A Tractable Class of Cooperative Games Defined by Directed Networks: Unanimity Decomposition and Shapley Value
For researchers in cooperative game theory and network economics, this provides a computationally tractable class of network-induced games with explicit solution formulas.
This paper introduces a class of cooperative games defined by weighted directed graphs, providing closed-form polynomial-time formulas for the Shapley and Banzhaf values via a unanimity decomposition. The games are shown to have a nonempty core and be totally balanced, offering a tractable example where stability and fairness concepts diverge.
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.