  In the theory of algorithmic randomness, several notions of random sequence
are defined via a game-theoretic approach, and the notions that received most
attention are perhaps Martin-Loef randomness and computable randomness. The
latter notion was introduced by Schnorr and is rather natural: an infinite
binary sequence is computably random if no total computable strategy succeeds
on it by betting on bits in order. However, computably random sequences can
have properties that one may consider to be incompatible with being random, in
particular, there are computably random sequences that are highly compressible.
The concept of Martin-Loef randomness is much better behaved in this and other
respects, on the other hand its definition in terms of martingales is
considerably less natural. Muchnik, elaborating on ideas of Kolmogorov and
Loveland, refined Schnorr's model by also allowing non-monotonic strategies,
i.e. strategies that do not bet on bits in order. The subsequent
``non-monotonic'' notion of randomness, now called Kolmogorov-Loveland
randomness, has been shown to be quite close to Martin-Loef randomness, but
whether these two classes coincide remains a fundamental open question. As
suggested by Miller and Nies, we study in this paper weak versions of
Kolmogorov-Loveland randomness, where the betting strategies are non-adaptive
(i.e., the positions of the bits to bet on should be decided before the game).
We obtain a full classification of the different notions we consider.
