Suppose I consider the following variant of BPP, which let us call E(xact)BPP: A language is in EBPP if there is a polynomial time randomized TG that accepts every word of the language with exactly 3/4 probability and every word not in the language with exactly 1/4 probability. Obviously EBPP is contained in BPP but are they equal? Has this been studied? What about the similarly definable ERP?
Motivation. My main motivation is that I wanted to know what the complexity theoretic analogue of the ``correct in expected value'' randomized algorithm of Faenza et al. (see http://arxiv.org/abs/1105.4127) would be. First I wanted to understand what decision problems such an algorithm can solve (with worst case polynomial running time). Let us denote this class by E(xpected)V(alue)PP. It is easy to see that USAT $\in$ EVPP. Also easy to see that EBPP $\subset$ EVPP. So this was my motivation. Any feedback about EVPP is also welcome.
In fact, their algorithm always outputs a nonnegative number. If we denote the decisions problems recognizable by such an algorithm by EVP(ositive)PP, then we still have USAT $\in$ EVPPP. While EBPP might not be a subset of EVPPP, we have ERP $\subset$ EVPPP. Maybe using these we can define a (nonnegative) rank for decision problems.