[andrewla]

Your proof for the EMH hinges on a very specific technicality -- whether the secret prime, P, is public knowledge, or to what degree it is public knowledge.

I think if we were to approach the EMH from a constructive or computability approach, we'd have to make that definition much crisper. That is, the feasibility of computing the factorization based on limited information determines whether the information is actually "available" in the sense of the EMH.

In the real world we can see things like drug trials that will determine the success of a pharmaceutical company. The information that the drug works or doesn't work, or has side effects, would be "apparent" to a hypothetical "ideal" expert in human biochemistry -- so simply knowing the chemical formula for a new drug, the information about its working is "available" in that sense.

[viraghr]

If you can show an inefficient market experimentally whenever you want, that's good enough to disprove EMH. The EMH doesn't say, "All information is normally factored into the price, except for technicalities."

I like your drug trial example but a "hypothetical ideal expert" isn't assured to exist. Do you have an existence proof?

On the other hand, unique prime factorization is assured, we know that given infinite time anyone can factor any large number.

[andrewla]

The EMH is usually restricted to "available information". This hinges on what "available" means.

I would argue that until the search space was reduced to the point where it would be feasible to factor the secret before another digit is revealed, the information is not "available" in any useful sense. So maybe, in some sense, this helps us narrow down a definition of what "available" means, but since real-world price signals are rarely something that is deterministically computable, I'm not sure that this would help to clarify the EMH.

In my drug trial example, such an expert does not exist, but if we're dealing with mundane real-world stuff, then the assurances that the proposed $1B gift to a random company would actually go through would have to be factored into any strategy -- why would I bother committing computing resources to factoring the number when in all likelihood the billion dollars would not handed over because why would anyone actually do that?

[viraghr]

100 possibilities are really easy to test, though, so P must be fully available near the end. We can all agree it's not available at the beginning (subject to the methodology.) So it becomes more and more available over time, but that is not what the EMH says -- at all. If you are saying market prices will reflect information as a function of how "readily" available it is (for example how smart you have to be to see it), that is just not EMH.

For the drug trial example, again, I like it but perhaps such an expert is theoretically excluded? Your statement might be like saying "suppose there's a hypothetical O(n) halting algorithm that answers the halting problem in constant time for the length of the program it's testing." That's nice but there's a proof that what you've just hypothesized is impossible[1].

I am not exactly saying that an ideal expert in human biochemistry who can predict the results of drug trials is impossible, but how do we know it's possible? I mean, is chemistry guaranteed to be fully computable or something?

For your last point, I think that what I asked to assume (that the $1B transfer is credible, that the computer is secure) are nowhere near the size of assumption of something like "assume there exists someone who can predict biochemical reactions in humans without testing." Basically the legal instruments and secure computers I included are easy, solved problems that I ask the reader to assume are being implemented following best practices.

[1] https://en.wikipedia.org/wiki/Halting_problem