A new value for cooperative games on intersection-closed systems

Abstract

In the model of cooperative games with restricted cooperation, certain coalitions are infeasible, meaning they cannot form, which directly influences payoff allocation. We consider a scenario where coalitions are no longer categorized as feasible or infeasible, but rather as known or unknown. In this setting, coalitions with unknown values are still feasible, but their values remain unknown in the allocation process. Classical allocation methods for games with restricted cooperation, such as the R-value, become unsuitable for such scenarios.

We introduce a new allocation rule called the uniform-dividend value (UD-value), designed specifically for cooperative games where some coalition values remain unknown (so-called incomplete cooperative games). The UD-value allocates payoffs by evenly distributing the total surplus within each group of indistinguishable coalitions. We demonstrate that for intersection-closed set systems, the UD-value is uniquely determined and can also be viewed as the expected Shapley value computed over all totally positive (i.e., nonnegative-surplus) extensions of the incomplete cooperative 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. Specifically, 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.