Integral Equations and Phase Transitions in Stochastic Games. An Analogy with Statistical Physics

Abstract

Maximization of the Kullback--Leibler information is known to result in general Esscher transformations. The Bose--Einstein and Fermi--Dirac statistics in a probability space~$(\Omega, {\cal{F}},P)$ give rise to another kind of information, namely, $$ S_B=\int \log\bigg(1+\frac{dP}{dQ}\bigg)\,dQ+ \int \log\bigg(1+\frac{dQ}{dP}\bigg)\,dP $$ for the Bose statistics and $$ S_F =\int\log\bigg(\frac{dP}{dQ}-1\bigg)\,dQ -\int\log\bigg(1-\frac{dQ}{dP}\bigg)\,dP, \qquad \frac{dP}{dQ} >1, $$ for the Fermi statistics. This information generates measure transformations corresponding to these statistics. In the presence of a payoff matrix, these transformations vary in accordance with the integral equations given in the paper. We give examples of financial games corresponding to Bose and Fermi statistics.