I am considering the following class of algorithms:
The algorithm has access to some probabilistic oracle (procedure) $f$ in addition to input.
The answer of procedure $f$ (we may assume it is boolean) may be incorrect with probability at most $p<1/3$. The error of different calls to $f$ are independent (thus we may increase the confidence of the result by repeatedly calling $f$ with the same argument and taking majority).
The algorithm can call multiple times $f$ with any values.
We request that whatever the bound on error probability $p$ (unknown to the algorithm), the algorithm itselft must return its correct answer with error probability at most $p$.
The question is: Is there any standard name for that class of 'confidence-preserving' algorithms?
For the sake of illustration, if we want to find the maximum of $3$ items and only have access to the items through some probabilistic oracle $f$ for comparing any pair of items, we can compare each pair $3$ times and take majority vote. The probability that each pair is ordered incorrectly is at most $3p^2$, hence the probability that we do not deduce the maximum correctly is at most $9p^2$. This is less than $p$ so the algorithm belongs to our class.
