# Randomness, stochasticity and approximations

## Abstract

Polynomial time unsafe approximations for intractable sets were introduced by Meyer and Paterson [9] and Yesha [19] respectively. The question of which sets have optimal unsafe approximations has been investigated extensively, see, e.g., [1, 5, 15, 16]. Recently, Wang [15, 16] showed that polynomial time random sets are neither optimally unsafe approximable nor *Δ*-levelable. In this paper, we will show that: (1) There exists a polynomial time stochastic set in **E**_{2} which has an optimal unsafe approximation. (2). There exists a polynomial time stochastic set in **E**_{2} which is *Δ*-levelable. The above two results answer a question asked by Ambos-Spies and Lutz et al. [3]: Which kind of natural complexity property can be characterized by *p*-randomness but not by *p*-stochasticity? Our above results also extend Ville's [13] historical result. The proof of our first result shows that, for Ville's stochastic sequence, we can find an optimal betting strategy (prediction function) such that we will never lose our own money (except the money we have earned), that is to say, if at the beginning we have only one dollar and we always bet one dollar that the next selected bit is 1, then we always have enough money to bet on the next bit. Our second result shows that there is a stochastic sequence for which there is a betting strategy such that we will never lose our own money (except the money we have earned), but there is no such kind of optimal betting strategy. That is to say, for any such kind of betting strategy, we can find another betting strategy which could be used to make money more quickly.

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