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We consider a natural generalization of Jackson and Wolinsky's (J Econ Theory 71:44-74, 1996) connections model where the quality or strength of a link depends on the amount invested in it and is determined by a non-decreasing function of that amount. The information that the nodes receive through the network is the revenue from investments in links. We prove that in this most general version of the connections model, the only possibly non-empty efficient networks, in the sense of maximizing the aggregate profit, are still the all-encompassing star and the complete network, with the sole and rare exception of a highly particular case where there is a draw between the allencompassing star, the complete network and a whole range of a particular type of nested split graph structures intermediate between them. Keywords Network formation • Connections model • Nested split graph • Efficiency JEL Classification A14 • C7 • D85 We thank the editor and two anonymous referees for their comments. This research is supported by the Spanish Ministerio de Economía y Competitividad under Projects ECO2015-66027-P and ECO2015-67519-P (MINECO/FEDER). Both authors also receive Basque Government Departamento de Educación, Política Lingüística y Cultura funding for Grupos Consolidados IT1367-19. N. Olaizola, F. Valenciano: BRiDGE Group (http://www.bridgebilbao.es).

This paper provides a new model of network formation that bridges the gap between the two benchmark models by Bala and Goyal, the one-way ‡ow model, and the two-way ‡ow model, and includes both as particular extreme cases. As in both benchmark models, in what we call an "asymmetric ‡ow"network a link can be initiated unilaterally by any player with any other, and the ‡ow through a link towards the player who supports it is perfect. Unlike those models, in the opposite direction there is friction or decay. When this decay is complete there is no ‡ow and this corresponds to the one-way ‡ow model. The limit case when the decay in the opposite direction (and asymmetry) disappears, corresponds to the two-way ‡ow model. We characterize stable and strictly stable architectures for the whole range of parameters of this "intermediate" and more general model. We also prove the convergence of Bala and Goyal's dynamic model in this context. JEL Classi…cation Numbers: A14, C72, D20, J00 Key words: Non-cooperative network formation, Asymmetric ‡ow, Stability, E¢ ciency, Dynamics.We thank Coralio Ballester and Francesco Feri for helpful comments. Of course, the usual disclaimer applies.

This work analyzes a managerial delegation model in which firms that produce a differentiated good can choose between two production technologies: a low marginal cost technology and a high marginal cost technology. For the former to be adopted more investment is needed than for the later. By giving managers of firms an incentive scheme based on a linear combination of profit and sales revenue, we find that Bertrand competition provides a stronger incentive to adopt the cost-saving technology than the strict profit maximization case. However, the results may be reversed under Cournot competition. We show that if the degree of product substitutability is sufficiently low (high), the incentive to adopt the cost-saving technology is larger under strict profit maximization (strategic delegation).JEL Classification: O31, L13