Another Concise Open Problem

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 $P (n) = p$ if $M$ halts on input $n$ in time $n^{k}$ and outputs a probability $p$ , and $P (n) = 1 / 2$ 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

$lim n \to \infty n \prod i = 0 \frac{| 1 - e_{i} - P (i) |}{| 1 - e_{i} - Q (i) |} = 0.$

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 $lim inf$ , 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)