Cooperative game theory aims to study how to divide a joint value created by a set of players. These games are often studied through the characteristic function form with transferable utility, which represents the value obtainable by each coalition. In the presence of externalities, there are many ways to define this value. Various models that account for different levels of player cooperation and the influence of external players on coalition value have been studied. Although there are different approaches, typically, the optimistic and pessimistic approaches provide sufficient insights into strategic interactions. This paper clarifies the interpretation of these approaches by providing a unified framework. We show that making sure that no coalition receives more than their (optimistic) upper bounds is always at least as difficult as guaranteeing their (pessimistic) lower bounds. We also show that if externalities are negative, providing these guarantees is always feasible. Then, we explore applications and show how our findings can be applied to derive results from the existing literature.

翻译：合作博弈理论旨在研究如何分配由一组参与人创造的联合价值。这类博弈通常通过具有可转移效用的特征函数形式进行研究，该函数表示每个联盟可获得的价值。在存在外部性的情况下，有多种定义该价值的方式。针对考虑不同参与人合作水平以及外部参与人对联盟价值影响的各种模型，已有广泛研究。尽管存在不同方法，但通常乐观与悲观方法能对战略互动提供充分见解。本文通过提供统一框架，厘清了这些方法的解读。我们证明：确保任何联盟所得不超过其（乐观）上界，至少与保障其（悲观）下界同等困难。同时指出，若外部性为负，则提供这些保障总是可行的。随后，我们探讨了应用场景，并展示了如何将我们的发现用于推导现有文献中的结论。