A coding theoretic study of homogeneous Markovian predictive games

TL;DR

This paper develops a game-theoretic framework for predictive games based on Markov processes, introducing a universal coding betting strategy and applying it to thermodynamics and Szilard's thought experiment.

Contribution

It presents a novel game-theoretic law of large numbers for Markovian predictions using a universal coding scheme, without diversified betting.

Findings

Established a law of large numbers for Markovian predictive games.

Introduced a betting strategy based on universal coding and martingale convergence.

Applied the framework to thermodynamics and Szilard's thought experiment.

Abstract

This paper explores a predictive game in which a Forecaster announces odds based on a time-homogeneous Markov kernel, establishing a game-theoretic law of large numbers for the relative frequencies of occurrences of all finite strings. A key feature of our proof is a betting strategy built on a universal coding scheme, inspired by the martingale convergence theorem and algorithmic randomness theory, without relying on a diversified betting approach that involves countably many operating accounts. We apply these insights to thermodynamics, offering a game-theoretic perspective on Le\'o Szil\'ard's thought experiment.