Power in Sharing Networks with a priori Unions

We introduce and analyze a novel family of power indices tailored for sharing networks in technological markets, where firms operate competitively within, but not across, distinct industrial sectors. In these settings, inter-firm collaboration structures emerge from formal technology licensing agreements. The proposed indices are defined over graphs with a priori unions and combine two key centrality measures - degree-based and rescaled eigenvector centrality - modulated by positive market coefficients that reflect sectoral dynamics. We first explore the monotonicity properties of these indices, highlighting their responsiveness to local changes in network structure. Interestingly, major economic actors exhibit structural stability when inter-sectoral technological spillovers are minimal. Building on these findings, we provide theoretical underpinnings by characterizing the indices as the Shapley values of a family of coherent and economically interpretable transferable utility (TU) games defined over such graphs. However, for a broad class of network structures, the core of these TU games is often empty, signaling inherent instability in technological sharing arrangements. Finally, we offer an axiomatic foundation for this family of indices, proving independence of the proposed axioms. This axiomatization extends naturally to exchange networks, even when stage-propagation coefficients are not positive.