Seek Mistakes in the Space Between Math and Reality

Epistemic status: Cranky grumbling about the kids these days and all of their math and how they are using it wrong

Note: I’ll be using a few examples as foils below. This isn’t intended as criticism of those foils, it’s because they provide good examples of what I think is a broader issue to be aware of.

Girl, I knew you were heaven-sent

When I’d explored the mate-space thirty-seven percent

Call off the search, come home with me

You’re the highest-rated option after one-over-e

(SMBC, I don’t have a link to the precise comic right now but will add if anyone finds it).

The last plane trip of Keltham’s first life starts out uneventful. He boards the aircraft, strolls a third of the way down the aisle with his eyes assessing all he passes, and then sits next to the first person who looks like a more promising seat-partner than all of the previous people he passed.

(mad investor chaos and the woman of Asmodeus, first lines).

The Secretary Problem is a famous result in mathematics. If you want to select the best option from a set, and you are presented these options one at a time and must choose either to accept or to reject, your optimal strategy is to auto-reject the first 1/​e (about 37%) of options, and then accept the first option you see that is better than all previous options.

This is obviously a valuable real-world strategy in various fields, such as dating. As you can see, it has been mathematically proven that the correct strategy is to spend 37% of your dating life rejecting anyone you see, no matter how desirable they may appear, and then accept people you meet thereafter only if they are more desirable than anyone you saw before. If you are currently in a relationship, and are young enough to have less than a third of your adult life spent, you should immediately break up with your partner.

What? What do you mean, that doesn’t sound right? It’s been mathematically proven! Surely you don’t think mathematicians have gotten this famous result wrong? Show me the flaw in the proof, then!

You can’t find a flaw? Well, in that case, I reiterate, it has been proven mathematically that marrying your college sweetheart is a foolish idea, that there exists a superior strategy. Sadly many foolish people, perhaps as a result of insufficient math education, implement such mathematically suboptimal strategies, claiming to find ‘true love’ even though they objectively reduce their odds of success. Perhaps some sort of intervention is required in order to correctly optimize their behavior.

This position is obviously stupid. However, the precise way in which it is stupid will be illuminating. It’s not that it’s wrong about the math. The math is correct. Instead, it’s wrong about the linkage between the math and the reality. The math uses certain assumptions that will turn out not to be true in reality, and without those assumptions the math doesn’t hold.

Assumptions made in the Secretary Problem, and ways in which they fail to align with reality:

  • The solution assumes that the only thing you care about is maximizing the probability that you end up with the best possible candidate. In reality this is not true:

    • You are not indifferent between the second-best possible candidate and the worst. Given a choice between ‘you are guaranteed the second-best candidate’ and ‘you have a 1% chance of getting the best candidate and a 99% chance of getting the worst’, you would pick the first. The math assumes you would prefer the second, and optimizes for the second.

    • There are costs to the process of searching (both costs of the search itself, and costs of the delay in selection). The math assumes you’re indifferent between finding your partner at the very beginning of the search process, or at the very end.

  • The solution assumes that you have no information regarding the distribution of candidates, and the only way you can evaluate candidates is by comparison to other candidates you have seen. In reality this is not true:

    • You have a reasonable idea of the distribution of candidates you may encounter. This lets you say ‘this is a strong candidate’ or ‘this is a weak candidate’ by reference not to other candidates but to your known distribution.

You may well be able to see more problems along these lines. I haven’t bothered to put much effort into exhaustively listing them—but I don’t need to, because one problem like this is sufficient to sink the whole argument.

Math proofs are fragile. If your math is entirely valid but you made one little mistake in the assumptions, your proof is not ‘nearly right’, it is entirely wrong. The Secretary Problem proof is an interesting piece of mathematics, and may be useful in some contexts, but it is completely worthless as a guideline for how to conduct yourself in real-world situations.

This is a useful illustration of a more general rule:

When someone says ‘Mathematical Result X proves Counter-Intuitive Thing Y’, do not look for problems in Mathematical Result X. First, there are unlikely to be problems with it; and second, it’s likely to be a very complicated proof, and may in fact be beyond your abilities to evaluate on any level beyond “Yup, Wikipedia sure says that that’s a theorem”.

Instead, look for problems in the linkage between that result and the real world.

An incomplete subset of things I’m thinking of here:

  • Claims that Arrow’s Impossibility Theorem proves that ‘the only stable form of government is a dictatorship’.

  • Every single real-world usage of the phrase ‘Modigliani-Miller’.

  • And a bonus complaint for mad investor chaos readers:

Consider a Prisoner’s Dilemma agent that Cooperates if it can prove its opponent will Cooperate with it, and Defects otherwise. Call it ‘Suspicious-FairBot.’

I assert that this agent will Cooperate with itself. When you ask me why, I mutter something about the ‘Assumable Provability Theorem’, and declare that this means ‘if something is provable within a proofsystem, starting from the premise that the quoted statement is provable inside the quoted proofsystem, then it’s thereby provable within the system’, and therefore that Suspicious-FairBot will Cooperate with itself.

Now consider another agent that Defects if it can prove its opponent will Defect against it, and Cooperates otherwise. Call it ‘Naive-FairBot’.

Naive-FairBot is strictly more Cooperative than Suspicious-FairBot—it will Cooperate unless it is proven that its opponent Defects, while Suspicious-FairBot will Defect unless it is proven that its opponent Cooperates. Both behave the same if their opponent’s move can be proven, but if the opponent’s move cannot be proven Naive-FairBot will Cooperate while Suspicious-FairBot will Defect.

It is therefore very strange that the Assumable Provability Theorem argument declares that Suspicious-FairBot will Cooperate with itself, while Naive-FairBot will Defect against itself.

Do not get me started on Neutral-FairBot, which Cooperates if it can prove its opponent Cooperates, Defects if it can prove its opponent Defects, and therefore must both Cooperate and Defect against itself.

What’s going on here? Is the math wrong? Does the math uncover a secret power by which we can guarantee Cooperation through Suspicion?

Or is there at least one difference between a proof system and a prisoner’s dilemma agent?