  We study the problem of whether a betting-strategy can be decomposed into an
equivalent set of simpler betting-strategies, such as betting-strategies that
bet on a restricted set of stages or bet on a restricted of favorable outcomes.
We show that the class of effectively enumerable betting-strategies has
irreducible members which cannot be decomposed into an equivalent set of
simpler betting-strategies. We answer questions of Kastermans and Hitchcock by
constructing a real on which no kastergale (which is a left-c.e.
supermartingale whose favorable outcomes are effectively determined) succeeds,
but some unrestricted left-c.e. supermartingale succeeds on it. We generalize a
result of Muchnick by showing that there is a non-1-random real such that no
muchgale (which is a left-c.e. supermartingale who does not bet on certain
stages) succeeds on it. Our methodology is then used to obtain further
irreducibility results, strongly supporting a conjecture that if a natural
class of left-c.e. supermartingales defines 1-randomness, then a single member
of that class can do so. In another word, the class of left-c.e.
supermartingales cannot be reduced to a simpler, natural subclass of it. For
example, we show that there is a non-1-random real such that no kastergale or
muchgale succeeds on it.

Probability;Logic