Before revision:  In our market mechanism, we specifically modify the Fisher market social optimization problem through a perturbation of the budgets of agents . The budget perturbations are set based on the dual variables of the additional physical constraints such that the KKT conditions of the newly defined social and individual optimization problems are equivalent.
After revision:  In our market mechanism, we specifically modify the Fisher market social optimization problem through a perturbation of the budgets of agents such that the KKT conditions of the newly defined social and individual optimization problems are equivalent.