If the tenant does not pay - they will revoke their voucher. They have a zero tolerance policy and you can complain to them about a tenant and they will act. And they do pay regularly, so that is all good.
The worst part of cook county housing authority is that they do not pay nearly enough for rent. Depending on the house, they are 20-30% below market value, but there is a guarantee on rent and behavior, so that is a trade off.
It really just depends on the landlord and tenant.