Commercially run prisons where the operators get paid based on a lack of recidivism in former inmates rather than the time those inmates spend at the facility could help with alleviating some of the problems with the current prison system in the US.
However, with these institutions in particular I'd say the metrics a pretty well-defined:
- both prisons and healthcare: Lack of recidivism, time until you're a productive member of society again
- prisons and universities: Economic welfare of former 'inmates'. Some universities for example operate on a business model where they get paid a share of the graduate's income for a few years after graduation.
Sure, like many metrics these probably could be gamed but they're way better than current metrics, including those state-run institutions of these kinds operate with (if they have any metrics at all, that is).
What is hard is not just finding good metrics (that cannot be gamed), it is finding good metrict (that cannot be gamed) that can also provide profits to good players.
Obviously, there's no guarantee that a former inmate won't commit another crime even if the rehabilitative measures he experienced while doing time were exemplary. However, there's also no guarantee that in the university example a graduate will find a well-paid job. Still, there are universities that have successfully implemented this business model.
There's a whole industry that's predicated on this kind of uncertainties: The insurance industry. While there's no guaranteed outcome risks and probabilities can be calculated and factored into the price of a service.
The question perhaps is: What value do we as a society allocate to a former convict likely not committing crimes in the future anymore but contributing to society instead?
Considering the alternative, i.e. a former convict not only being a liability to society but also being highly likely to commit crimes in the future, I'd that service should come with quite a high price tag.