Game Derandomization

TL;DR

The paper develops Kolmogorov Game Derandomization to upper-bound the complexity of deterministic winning players across deterministic, probabilistic, computable, lower computable, and uncomputable environments, extending from a Crete-like illustrative game to zero-sum repeated games. It leverages the Monotone EL theorem, game fragments, and semi-agent constructions to derive bounds of the form $\mathbf{K}(\cdot) <^{\log} \text{(environment-information)}$, often including the halting-sequence term $\mathbf{I}(\cdot;\mathcal{H})$. It delivers new bounds, resource-bounded variants, and partial-derandomization results, and it discusses fundamental obstacles such as Lose/No-Halt games and the role of exotic information. The contributions provide the first resource-bounded upper bounds for deterministic players in such game settings and illuminate how algorithmic information theory informs the feasibility and limits of derandomization in strategic environments.

Abstract

Using Kolmogorov Game Derandomization, upper bounds of the Kolmogorov complexity of deterministic winning players against deterministic environments can be proved. This paper gives improved upper bounds of the Kolmogorov complexity of such players. This paper also generalizes this result to probabilistic games. This applies to computable, lower computable, and uncomputable environments. We characterize the classic even-odds game and then generalize these results to time bounded players and also to all zero-sum repeated games. We characterize partial game derandomization. But first, we start with an illustrative example of game derandomization, taking place on the island of Crete.

Theorems & Definitions (68)

• Theorem 1: Epstein23
• Definition 1: Stochasticity
• Lemma 1: EpsteinOutliers23Levin16
• Lemma 2: Epstein22Exm22
• Theorem 2: Monotone EL Theorem, Epstein24
• Theorem 3
• proof
• Claim 1
• Theorem 4: Probabilistic Environments
• proof