Given the success of the last open problem I posted, (Janos Kramer disproved my conjecture) I decided to post another one. Again, I will cut off the philosophy, and just give the math. The reason I want this result is to solve the problem described here.

Let $S$ be the set of all paris $(M,k)$ where $M$ is a Turing machines which and $k$ is a natural number.

For $P=(M,k)\in S$, we define the function $P\left(n\right)=p$ if $M$ halts on input $n$ in time $n^k$ and outputs a probability $p$, and $P\left(n\right)=1/2$ otherwise.

Let $E=e_1,e_2,\dots$ be an infinite bit string. We say $Q\in S$ defeats $P\in S$ on $E$ if

$$\underset{n\to \infty }{lim}\prod _{i=0}^n\frac{|1-e_i-P\left(i\right)|}{|1-e_i-Q\left(i\right)|}=0.$$

If there is no $Q$ in $S$ which defeats $P$ on $E$, we say that $P$ is undefeated on $E$.

Let $A$ be an algorithm which takes as input a finite bit string, and whose output encodes an element of $S$.

Does there exist an algorithm $A$ such that for all $E$, if there exists a $P$ which is undefeated on $E$, then for all sufficienty large $n$, $A(e_1,\dots ,e_n)$ is undefeated on $E$?

Note that $A$ can run as long as it wants. I would also be very interested in a positive result that requires that $P\in S$ is undefeated by every function computable in exponential time. ($P$ still runs in polynomial time.)

Also, note that if I replace the $lim$ with $liminf$, and require that $P\in S$ is undefeated by every function computable in exponential time, then such an $A$ does in fact exist. (The algorithm that does this has not been posted on the forum yet, sorry. If anyone wants to work on this problem, and is curious, I can explain it to them)