Balanced Externalities and the Proportional Allocation of Nonseparable Contributions

In this paper, we study the implications of extending the balanced cost reduction property from queueing problems to general games. As a direct translation of the balanced cost reduction property, the axiom of balanced externalities for solutions of games, requires that the payoff of any player is equal to the total externality she inflicts on the other players with her presence. We show that this axiom and efficiency together characterize the Shapley value for 2-additive games. However, extending this axiom in a straightfoward way to general games is incompatible with efficiency. Keeping as close as possible to the idea behind balanced externalities, we weaken this axiom by requiring that every player's payoff is the same fraction of its total externality inflicted on the other players. This weakening, which we call weak balanced externalities, turns out to be compatible with efficiency. More specifically, the unique efficient solution that satisfies this weaker property is the proportional allocation of nonseparable contribution (PANSC) value, which allocates the total worth proportional to the separable costs of the players. We also provide characterizations of the PANSC value using a reduced game consistency axiom.