other can be regarded as a group and they will update strategies simultaneously. First, the concept of (strongly) GSPG is given and how to design two kinds of games is shown through the semi-tensor product (STP) of matrices. Subsequently, it is confirmed that their corresponding dynamics will converge almost surely to a strategy invariant set of (strongly) group-based recurrent state equilibriums and remain stable. Meanwhile, the equilibriums are acquired based on the algebraic forms of the games' dynamics, and the algorithm of the (strongly) GSPG process is established. Finally, the results in this article are demonstrated effectively by a typical example.