  We consider any network environment in which the "best shot game" is played.
This is the case where the possible actions are only two for every node (0 and
1), and the best response for a node is 1 if and only if all her neighbors play
0. A natural application of the model is one in which the action 1 is the
purchase of a good, which is locally a public good, in the sense that it will
be available also to neighbors. This game typically exhibits a great
multiplicity of equilibria. Imagine a social planner whose scope is to find an
optimal equilibrium, i.e. one in which the number of nodes playing 1 is
minimal. To find such an equilibrium is a very hard task for any non-trivial
network architecture. We propose an implementable mechanism that, in the limit
of infinite time, reaches an optimal equilibrium, even if this equilibrium and
even the network structure is unknown to the social planner.
