One Night in Delphi

post by Eggs (donald-sampson) · 2025-04-18T02:17:04.957Z · LW · GW · 2 comments

Contents

2 comments

A rationality exercise [LW · GW].

 

Mikhail is a mathematician working on a particular conjecture. His goal is to prove it if true or find a counterexample if false. After thoroughly reviewing the evidence, he estimates — fairly and with well-calibrated confidence — that there's a 75% chance the conjecture holds.

One night, he dreams of the Oracle at Delphi.

After making proper obeisance Mikhail takes advantage of the opportunity and asks if the conjecture he is working on holds true. the Oracle delivers a surprisingly clear and specific prophecy: the conjecture is unequivocally true, and if Mikhail dedicates himself exclusively to proving it, he’ll succeed within three months. Upon realizing that he doesn't actually speak Greek, Mikhail wakes up.

Mikhail has had Oracle dreams before. But this one lingers. He spends the next day carefully re-evaluating the evidence. Did he subconsciously realize something his waking mind missed? Is there any new angle, any previously overlooked implication? Were the spirits of dead mathematicians trying to communicate with him? After a detailed review, he finds nothing new — and reaffirms his original estimate: 75% chance the conjecture is true.

So now he faces a different kind of question.

Is there value in determining if the Oracle was right?

 

Let's make some simplifying assumptions. 

• Each day, Mikhail can split his time between trying to prove or disprove the conjecture.
• If the conjecture is true, every day spent trying to prove it has a 1% chance of success and every day spent trying to disprove it has a 0.1% chance (perhaps through discovering a related lemma).
• If the conjecture is false, the situation reverses: 1% success per day when searching for counterexamples, 0.1% when trying to prove it (and stumbling on contradictions).
• Mikhail values proof or disproof equally.
• The problem is mathematically decidable, but he might never succeed in either approach if he is unlucky.

With this information we can calculate the expectation for success on any given day if he spends fraction $x$ of the day trying to prove the conjecture and $1-x$ working on disproof. 

$$E\left[S\right]=0.75(0.01x+0.001(1-x\left)\right)+0.25\left(0.01\right(1-x)+0.001x)$$$$=0.0075+0.00025x.$$

This estimate is maximized by selecting $x=100\%$, or spending all his time trying to prove the conjecture gives the best chance of success on any given day. But Mikhail doesn’t care about today. He cares about eventual success. We can instead calculate the expected number of days until success:

$$E\left[D\right]=0.75\frac{1}{0.01x+0.001(1-x)}+0.25\frac{1}{0.01(1-x)+0.001x}$$

This expectation is minimized by choosing x=0.664, yielding a mean of about 170 days. This is Mikhail’s baseline strategy — the best he can do, given his current understanding.

On the face of it, trusting the Oracle’s claim sounds absurd. Ockham’s razor suggests that “dreams be weird” is far more plausible than prophetic revelation. (Let alone whether the Oracle at Delphi would ever make an unambiguous prophesy.) But we can treat the Oracle’s claim as a testable hypothesis. Suppose Mikhail follows the Oracle’s advice and spends 100 days devoted exclusively on proving the conjecture, then reverts to the optimal strategy if unsuccessful. (Mikhail rounded up in case the Oracle's three months didn't quite line up with modern calendars.) What’s the cost?

We can compute the expected extra days of work this detour might cost, using Mikail's current expectations. Each day is an independent chance of success so we can model this using the memoryless property of exponential waiting times. Either he succeeds in the 100 days, or we revert to the above expectations plus 100 days:

$$\begin{array}{cc}E\left[C\right]=& 0.75(\frac{1-e^{-100\left(0.01\right)}}{0.01}+e^{-100\left(0.01\right)}(100+\frac{1}{0.01\left(0.664\right)+0.001\left(0.336\right)}))\\ & +0.25(\frac{1-e^{-100\left(0.001\right)}}{0.001}+e^{-100\left(0.001\right)}(100+\frac{1}{0.001\left(0.664\right)+0.01\left(0.336\right)}))\\ =& 217.18\text{days}\end{array}$$

So if the Oracle is wrong, Mikhail expects to waste 217 - 170 = 47 days on average.

But if the Oracle is right, he finishes in 100 days instead of 170 — a 70-day gain. 

Together this means it’s rational to test the Oracle if there's more than a 40.3% prior chance that the dream was correct. Even though we are ignoring the chance that Mikhail might get an unambiguous prediction once every decade or so.

Is that plausible? Mikhail has to decide for himself, balancing his priors against the relative tradeoffs. But from a purely evidential and probabilistic standpoint, it’s at least thinkable. And most importantly: it’s testable. This isn’t a commitment to metaphysics — it’s just a bounded, reversible experiment.

As good Bayesians we also know that even if the Oracle is correct on all counts, it doesn't prove that Mikhail has an in with the supernatural. The likelihood ratio of this experiment is only 1:0.4979 ~ 2:1. After this update Ockham's vanishingly small prior is still a vanishingly small posterior. This is hardly enough on its own to change future expectations. But it might be just enough for Mikhail to look up how to say “thank you” in ancient Greek, just in case.

2 comments

Comments sorted by top scores.