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The classic model of computable randomness considers martingales that takereal or rational values. Recent work by Bienvenu et al. (2012) and Teutsch(2014) shows that fundamental features of the classic model change when themartingales take integer values.  We compare the prediction power of martingales whose wagers belong to threedifferent subsets of rational numbers: (a) all rational numbers, (b) rationalnumbers excluding a punctured neighbourhood of 0, and (c) integers. We alsoconsider three different success criteria: (i) accumulating an infinite amountof money, (ii) consuming an infinite amount of money, and (iii) making theaccumulated capital oscillate.  The nine combinations of (a)--(c) and (i)--(iii) define nine notions ofcomputable randomness. We provide a complete characterization of the relationsbetween these notions, and show that they form five linearly ordered classes.  Our results solve outstanding questions raised in Bienvenu et al. (2012),Teutsch (2014), and Chalcraft et al. (2012), and strengthen existing results.

Effective Martingales with Restricted Wagers