Effective martingales with restricted wagers

The classic model of computable randomness considers martingales that take real or rational values. Recent work shows that fundamental features of the classic model change when the martingales take integer values.We compare the prediction power of martingales whose wagers belong to three different subsets of rational numbers: (a) all rational numbers, (b) rational numbers excluding a punctured neighborhood of 0, and (c) integers. We also consider three different success criteria: (i) accumulating an infinite amount of money, (ii) consuming an infinite amount of money, and (iii) making the accumulated capital oscillate.The nine combinations of (a)-(c) and (i)-(iii) define nine variants of computable randomness. We provide a complete characterization of the relations between these notions, and show that they form five linearly ordered classes.