Wal-Mart *is* the third party. It is neither the taxing authority nor the taxpayer. Wal-Mart does, however, set the prices at which the taxpayer is taxed. The analogy is solid.
Wal-Mart sells a gizmo for $1.00.
The Gizmo Appraisal District says the Gizmo has a value of $1.25 and the tax will be figured on that amount.
Just as Wal-Mart sets the price it sells at, I can set the price for my house. But, it is taxed at the value the unelected HCAD determines.