The two-body problem is solvable in physics. If we imagine a solar system that consists of exactly one star and one planet, it is possible to predict the planet’s orbit. It’s not at all clear that the three-body problem is solvable. Put one more planet in there, and it becomes very difficult to predict the orbit of either one. Maybe impossible.

In economics, we are looking to optimize a transaction. Note, a transaction doesn’t have to be perfectly optimal, it just needs to be optimal enough, in the sense that all parties are happy with the result. So it’s not a question of whether I could have saved another $300 on the car I just bought, but whether me and the dealer didn’t feel we got ripped off.

The two-body problem in economics — where we are considering a transaction between one seller and one buyer — can be optimized. If we assume that no party can hurt himself simply by walking away from a transaction, then an optimal solution is inevitable. If the transaction happens, and both walk away happy, we call that strongly optimal. A weakly optimal result happens when at least one party breaks even, and nobody profits at the other’s expense. If both parties walk away, no-harm-no-foul. That’s a weakly optimal outcome.

Question is, is the three-body problem solvable in economics? Here, there are three parties to the transaction. Of course it can; it’s entirely possible three people could sit down at the negotiation table and come up with a win-win-win. Question is, is it guaranteed?

Let’s think for a minute about a reasonable set of rules for a three-party transaction:

- Completeness. For now we will assume that, in aggregate, the three parties have enough information to solve the problem.
- Asymmetry. Each party has access to some of that information, but no one has access to all of it.
- Non-coercivity. In order for a transaction to occur, all parties must consent. You can veto, or walk away as you desire.
- Subjectivity. Each party is allowed to use personal criteria for beneficence; there’s no requirement that subjective benefit be distilled into terms actionable by the other parties. So turning the cards face-up doesn’t help.
- Good faith. You can’t cheat.

Here’s an example I came up with for healthcare, which for all practical purposes is a three-party transaction, between a doctor, a patient, and an insurance company.

Player 1: The doctor, who knows the diagnosis (imperfectly) and proposes a procedure. He knows there is a x% chance of success, y% chance of complications. He wins if the transaction occurs, and collects a fee. Let’s say he is unharmed if the procedure is not consummated.

Player 2: the patient, who knows his own risk tolerance. His job is to weight the probabilities the doctor gives him. His breakeven point is where

Ax = By

Where A is the weighted value of success, and B is the weighted (negative) value of complications. Completely subjective of course. Likewise, let’s say the patient is unharmed if he decides of his own accord not to have the procedure.

Player 3: the Insurance company, which has no idea what’s going on with the patient, but knows sound actuarial procedure. Basically, paying a fair price for a procedure that is medically necessary (in other words, not futile). Let’s say the insurance company wins if it follows sound actuarial procedure, and loses if it does not. And is unharmed if it vetoes the procedure.

First, let’s assume all players agree to the transaction.

There is one strongly optimal outcome:

- Procedure carries a positive expectation for the patient.
- Surgeon collects a fee.
- Insurance company follows sound procedure and gets to stay in business a little longer.

I can think of a couple of possible suboptimal outcomes:

- Patient gets talked into a procedure he would have refused, if with perfect knowledge he knew how risky it was. Even if it was sound actuarially.
- Insurance company pays for procedure that, with perfect knowledge, it would have known to be futile.

Now let’s consider if one party vetoes the transaction. Unlike the two-body problem, where vetoing a transaction results in a weakly optimal outcome, it is possible to have negative outcomes.

- If doctor vetoes a procedure because she decides it’s too risky, the patient could be harmed if the doctor’s risk assessment is more conservative than the patient’s
- If insurance company vetoes due to relative futility, the patient might lose, if the insurance company makes its decision based on a non-ergodic distribution of benefit

That last one deserves some explanation. Let’s say the proposed procedure has a likelihood of success of about 4-6%, and there is likewise a 2-4% likelihood of spontaneous remission. Inasmuch as there is a lot of overlap in those ranges, vetoing the procedure would be sound actuarial practice. The patient, on the other hand, may be facing as much as 98% probability of death, and a 4% likelihood of survival may sound like a good bet, regardless of risk. I think the patient loses there.

There’s one final thing to consider, which is, if either the doctor or the insurance company fail to agree on a fee. For the most part, this is an weakly optimal outcome — no harm, no foul — provided the patient has the option to find an alternative provider. Sometimes the insurance company miscalculates. The patient may not have the time or money to drive all the way across the state to find a provider who charges a reasonable fee. I think this falls into the general category of information incompleteness, of which we can think of several examples:

- The insurance company may not know the market as well as it should.
- The doctor may have misdiagnosed the patient.
- The patient might be a hypochondriac.

All I can say is, you can’t win em all.

So. In this particular scenario, there is no guarantee of an optimal outcome. The question is, is this a general principle?

The tricky thing about this scenario is the non-ergodic distribution of risk and benefit. Something may seem not risky from one subjective viewpoint, and prohibitively risky from another. A procedure may seem futile from one perspective, but may carry a thin margin of benefit from another.

This type of subjectivity is characteristic of healthcare decision making, but isn’t necessarily a part of all three-way transactions.

Still, it’s not at all clear to me that you can have both beneficence and subjectivity, both at the same time. There may be situations where you have to rely on information somebody else has, that is filtered through their perceptual system rather than yours.