Implementation of optimal connection networks

We consider a connection networks model. Every agent has a demand in the form of pairs of locations she wants connected, and a willingness to pay for connectivity. A planner aims at implementing a welfare maximizing network and allocating the resulting cost, but information is asymmetric: agents are fully informed, the planner is ignorant. The options for full implementation in Nash and strong Nash equilibria are studied. We simplify strategy sets without changing the set of Nash implementable correspondences. We show the correspondence of consisting of welfare maximizing networks and individually rational cost allocations is implementable. We construct a minimal Nash implementable desirable solution in the set of upper hemi-continuous and Nash implementable solutions. It is not possible to implement solutions such a the Shapley value unless we settle for partial implementation.