Balanced Contributions in Networks and Games with Externalities

TL;DR

This paper introduces the BCE rule for balanced contributions in network games with externalities, characterizing a unique, fair, and component-efficient allocation that generalizes existing values.

Contribution

It characterizes the BCE rule satisfying balanced contributions in networks with externalities, bridging gaps with cycle-sum identities and connecting to known solution concepts.

Findings

BCE rule coincides with Myerson value for TU games.

Cycle-sum identity reduces non-tree edge contributions.

BCE recovers externality-free value on complete networks.

Abstract

For networks with externalities, where each component's worth may depend on the full network structure, balanced contributions and fairness lead to distinct component-efficient allocation rules. We characterize the unique component-efficient allocation rule satisfying balanced contributions -- the BCE rule. Existence is the main challenge: balanced contributions must hold on every edge, but the construction uses only spanning-tree edges. A cycle-sum identity bridges this gap by reducing balanced contributions on non-tree edges to relations in proper subnetworks. The BCE rule coincides with the Myerson value for TU games and with its generalization by Jackson--Wolinsky for network games without externalities, it recovers the externality-free value on the complete network, and -- unlike the fairness-based FCE rule -- it does not reduce to a graph-free formula applied to the graph-restricted game.