Cooperative game theory is a specialized field within game theory that explores how individuals can collaborate to achieve mutually beneficial outcomes. Over time, researchers have expanded this concept to encompass higher dimensions, offering new perspectives. In 2000, Bilbao et al. introduced the notion of bi-cooperative games, which represents a broader extension of traditional cooperative games. Unlike the conventional approach of utilizing functions defined from \(2^N\) to \({\mathbb {R}}\) , bi-cooperative games introduce a distinctive viewpoint by employing functions defined from \(3^N\) to \({\mathbb {R}}\) . In 2024, M. Slime et al. further expanded cooperative games, focusing on scenarios where players choose actions from the Cartesian product of two sets. They extended the scope of cooperative games by utilizing functions defined from \(2^N\) to \({\mathbb {R}}\) , to a bi-dimensional cooperative game framework with functions defined from \(2^{N \times N'}\) to \({\mathbb {R}}\) . Building upon these developments, our paper contributes to the advancement of bi-cooperative games by extending their application to the Cartesian product \(N \times N'\) . This extension introduces a new layer of complexity, capturing interactions not only within a single set N, but also between two sets N and \(N'\) . The functions are now defined from \(3^{N \times N'}\) to \({\mathbb {R}}\) , reflecting the expanded range of possibilities and scenarios.
