Resumen de Potential, Value and Probability

Federico Valenciano Llovera , Annick Laruelle

• From Shapley�s (1953) seminal paper a copious family of �solutions� has grown in different directions with many ramifications1 : nonsymmetric values, semivalues, weak semivalues, weighted weak semivalues, probabilistic values, least square values, coalitional values, coalitional semivalues, etc., as well as NTU or non atomic extensions of some of these notions, or their restriction to special subdomains as, e.g., simple games. In fact, a lot of energy in cooperative game theory has been and is still devoted to the study of these ramifications, and the search of new �axiomatic� characterizations of these objects, be it the very seminal concept or any of its extensions around. A weak point in the literature concerned with these extensions is the lack of a clear motivating goal, apart from the purely mathematical motivation (dropping axioms and seeing what happens, finding weaker or more appealing ones, etc.). And as a rule the �stories� behind these notions are not very compelling to tell the truth.

  One of the stories that are told in connection with most of these notions is one in probabilistic terms. Some of these notions are covered by Weber�s notion of �group value�. Nevertheless this covering requires a different probabilistic model for every case: the formation of the grand coalition in a certain order, all orders being or not equally probable; a vector of probabilistic assessments of the players of the coalition every player will join with different �consistency� requirements, etc.