A pair of universal sequence-set betting strategies

TL;DR

This paper introduces the sequence-set betting game, a generalization of a non-monotonic betting game, and demonstrates that two strategies can collectively predict all non-Martin-Löf random strings, unlike any single computable strategy.

Contribution

It presents a new sequence-set betting framework and shows that a pair of strategies can predict all non-Martin-Löf random strings, unlike single strategies.

Findings

No single computable strategy predicts all non-Martin-Löf random strings.

Two strategies together predict every non-Martin-Löf random string.

The sequence-set betting game generalizes previous non-monotonic betting models.

Abstract

We introduce the sequence-set betting game, a generalization of An. A. Muchnik's non-monotonic betting game. Instead of successively partitioning the infinite binary strings by their value of a bit at a chosen position, as in the non-monotonic game, the player is allowed to partition the strings into any two clopen sets with equal measure. We show that, while there is no single computable sequence-set betting strategy that predicts all non-Martin-L\"of random strings, we can construct two strategies such that every non-Martin-L\"of random string is predicted by at least one of them.