Comparing disorder and adaptability in stochasticity

Abstract: In the literature, there are various notions of stochasticity which measure how well an algorithmically random set satisfies the law of large numbers. Such notions can be categorized by disorder and adaptability: adaptive strategies may use information observed about the set when deciding how to act, and disorderly strategies may act out of order. In the disorderly setting, adaptive strategies are more powerful than non-adaptive ones. In the adaptive setting, Merkle et al. showed that disorderly strategies are more powerful than orderly ones. This leaves open the question of how disorderly, non-adaptive strategies compare to orderly, adaptive strategies, as well as how both relate to orderly, non-adaptive strategies. In this paper, we show that orderly, adaptive strategies and disorderly, non-adaptive strategies are both strictly more powerful than orderly, non-adaptive strategies. Using the techniques developed to prove this, we also make progress towards the former open question by introducing a notion of orderly, ``weakly adaptable'' strategies which we prove is incomparable with disorderly, non-adaptive strategies.