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.

💡 Research Summary

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This paper investigates how to define coalition values in cooperative games when the actions of some players affect the feasible actions of others—a situation the authors term “feasibility externalities.” Each player i chooses an action from a set A_i; the set of actions that a coalition S can jointly implement, given the actions of the complement N\S, is denoted f_S(a_{N\S}) ⊆ A_S. The authors distinguish two types of feasibility externalities: negative (others’ actions shrink a coalition’s feasible set) and positive (others’ actions enlarge it).

Within this general framework the authors introduce two extreme value functions: the optimistic value, obtained when the complement behaves in the most favorable way for the coalition (i.e., the coalition enjoys the largest feasible set), and the pessimistic value, obtained when the complement behaves in the most adverse way (i.e., the coalition is left with the smallest feasible set). A central question is whether these optimistic/pessimistic values coincide with the outcomes of a sequential game in which a coalition moves first (chooses its actions before the complement) or last (chooses after the complement).

The main theoretical contributions are as follows:

1. Negative feasibility externalities – When other players’ actions can only reduce a coalition’s feasible set, the optimistic value exactly equals the “first‑move” outcome, while the pessimistic value equals the “last‑move” outcome. Consequently, the anti‑core of the optimistic game (allocations that never exceed the optimistic upper bounds) is a subset of the core of the pessimistic game (allocations that never fall below the pessimistic lower bounds). Moreover, both sets are guaranteed to be non‑empty, providing a universal existence result for stable allocations under negative externalities.

2. Positive feasibility externalities – Here the pessimistic value corresponds to the first‑move outcome, but the optimistic value does not always correspond to the last‑move outcome. The inclusion A(optimistic) ⊆ C(pessimistic) holds only if, for every coalition, the optimistic value does happen to be the last‑move outcome; otherwise the relationship can fail. In addition, the core and anti‑core may be empty, so existence of stable allocations cannot be assured without further assumptions.

3. Duality condition – The authors give a sufficient condition for the optimistic and pessimistic games to be dual to each other: the sequential decisions of a coalition moving first and its complement moving last must always lead to the grand‑coalition optimum. When this holds, the anti‑core of the optimistic game coincides with the core of the pessimistic game, mirroring classic duality results for TU games.

4. Implications for classic applications – The framework is applied to a variety of well‑studied problems: queueing systems, minimum‑cost spanning‑tree problems, river‑sharing, bankruptcy‑claims, and airport‑cost allocation. In each case the authors either recover known core results (e.g., the classic queueing core) or derive new insights (e.g., the possibility of an empty core under positive externalities in joint‑production with increasing returns). The paper also shows how optimistic/pessimistic approaches can generate dual games in the bankruptcy and airport settings, provided the duality condition is satisfied.

5. Relation to existing literature – The work unifies and extends earlier concepts such as the α‑ and β‑games, the minimarg/maximarg operators of Curiel and Tijs, and the partition‑function form literature. Unlike those approaches, which often start from a given value function and then construct operators, this paper starts from the underlying optimization problem and asks how to define the value functions themselves in the presence of feasibility externalities.

Overall, the paper delivers a comprehensive theory linking the sign of feasibility externalities to the correspondence between optimistic/pessimistic values and sequential move order, to inclusion relations between core and anti‑core, and to existence guarantees. By doing so, it provides both a clean analytical tool for researchers studying cooperative games with externalities and practical guidance for designing allocation mechanisms in settings where the timing of actions matters.