Combining the Shapley value and the equal division solution: an overview

We give an overview of several results on the axiomatization and strategic implementation of cooperative game solutions that combine marginalism and egalitarianism. The best-known marginalist solution for cooperative transferable utility games is the Shapley value, while the equal division solution is the most direct egalitarian solution. Both solutions have been axiomatized in various ways, starting with Shapley (1953)’s original axiomatization of his value. His axiomatization heavily relies on a basis for the game space, specifically the unanimity (game) basis. We review several contributions from the literature where using a different basis, and using similar axioms as for the Shapley value, gives the equal division solution or other solutions that combine features of the Shapley value and the equal division solution. We specifically focus on the classes of egalitarian Shapley values and discounted Shapley values. Besides axiomatizations, we discuss strategic implementation of these solutions, where the difference between the mechanisms implementing the different solutions is with respect to the possibility of the negotiations to break down.