Modified potential approach to efficient, linear and symmetric values for TU-games

The potential approach is a useful tool in physics, and the successful treatment of such concept turned out to be reproducible, in the late eighties, in the field of cooperative game theory. This report is devoted to the modified potential approach for values on cooperative games, particularly for the Shapley value, the Solidarity value, and the class of values satisfying efficiency, linearity and symmetry (ELS values). Chapter 1 introduces the basic concepts in the field of game theory, especially in the cooperative part. Besides the set-valued solutions, such as the imputation set and the stable set, we primarily discuss the Shapley value and the Solidarity value, both of which are single-valued solutions. After presenting several properties of solutions using in axiomatization, two equivalent forms of ELS values are studied in particular. Concerning the noncooperative part, games in normal form and the famous solution concept–Nash equilibrium are discussed. In Chapter 2, the classical potential approach, which depict the equivalence between the classical gradient and the Shapley value, is introduced. In view of the classical potential, we consider a more general concept called the modified potential, which also satisfies the 0-normalized property, but with a modified gradient in the efficiency condition. Concerning this new concept, a value possesses a modified potential representation only when it equals to the modified gradient. For the ELS value, in which the Shapley value and the Solidarity value are two special cases, we discuss its sufficient and necessary condition when it admits a modified potential representation, especially in the separable case. In Chapter 3, the Shapley value can be written as a linear combination of the corresponding coordinates based on the unanimity game. In order to simplify such expression, we define another basis analogously to the collection of unanimity games, associated with the Shapley value. Similarly, two new basis of the game space, with respect to the Solidarity value and ELS values respectively, are defined, such that these two values admit a simple sum expression concerning their coordinates. According to these basis, the modified potential admits a new expression, thus the modified potential representation of different values can be verified. In the last section, the concept of the potential game, which belongs to the field of noncooperative game theory, is introduced. We investigate the Solidarity value and the ELS value representations of potential games, which are closely related to its Shapley value representation. Chapter 4 presents the reduced game and the corresponding reduced game property, which says the payoffs of players in a subset should not change or they should have no reason to renegotiate, if they apply the same solution rule among themselves as in the original game. For the Shapley value, Sobolev defined a special reduced game, such that the Shapley value in such (n − 1)-person reduced game equals to that in the original nperson game. By the modified potential approach discussed in Chapter 2, we find the