The Shapley value

Abstract

The paper is a review of most interesting topics on the most renowned one element solution of cooperative games with side payments – the Shapley value. After an elementary introduction, we present the original definition of the Shapley value, basic formula and properties as well as main theorems characterizing the Shapley value by systems of axioms. The form and behaviour of the value of some specific games and types of games is demonstrated. Some applications are described, including those in cost sharing problems and as a power index in group decision making problems modelled as simple games. Also some particularly typical or interesting extensions and generalizations of the Shapley value are presented.