We classify the extreme points of a polytope of probability distributions in the (2,2,2) CHSH-Bell setting that is induced by a single Tsirelson bound. We do the same for a class of polytopes obtained from a parametrized family of multiple Tsirelson bounds interacting non-trivially. Such constructions can be applied to device-independent random number generation using the method of probability estimation factors [Phys. Rev. A, 98:040304(R), 2018]. We demonstrate a meaningful improvement in certified randomness applying the new polytopes characterized here.