Three gamblers enter the casino. They gamble on a sequences of *Red* or *Black*. Gambler 1 is allowed to place any real valued wagers, Gambler 3 may only wager integer sums of money, and Gambler 2, a hybrid of 1 and 3, may wager any real number outside the open interval (0,1). Is there a difference between their gambling power? Can the casino make Gambler 1 (or 1 and 2) win while 2 and 3 (or just 3) lose, no matter how the latter specify their strategies?
The answers to the above questions depend on how you define win, but not so much on how you define specify (as long as there are only countably many different specifications). We consider three success criteria: (1) accumulation of an infinite amount of money, (2) saving of an infinite amount of money, and (3) oscillation.

The three types of gamblers (1, 2, 3) combined with the three success criteria give rise to nine notions of pseudo-randomness. A binary sequence is random with respect to gambler-type *T* and success criterion *S*, if no finitely specified *T*-martingale achieves *S* on that sequence. It is shown that the nine combination define 5 different classes of pseudo-randomness with strict linear containment relation between them. This result is robust to merely any specification language. For example, once specification language allows only recursive (computable) definitions, whereas another possible specification language allows the usage of uncomputable predicates, such as the truth value of any mathematical statement.

Our results include solving questions raised in Bienvenu et al. (2012) and Teutsch (2012).