Suppose we have a network in which agents request access to its resources. Thus we have a resource allocation problem.
Ideally, we want to incentivize agents to send social-welfare supporting messages. To do that the mechanism designer must design monetary incentives that align each agent's objective with the social objective.
I read in a paper that one way to do this is to utilize the Lagrange multipliers associated with the optimal solution of the optimization problem. That way we can determine the per unit price of the request per each link, so that the total payment of each agent over all links internalizes her effect on the social welfare.
I don't understand why that's even a way to do it. Could anyone provide some input on this? Do we use the Lagrange multipliers in mechanism design problems (by associating them to the prices they agents have to pay)?