You guys are trying to make it too complicated.
HUd will pay a fixed rate for a 3 bedroom. say $800.. A family with 4 kids comes in and qualifies for a 3 bedroom house. So they get a HUD voucher for 3 bedroom house.
Next, HUD looks at the family's finances and determines how much they can afford to pay for a house (about 1/3 their income.)
HUD determines that the family can afford to pay $200 a month. So the family gets the $800 house that they qualify for, but they must pay the first $200 of the rent and HUD picks up the rest of it.
That is why landlords get part of the rent from the tenant and part from HUD. The landlord is NOT getting $600 from HUD, plus the $200 tenant share, plus another $200 a month extra, unless he like the thought of jail time for himself. If you agrtee to accept teh program, you take it on their terms, not your own.
The HUD fair market remt includes the cost of utilities, by the way, so if you have a3 bedroom house in an area where HUD pays $800 for a 3 bedroom house, you are NOT getting $800 a month for your house unless utilities are included. There is a deduction from the rent amount if utilities are not oncluded.