A pair of universal sequence-set betting strategies

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.

💡 Research Summary

The paper introduces a new betting framework called the “sequence‑set betting game,” which generalizes Muchnik’s non‑monotonic betting game. In the classic non‑monotonic game a player refines the space of infinite binary sequences by fixing a bit position and splitting according to the value of that bit. The sequence‑set game removes this restriction: at each round the player may partition the current set of sequences into any two clopen (simultaneously closed and open) subsets of equal Lebesgue measure (½ each). This richer partitioning power makes the game more expressive while preserving fairness, because each side of the split carries the same measure.

The authors first recall the connection between betting games and algorithmic randomness, especially Martin‑Löf randomness (ML‑randomness). It is known that no single computable betting strategy can succeed on every non‑ML‑random sequence; such a universal strategy does not exist because the class of non‑random sequences is too complex to be captured by a single effective betting rule.

The main contribution is the construction of two computable sequence‑set betting strategies, denoted (S_1) and (S_2), with the property that for every binary sequence that fails the Martin‑Löf test (i.e., is not ML‑random), at least one of the two strategies will succeed. Success is defined in the usual betting‑game sense: the capital of the strategy grows without bound along the infinite play on that particular sequence.

The construction proceeds as follows. For each stage of a universal Martin‑Löf test ({U_n}), the authors consider the clopen components of (U_n) that have measure (2^{-n}). They design a priority function that, given the current history of bets, selects a clopen split that “targets” the part of the space where the opponent sequence is likely to lie. Strategy (S_1) focuses on a certain family of splits (for example, those that isolate strings with a 0 in a specific position), while (S_2) is defined to use the complementary family (e.g., strings with a 1 in that position). Both strategies are computable because the priority function can be evaluated effectively from the finite history.

The key technical lemma shows that for any non‑ML‑random sequence (x), there exists a stage (n) such that (x\in U_n). At that stage, the clopen component containing (x) will be selected by at least one of the two priority functions, guaranteeing that the corresponding strategy places its bet on the correct side of the split. Consequently the capital of that strategy is multiplied by a factor greater than one, and iterating this argument yields unbounded capital growth. If the first strategy fails to capture the component, the complementary design of the second strategy ensures it will capture it, establishing the “covering” property of the pair.

The authors term the pair ((S_1,S_2)) a universal pair of sequence‑set betting strategies. This notion is novel: while a single universal betting strategy is impossible, a small finite collection can achieve universality when the game’s partitioning power is sufficiently rich. The paper also discusses the measure‑theoretic intuition behind the result: the two strategies together partition the space of non‑random sequences into two effectively enumerable subclasses, each of which is captured by one strategy.

In the discussion section the authors raise several open problems. One is whether a single strategy could be obtained by allowing more sophisticated betting rules (e.g., non‑uniform capital allocation) or by relaxing the requirement that the two subsets have exactly equal measure. Another direction is to explore analogous universal pairs in other spaces (such as the unit interval with binary expansions) or under different randomness notions (e.g., Schnorr randomness). The paper also suggests investigating the minimal number of strategies required for universality; the current result shows that two suffice, but it remains unknown whether one is ever possible.

Overall, the work deepens the relationship between algorithmic randomness and betting games by showing that the limitation of a single computable betting strategy can be overcome through a carefully coordinated pair. It expands the toolkit for studying randomness via game‑theoretic methods and opens new avenues for research on the interplay between computability, measure theory, and probabilistic betting models.