Resumen de A survey on the potential approach in cooperative game theory

Theo S.H. Driessen

• In 1989, the discrete potential approach to the solution theory for cooperative TU-games was introduced by Hart and Mas-Colell resulting in the characterization of the known Shapley value as the unique efficient value that admits a potential representation (prescribed by the discrete gradient of a so-called potential function). In 1997, Calvo and Santos established that the Shapley value is the very best representative of the family of (not necessarily efficient) values that admit the (standard) potential representation. Our main goal is to survey the main recent results in the next two different extensions of the standard potential approach:

  (1) In the framework of the so-called weighted pseudo-potential approach to efficient values, the revised representation of the value may incorporate, besides a fraction of the discrete gradient, a fraction of the underlying pseudo-potential function itself, as well as a fraction of the average of all the components of the gradient. Concerning the class of efficient values that admit a pseudo-potential representation, a number of appealing examples are put forward and moreover, a characterization of the full class of such values is stated (e.g., by linearity).

  (2) The standard efficiency principle refers to the addition (sum) of payoffs to the single players, whereas the (standard) potential representation of a player�s value in a game refers to the subtraction (difference) of two evaluations by the potential function (at the initial game and an induced subgame). It is shown that the equivalence theorem by Calvo and Santos can be revised whenever the efficiency principle and potential representation are revised in the sense that the addition (and subtraction) are replaced by multiplication (and quotient).

  In even stronger words, no change of the several equivalence theorems happens in case an arbitrary binary group operator on the set of real numbers (which guarantees the existence of the inverse of any real number) replaces the additive or multiplicative operator. Hence, in the setting of a group structure, a counterpart of the standard Shapley value arises as the very best representative of the family of (not necessarily group-efficient) values that admit the potential representation (with reference to the underlying group structure).