A Tractable Class of Cooperative Games Defined by Directed Networks: Unanimity Decomposition and Shapley Value

David Ryz\'ak and Tom\'a\v{s} Kroupa

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Abstract: We introduce a class of cooperative games induced by weighted directed graphs. Specifically, the coalitional value combines an internal interaction term given by the induced subgraph game with an external component based on minimal incoming edges from outside the coalition. The resulting game has a convenient representation in terms of unanimity games. This representation enables closed-form polynomial-time formulas for the Shapley and Banzhaf values. We further establish that the game has a nonempty core and is totally balanced. The class of such games therefore provides an analytically and computationally tractable example of structured network- induced cooperative games in which stability-based allocations and fairness-based solution concepts do not coincide.

Date: 2026-05
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