Generalized second law of thermodynamics in the Glosten-Milgrom model

TL;DR

This paper establishes a generalized second law of thermodynamics for the Glosten-Milgrom model, providing an upper bound on informed traders' gains, supported by Bayesian inference and entropic inequalities, with numerical validation.

Contribution

It extends previous thermodynamic bounds to the finite-horizon Glosten-Milgrom model using novel Bayesian and entropic methods.

Findings

Derived an upper bound for expected trader gains.

Identified a characteristic timescale in the model.

Numerical results illustrate gain fluctuations.

Abstract

We derive an upper bound for the expected gain of informed traders in the Glosten-Milgrom model with finite horizon, fully analogous to a generalized second law of thermodynamics. This result extends that obtained by Touzo et al. a couple of years ago. The proof relies on Bayesian inference (exploiting the invariance of the problem under consecutive game sequences) and an interesting entropic inequality. We also provide numerical results both supporting the existence of a characteristic timescale in the model and illustrating the magnitude of gain fluctuations. Other possible extensions are discussed.