How to Fix Private Prisons and Immigration

by FCCC 6 min read19th Jun 202035 comments


The goal of prisons

Prisons are widely criticized, and for good reason. Crime is rife within prisons, recidivism is high (roughly half of former inmates return to prison), and incarcerating inmates may, at the margin, increase the number of committed crimes. The root of the problem? The prison system doesn’t satisfy good goals.

Selecting a good goal for a system is hard. The goal that a few people have suggested to me is “Prisons should minimize crime”. This goal is bad. It implies that prisons might spend a million dollars to prevent the theft of a candy bar. Good goals account for benefits and costs, and allow an action if and only if it is in society’s interests.

One such goal might be to maximize the total societal contribution of any given set of inmates within the limits of the law (limits such as “Don’t restrict the freedom of inmates after their release”).

Let’s see what kind of prison system satisfies this goal.

The proposed prison system

Prisons enter a second-price auction for each convict’s contract: The highest-bidding prison incarcerates the convict and pays the second-highest bid. After the auction, whenever the convict pays tax, the government passes it on to the prison; whenever he uses a government service, the government passes the cost on to the prison; whenever he commits a crime, the prison must pay the government the “equivalent compensation” of the crime (e.g. a former inmate committing a murder might cost his former prison whatever the value of a human life is).

Note that second-price auctions allow negative bids, which will be necessary to sell the contracts of inmates that are likely to reoffend.

If a former inmate is reincarcerated, their new contract goes up for bid, and the sale price goes to his former prison (rather than the government). If the sale price is negative, the old prison pays the new prison.

Once a convict leaves prison, the prison has no control over the him. He abides by the same tax rules as any non-convict. He can access the same government services. He would not be affected at all if the government suddenly revoked his contract after his release (i.e. the government stopped paying the prison his taxes, and so on). This system affects the convict only by changing what occurs during his incarceration.

Defining “future societal contribution”

Given we want to maximize the societal contribution of all inmates and we can’t change contributions from the past, we want to maximize the future contribution of all inmates. So how do we define “future societal contribution”?

Bob has been convicted of a crime. His future societal contribution () can be split into his future monetary and nonmonetary contributions. His future monetary contribution can be split into

  • the government’s cost of funding his incarceration (),
  • his tax incidence (),
  • the amount of welfare and government subsidies he is granted (),
  • the monetary effects of Bob’s future crimes (if any), such as cost of replacement and repair.

We will define equivalent compensation as the payment amount that would make us as well off as we would have been had the crime not occurred. Equivalent compensation for the monetary effects of crimes always exists because those effects are, by definition, a monetary amount. Bob’s future nonmonetary contribution includes the nonmonetary effects of the crimes he will commit. Suppose the equivalent compensation of a crime is equal to the maximum amount that we (as a society) would spend to prevent it. Thus, since we do not have infinite money, equivalent compensation for the nonmonetary effects of a crime must always exist. Therefore the monetary and nonmonetary effects of his future crimes can be summed into a single value ().

Let us ignore Bob’s other nonmonetary contributions for simplicity.

Under these assumptions, Bob’s contribution to society can be represented as a dollar amount:

For our proposed system (with the price of Bob’s contract being ),

The prison’s profit

Suppose that the profit a prison would generate from a given convict is independent from the profit it would generate from its other convicts. Then, if Bob is incarcerated by prison , the prison’s profit from Bob () under any system is the funding it receives for him minus its cost in imprisoning him ():

Under our proposed system, the prison’s profit is found by substituting our previously defined level of funding:

(Not true in the case of bankruptcy: A company can’t make unbounded negative profits. To prevent prisons bidding on negative-sale contracts, declaring bankruptcy, and running away with the money, we probably want capital-rich intermediates to underwrite the prisons.)

The profit-maximizing bid of each prison

Given our assumptions, if prison wants to maximize its total profit (), it must maximize the profit it generates from each of its convicts (all ) simultaneously:

(This is, strictly speaking, not true. The error in the argument is the “profit independence” assumption. But it doesn’t affect anything. Prisons want to maximize their profit over all inmates, and they’ll select a set of inmates that does so, which will tend towards maximizing the total societal contribution of the given set of inmates. We can probably avoid the assumption by having a continuous simultaneous auction of all inmate contracts.)

Let’s suppose that prisons do maximize profits. Thus, each prison bids its profit-maximizing bid (), which, in a second-price auction with non-colluding bidders, is the bid () that would make its profit zero if it paid its bid:

Then, the profit-maximizing bid can be found by making the relevant substitutions:

Note that each prison’s profit-maximizing bid is independent of the other prisons’ bids. Also note that the bid is equal to the sum of society’s and the prison’s benefit:

Sufficient criteria to maximize the contribution

Each inmate’s future contribution to society is maximized if

1) each inmate is allocated to the prison that can create the greatest for that inmate, and

2) all prisons, to their best of their ability, maximize of each of their inmates, and

3) the incarcerating prison’s profit (i.e. its share of ) is zero.

The proposed system satisfies the sufficient criteria

Criterion 1: The inmate is allocated to his best prison

Prison imprisons Bob if and only if its bid, its profit-maximizing bid, is the highest.

The prisons reveal which prison is best just by submitting their bids. Since the profit-maximizing bids happen to equal , the highest bidding prison is the best prison, and Bob gets allocated to it.

I.e. criterion 1 is satisfied.

Criterion 2: Prisons maximize the total contribution

Recall that the winning bidder pays the second-highest bid:

Recall that, barring collusion, a profit-maximizing prison will bid independently of the bids of any other prison: The winning prison cannot affect the second-highest bid, despite that bid being a factor in the prison’s profit. Therefore, the winning prison maximizes its profits by maximizing the sum of the other factors of its profit function, which happen to equal :

I.e. criterion 2 is satisfied.

Criterion 3: The prison’s profit is zero

The winning prison’s profit equals the difference between the total benefit it would provide and that of the second-best prison:

If the best two prisons are equally capable, the profit is zero. I.e. criterion 3 is satisfied.

(Well-run companies can have zero economic profit. I should have included the opportunity costs of prisons and society at the beginning of the proof.)

Poorly estimating equivalent compensation doesn’t always create inefficiencies

We want to rehabilitate if and only if the benefits exceed the costs. The costs are a dollar figure, so when choosing whether to rehabilitate, we need to put a dollar value on the benefits: Estimating each crime’s equivalent compensation is necessary under any well-functioning prison system. Since the estimates are ultimately a normative decision, it’s impossible to solve using empirical data alone. Not that that’s stopped anyone from trying.

Fortunately, our valuations of equivalent compensation do not need to be accurate. To understand this, there are three figures to keep in mind:

  • The true value of the crime’s equivalent compensation,
  • Our valuation of the crime’s equivalent compensation, and
  • The prison’s cost of rehabilitation to prevent that crime.

If the cost of rehabilitation is lower than the true value, we want the rehabilitation to occur. And it does as long as the cost of rehabilitation is lower than both the true value and our valuation. And since prisons earn zero profit, the cost to society is the cost of rehabilitation.

If the cost of rehabilitation is higher than the true value, we don’t want the rehabilitation to occur. And it does not as long as the cost of rehabilitation is higher than both the true value and our valuation.

Inefficiencies occur only when cost of rehabilitation falls between the other two values: Either the rehabilitation will occur when it shouldn’t, or it won’t occur when it should.

Measuring tax contributions and usage of government services

We don’t need to measure everything, only the things that are significant and can be changed by the prisons. For example, we don’t need to measure the past (since it cannot be changed). We don’t need to calculate the wear-and-tear a convict causes to a road’s surface (since it’s not significant). We only want to measure the big things, such as welfare payments, public housing, education and healthcare subsidies, and tax incidence. The amount of welfare a person is paid is relatively easy to get (I’ve written code to retrieve that value from government databases). I don’t know how hard it is to measure the other values, but calculating “Net tax contribution” is essential to solving many policy issues.

Immigration should use a similar mechanism

Same strokes, different folks

When I tell people that prisons and immigration should use a similar mechanism, they sometimes give me a look of concern. This concern is based on a misconception: Using the same mechanism does not imply goods are related in any meaningful way. The transaction mechanism for computers and peanut butter is the same, but that’s the only thing those goods have in common.

Our mechanism applies to any transaction between two parties where either one party doesn’t have a choice in the matter (e.g. inmates), or one side of the transaction is always identical, so one party doesn’t care who is on the other side of the transaction. The latter is true for immigrants: It doesn’t matter who rubber stamps an immigration form.

A decentralized immigration system

Applying our system to immigration is essentially a prediction market for whether any given prospective immigrant will be a positive contributor to society. For our (non-refugee) immigration system, we can copy and paste our prison funding mechanism, with two main differences: We're not imprisoning immigrants, and negative bids are not allowed.

The rest of the system is as you’d expect. Companies enter a second-price auction for the contract of each immigrating family unit. If bids are submitted, the winning bidder pays the second highest bid and the family is allowed into the country. After the auction, whenever the family pays tax, the government passes it on to the company; whenever the family uses a government service, the government passes the cost on to the company; whenever the family commits a crime, the company pays the government the equivalent compensation of the crime.

Optional modifications

I'm not sure about the following modifications.

  • If the family commits a crime, their contract goes up for auction again, no negative bids allowed. If no one bids on their contract, the family is deported. If there are bidders, the transaction amount (i.e. the second-highest bid) is transferred to the previous holder of the contract (the previous contract holder is allowed to bid, in which case no transaction occurs between companies). This modification reduces the potential downside of allowing an immigrant into the country, which results in more people becoming viable immigrants. It results in bids being higher, which means more money to the government. It also deters immigrants from committing crime.
  • If the government wanted to control the number of immigrants, it could demand a fixed payment from the winning bidder of each immigrant’s contract. This would reduce the number of immigrants that companies would be willing to accept.
  • We might also want to allow public donations to act as a decentralized valuation of a potential immigrant’s desirable nonmonetary contributions to society.