Before revision:  We then propose a family of welfare functions that are governed by a single inequity aversion parameter which allows a decision maker to study task-dependent trade-offs between fairness and total influence and effectively trade off quantities like influence gap by varying this parameter. We use these welfare functions as a fairness notion to rule out undesirable allocations. We show that the resulting optimization problem is monotone and submodular and can be solved  with optimality guarantees.
After revision:  The resulting optimization problem is monotone and submodular and can be solved with optimality guarantees.