Optimistic and pessimistic approaches for cooperative games

Abstract

Cooperative game theory explores how to fairly allocate the joint value generated by a group of decision-makers, but its application is compromised by the large number of counterfactuals needed to compute the value of all coalitions, a problem made even more complicated when externalities are present. We provide a theoretical foundation for a simplification used in many applications, in which the value of a coalition is computed assuming that they either select before or after the complement set of agents, providing optimistic and pessimistic values on what a coalition should receive. In a vast set of problems exhibiting what we call feasibility externalities, we show that ensuring a coalition does not receive more than its optimistic value is always at least as difficult as ensuring it receives its pessimistic value. Furthermore, under the presence of negative externalities, we establish the existence of stable allocations that respect these bounds. Finally, we examine well-known optimization-based applications and their corresponding cooperative games to show how our results lead to new insights and allow the derivation of further results from the existing literature.