Optimistic and pessimistic approaches for cooperative games

TL;DR

This paper analyzes optimistic and pessimistic approaches in cooperative game theory with externalities, revealing their relationships and implications for allocation sets, especially under negative externalities, with applications illustrating these concepts.

Contribution

It establishes the relationship between optimistic and pessimistic values in externality-rich cooperative games and provides conditions for the non-emptiness of allocation sets under negative externalities.

Findings

In negative externalities, the optimistic and pessimistic bounds always exist.

Allocation sets bounded by these extremes are non-empty under negative externalities.

The paper offers new insights and tools for analyzing cooperative games with externalities.

Abstract

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.