Analyzing cooperative game theory solutions: core and Shapley value in cartesian product of two sets

Abstract

The core and the Shapley value stand out as the most renowned solutions for addressing sharing problems in cooperative game theory. These concepts are widely acknowledged for their effectiveness in tackling negotiation, resource allocation, and power dynamics. The present study aims to extend various notions of cooperative games from the standard set N to a new class of cooperative games defined on the cartesian product N×N′ (utilizing the specific coalition A*B). This extension encompasses fundamental concepts such as rationality, core, and Shapley value. The findings presented in this study demonstrate that the core concept as a solution yields a set of imputations without favoring any specific point within the set, in contrast to the Shapley value, which offers a singular solution. Moreover, the results confirm that the Shapley value satisfies the conditions defining the core of a game. Through both theoretical analysis and numerical findings, employing a practical example, it becomes evident that the Shapley value offers a more distinct solution to the sharing problem compared with the core solution.