The Shapley value for bicooperative games based on the semi-tensor product

Abstract

In cooperative games, players can choose whether or not to participate in the coalition based on their own strategies. Bicooperative games are the generalization of cooperative games. In bicooperative games, players have the option of abstention in addition to “yes” and “no”. Therefore, in cooperative games, distributing the total profit obtained by the participant alliance is one of the most important issues in bicooperative game theory. In this paper, the calculation of the Shapley value for bicooperative games is discussed by using the semi-tensor product of matrices. Firstly, the Shapley matrix of bicooperative games is constructed. Secondly, the Shapley value formula for a bicooperative game is transformed into the product of the characteristic function matrix and the Shapley matrix. Finally, an example is given to demonstrate the main results. The matrix form of Shapley value obtained in this paper simplifies the calculation and provides a new tool for researching bicooperative games.