  This paper explores strategic network formation under incomplete information
using data from a single large network. We allow the utility function to be
nonseparable in an individual's link choices to capture the spillover effects
from friends in common. In a network with n individuals, the nonseparable
utility drives an individual to choose between 2^{n-1} overlapping portfolios
of links. We develop a novel approach that applies the Legendre transform to
the utility function so that the optimal decision of an individual can be
represented as a sequence of correlated binary choices. The link dependence
that results from the preference for friends in common is captured by an
auxiliary variable introduced by the Legendre transform. We propose a two-step
estimator that is consistent and asymptotically normal. We also derive a
limiting approximation of the game as n grows large that can help simplify the
computation in very large networks. We apply these methods to favor exchange
networks in rural India and find that the direction of support from a mutual
link matters in facilitating favor provision.
