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A coding theoretic study of homogeneous Markovian predictive games

Takara Nomura, Akio Fujiwara

TL;DR

The paper develops a time-homogeneous $k$th-order Markovian predictive game in which Forecaster's predictions arise from a Markov kernel and Skeptic uses a universal, Lempel-Ziv–based betting strategy to test Reality's sequence. The main result shows that Skeptic's capital $K_n$ diverges to infinity unless the empirical block frequencies $S_n(a^\ell)/n$ converge to the stationary distribution $P(a^\ell)$ for all $\ell\ge k$, establishing a game-theoretic law of large numbers tied to the Markovian structure. The authors connect coding theory with game-theoretic probability by interpreting the capital process as a likelihood ratio and by constructing a computable universal coding scheme via $Q_{LZ}$, deriving entropy-rate connections akin to LZ-78 and Brudno-type theorems. Applications to Szilárd's engine and entropy illustrate how deviations from the Forecaster's law enable extractable work and quantify information content through compression rates, highlighting implications for thermodynamics and information theory within a predictive-game framework.

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 Leó Szilárd's thought experiment.

A coding theoretic study of homogeneous Markovian predictive games

TL;DR

The paper develops a time-homogeneous th-order Markovian predictive game in which Forecaster's predictions arise from a Markov kernel and Skeptic uses a universal, Lempel-Ziv–based betting strategy to test Reality's sequence. The main result shows that Skeptic's capital diverges to infinity unless the empirical block frequencies converge to the stationary distribution for all , establishing a game-theoretic law of large numbers tied to the Markovian structure. The authors connect coding theory with game-theoretic probability by interpreting the capital process as a likelihood ratio and by constructing a computable universal coding scheme via , deriving entropy-rate connections akin to LZ-78 and Brudno-type theorems. Applications to Szilárd's engine and entropy illustrate how deviations from the Forecaster's law enable extractable work and quantify information content through compression rates, highlighting implications for thermodynamics and information theory within a predictive-game framework.

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 Leó Szilárd's thought experiment.

Paper Structure

This paper contains 17 sections, 16 theorems, 154 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Betting strategy inspired by Lempel-Ziv coding scheme
  4. Properties of Lempel-Ziv betting strategy
  5. Proof of Theorem \ref{['thm:kth_Markov_normal_number']}
  6. Applications
  7. Szilárd's engine game
  8. Entropy
  9. Concluding remarks
  10. Proof of lemmas
  11. Proof of Lemma \ref{['lem:liminf_comp_and_Q']}
  12. Proof of Lemma \ref{['lem:cond_empirical']}
  13. Proof of Lemma \ref{['lem:kth_Markov_stationary']}
  14. Proof of Lemma \ref{['lem:Ziv_ineq']}
  15. Proof of Lemma \ref{['lem:lth_Markov_divergence']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

In the simple predictive game, Skeptic has a prudent strategy $\beta : \Omega^\ast \rightarrow \mathbb{R}^\Omega$ that ensures $\lim_{n \to \infty} K_n = \infty$ unless for all $a \in \Omega$. Here, a strategy is called prudent if $K_n > 0$ for all $n\in\mathbb{Z}_{>0}$ and every sequence $\omega_1^n \in \Omega^n$ chosen by Reality.

Figures (2)

  • Figure 1: Szilárd's engine game.
  • Figure 2: Generalized Szilárd's engine game having three chambers. The pulleys can move horizontally and are assumed to be negligibly small.

Theorems & Definitions (29)

  • Theorem 1.1: Game-theoretic law of large numbers
  • Theorem 1.2
  • Remark
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • ...and 19 more