Now say the majority divides the department into 4 line items. They give 9 points to each and the opponents put 4 against each. This lets the majority double its affect. The fair-share version reduces that incentive by making the tax rate a function of the percentage change. Under IP there is an advantage in sub-dividing agencies if people place more chips for agencies than against.
Most reps hesitate to place chips against another's fiefdom.
The FIP rule therefore requires each rep to balance positive and negative changes. A column for tax relief or increase still would allow a rep to influence the overall budget. And if she adds money to tax relief, she must take money away from another budget. A rep needs to see the current budgets and repeatedly move her chips. A network-based ballot like the one pictured below is a good method.