No one is going to run a prison for free--there has to be some exchange of money (even in public prisons, you must pay the employees). Whether that exchange is moral or not, depends on whether it is facilitated by a system that has good consequences.
In the predominant popular consciousness, this is not sufficient for the exchange to be moral. Buying a slave and treating them well is not moral, even if they end up with a happier life than they otherwise would have had. Personally, I'm consequentialist, so in some sense I agree with you, but even then, "consequences" includes all consequences, including those on societal norms, perceptions, and attitudes, so in practice framing effects and philosophical objections do still have relevance.
Of course there has to be an exchange of money, but it's still very relevant what, conceptually or practically, that money buys. We have concepts like "criminal law" and "human rights" because we see benefits to not permitting everything to be bought or sold or contracted, so it's worth considering whether something like this crosses one of those lines.
Under this system, I think prisons will treat their inmates far better than they currently do: allowing inmates to get raped probably doesn't help maximize societal contribution.
I agree that seems likely, but in my mind it's not the main reason to prevent it, and treating it as an afterthought or a happy coincidence is a serious omission. If your prison system's foundational goal doesn't recognize what (IMO) may be the most serious negative consequence of prison as it exists today, then your goal is inadequate. Indirect effects can't patch that.
As a concrete example, there are people that you might predict are likely to die in prison (e.g. they have a terminal illness with a prognosis shorter than their remaining sentence). Their expected future tax revenue is roughly zero. Preventing their torture is still important, but your system won't view it as such.
Now that I'm thinking about it, I'm more convinced that this is exactly the kind of thing people are concerned about when they are concerned about commodification and dehumanization. Your system attempts to quantify the good consequences of rehabilitation, but entirely omits the benefits for the person being rehabilitated. You measure them only by what they can do for others – how they can be used. That seems textbook dehumanization to me, and the concrete consequence is that when they can't be used they are worthless, and need not be protected or cared for.
As my other comment promised, here's a couple of criticisms of your model on its own terms:
My instinctive emotional reaction to this post is that it worries me, because it feels a bit like "purchasing a person", or purchasing their membership in civil society. I think that a common reaction to this kind of idea would be that it contributes to, or at least continues, the commodification and dehumanization of prison inmates, the reduction of people to their financial worth / bottom line (indeed, parts of your analysis explicitly ignore non-monetary aspects of people's interactions with society and the state; as far as I can tell, all of it ignores the benefits to the inmate of different treatment by different prisons).
This is in a context where the prison system is already seen by some as effectively legal slavery by the back door, where people believe that (for example) black people have been deliberately criminalized by the war on drugs as part of an explicit effort to exploit and suppress them, and where popular rhetoric around criminals has always been eager to treat them as less than human.
This perspective explicitly doesn't address the specific content of your proposal, but I think there's a few reasons why it's important to pay attention to it:
I'm posting this as a "frame challenge", but I think I also have some critiques of your model on its own terms, which I'll post as a separate comment.
As an offtopic aside, I'm never sure how to vote on comments like this. I'm glad the comment was made and want to encourage people to make comments like this in future. But, having served its purpose, it's not useful for future readers, so I don't want to sort it to the top of the conversation.
The number of possible pairs of people in a room of n people is about n^2/2, not n factorial. 10^2 is many orders of magnitude smaller than 10! :)
(I think you are making the mistake of multiplying together the contacts from each individual, rather than adding them together)
I strongly agree with both this specific sentiment and the general attitude that generates sentiments like this.
However, I think it's worth pointing out that you don't have to agree with the Labour Party's current positions, or think that it's doing a good job, to be a good (honest) member. I think as long as you sincerely wish the party to perform well in elections or have more influence, even if you hope to achieve that by nudging its policy platform or general strategy in a different direction from the current one, then I wouldn't think you were being entryist or dishonest by joining.
(I feel like this criterion is maybe a bit weak and there should be some ideological essence of the Labour Party that you should agree with before joining, but I'm not sure it would be productive to pin down exactly what it was and I expect it strongly overlaps with "I want the Labour Party to do well" anyway)
I actually thought the "of course I'd rather you'd stay a member" part was odd, since nowhere in the post up to that point had you said anything to indicate that you supported Labour yourself. The post doesn't say anything about whether Labour itself is good or bad, or whether that should factor into your decision to join it at all, but in this comment it sounds like those are crucial questions for whether this step is right or not.
Yeah I think you have to view this exercise as optimizing for one end of the correctness-originality spectrum. Most of what is submitted is going to be uncomfortable admitting in public because it's just plain wrong, so if this exercise is to have any value at all, it's in sifting through all the nonsense, some of it pretty rotten, in the hope of finding one or two actually interesting things in there.
GiveWell used to solicit external feedback a fair bit years ago, but (as I understand it) stopped doing so because it found that it generally wasn't useful. Their blog post External evaluation of our research goes some way to explaining why. I could imagine a lot of their points apply to CEA too.
I think you're coming at this from a point of view of "more feedback is always better", forgetting that making feedback useful can be laborious: figuring out which parts of a piece of feedback are accurate and actionable can be at least as hard as coming up with the feedback in the first place, and while soliciting comments can give you raw material, if your bottleneck is not on raw material but instead on which of multiple competing narratives to trust, you're not necessarily gaining anything by hearing more copies of each.
Certainly you won't gain anything for free, and you may not be able to afford the non-monetary cost.
The convention in a lot of public writing is to mirror the style of writing for profit, optimized for attention. In a co-operative environment, you instead want to optimize to convey your point quickly, to only the people who benefit from hearing it. We should identify ways in which these goals conflict; the most valuable pieces might look different from what we think of when we think of successful writing.
Writing to persuade might still be best done discursively, but if you anticipate your audience already being sold on the value of your information, just present the information as you would if you were presenting it to a colleague on a project you're both working on.