How to build a probability-free casino

Casinos operate by generating sequences of outcomes which appear unpredictable, or random, to effective gamblers. We investigate relative notions of randomness for gamblers whose wagers are restricted to a finite set. Some sequences which appear unpredictable to gamblers using wager amounts in one set permit unbounded profits for gamblers using different wager values. In particular, we show that for non-empty finite sets A and B, every A-valued random is B-valued random if and only if there exists a k⩾0 such that B⊆A⋅k.