Suppose that $P \neq NP$, and $P = BPP$.
Assume one is given a decision language $L \in NPC$, and she has only polynomial time turing machines. Additionally, she can't use randomness (not sure that's relevant since I assumed $P= BPP$).
How can she compare between two algorithms $A_1, A_2$? Clearly, $A_1$ and $A_2$ cannot be correct for every input (they must run in polynomial time). Does it make sense to define the better algorithm as the one which answers correctly on more inputs? (And two algorithms will be equal when they answer correctly on the same number of instances).
Are there known methods to do so?