Game theory provides valuable tools to examine expert multi-agent systems. In a cooperative game, collaboration among agents leads to better outcomes. The most important solution for such games is the Shapley value, that coincides with the expected marginal contribution assuming equiprobability. This assumption is not plausible when externalities are present in an expert system. Generalizing the concept of marginal contributions, we propose a new family of Shapley values for situations with externalities. The properties of the Shapley value offer a rationale for its application. This family of values is characterized by extensions of Shapley’s axioms: efficiency, additivity, symmetry, and the null player property. The first three axioms have widely accepted generalizations to the framework of games with externalities. However, different concepts of null players have been proposed in the literature and we contribute to this debate with a new one. The null player property that we use is weaker than the others. Finally, we present one particular value of the family, new in the literature, and characterize it by two additional properties.

Introduction

There are many successful applications of game theoretical tools to study expert or intelligent multi-agent problems (see for instance Parsons and Wooldridge, 2002; Pendharkar, 2012). The classic model of games with transferable utility has been thoroughly studied and today it is a theory with solid foundations. It has been widely applied to economic, social, or political problems binding the gap between these fields and mathematics. In particular, it has endowed social sciences with a formal framework in which meaningful statements can be done. One of the main research questions is how to distribute the gains obtained by a given group of agents. In this regard, the Shapley value (Shapley, 1953) is probably the most popular solution and has been used to study a variety of expert systems (Alonso-Meijide and Carreras, 2011; Torkaman et al., 2011). It is defined as the average contribution of a player to its predecessors in a permutation and supported by appealing axiomatic characterizations. The characterizations provide a normative foundation of the value and play an important role in its applications. Most of the contributions in the literature overlook a key fact in today’s globally interconnected societies, decisions within a group of agents can affect the outcomes of other groups of agents. Thrall and Lucas (1963) devised the partition function to incorporate coalitional externalities to classic cooperative games.