A New Value for Cooperative Games on Intersection-Closed Systems

TL;DR

This paper introduces the uniform-dividend value (UD-value), a new allocation rule for incomplete cooperative games on intersection-closed systems, providing a unique and interpretable solution with desirable fairness properties.

Contribution

The paper proposes the UD-value, a novel allocation rule for incomplete cooperative games, with axiomatic characterization and analysis of its properties and relationships to existing rules.

Findings

UD-value is uniquely determined for intersection-closed systems.

UD-value can be interpreted as the expected Shapley value over positive extensions.

Experiments show UD-value and R-value are often closer to each other than to IC-value.

Abstract

We introduce a new allocation rule, the uniform-dividend value (UD-value), for cooperative games whose characteristic function is incomplete. The UD-value assigns payoffs by distributing the total surplus of each family of indistinguishable coalitions uniformly among them. Our primary focus is on set systems that are intersection-closed, for which we show the UD-value is uniquely determined and can be interpreted as the expected Shapley value over all positive (i.e., nonnegative-surplus) extensions of the incomplete game. We compare the UD-value to two existing allocation rules for intersection-closed games: the R-value, defined as the Shapley value of a game that sets surplus of absent coalition values to zero, and the IC-value, tailored specifically for intersection-closed systems. We provide axiomatic characterizations of the UD-value motivated by characterizations of the IC-value and discuss further properties such as fairness and balanced contributions. Further, our experiments suggest that the UD-value and the R-value typically lie closer to each other than either does to the IC-value. Beyond intersection-closed systems, we find that while the UD-value is not always unique, a surprisingly large fraction of non-intersection-closed set systems still yield a unique UD-value, making it a practical choice in broader scenarios of incomplete cooperative games.