Optimistic and pessimistic approaches for cooperative games

Cooperative game theory studies how to allocate the joint value generated by a set of players. These games are typically analyzed using the characteristic function form with transferable utility, which represents the value attainable by each coalition. In the presence of externalities, coalition values can be defined through various approaches, notably by trying to determine the best and worst-case scenarios. Typically, the optimistic and pessimistic perspectives offer valuable insights into strategic interactions. In many applications, these approaches correspond to the coalition either choosing first or choosing after the complement coalition. In a general framework in which the actions of a group affects the set of feasible actions for others, we explore this relationship and show that it always holds in the presence of negative externalities, but only partly with positive externalities. We then show that if choosing first/last corresponds to these extreme values, we also obtain a useful inclusion result: allocations that do not allocate more than the optimistic upper bounds also do not allocate less than the pessimistic lower bounds. Moreover, we show that when externalities are negative, it is always possible to guarantee the non-emptiness of these sets of allocations. Finally, we explore applications to illustrate how our findings provide new results and offer a means to derive results from the existing literature.