But yes, there are incentives that do reward cells and even individuals for number of imprisonments. And no, they do not check nor punish "administrative errors".
Taxpayers are paying the private companies, but they're very much for profit.
About that:
https://www.forbes.com/sites/thomasbrewster/2025/01/29/billi...
https://www.motherjones.com/politics/2025/03/private-prison-...
https://afsc.org/newsroom/ice-signs-massive-contract-profit-...