Jarzynski-Type Equalities in Gambling: Role of Information in Capital Growth

Abstract

We study the capital growth in gambling with (and without) side information and memory effects. We derive several equalities for gambling, which are of similar form to the Jarzynski equality and its extension to systems with feedback controls. Those relations provide us with new measures to quantify the effects of information on the statistics of capital growth in gambling. We discuss the implications of the equalities and show that they reproduce the known upper bounds of average capital growth rates.

Appendices

Appendix 1: Notations and Definitions

Here we summarize notations and definitions of information-theoretical quantities used in the text.

A realization of a stochastic variable X is represented by its small letter, x in this case.

Let $\{ x_i\}$, $\{y_i\}$ be time-sequences of stochastic variables. A variable with a superscript n indicates variables from 1 to n collectively,

$$\begin{aligned} x^n \equiv \{ x_1, \ldots , x_n \}. \end{aligned}$$

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We assume that the dependences of the variables $x^n$ are causal, by which we mean that the probability distribution $P(x_i)$ is dependent on $x_j$ only if $j<i$. The joint probability $P(x^n)$ is decomposed as

$$\begin{aligned} P(x^n) = \prod _i P(x_i | x^{i-1}) , \end{aligned}$$

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where P(x|y) is the conditional probability.

The average over variables $\{x,y,\ldots \}$ is expressed by $\langle \cdots \rangle _{x,y,\ldots }$. Subscript may be omitted, in that case the average is taken over all the stochastic variables.

The Shannon entropy of $X^n = \{X_1, \ldots , X_n \}$ is

$$\begin{aligned} S(X^n) \equiv - \langle \ln P(x^n) \rangle _{x^n} = - \sum _{x^n} P(x^n) \ln P(x^n) . \end{aligned}$$

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The Kullback-Leibler divergence of a distribution Q(y) from another distribution P(y) is defined by

$$\begin{aligned} D_\mathrm{KL}\left( P(y)||Q(y) \right) \equiv \sum _y P(y) \ln \frac{P(y)}{Q(y)}. \end{aligned}$$

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The mutual information between the stochastic variable X and Y is defined by

$$\begin{aligned} I(X:Y) \equiv \left\langle \ln \frac{P(x,y)}{P(x)P(y)} \right\rangle _{x,y} = \sum _{x,y} P(x,y) \ln \frac{P(x,y)}{P(x)P(y)}. \end{aligned}$$

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We used the following causal conditioning notations developed by Kramer [32]. The probability distribution of $x^n$ causally conditioned on $y^{n-d}$ is denoted as

$$\begin{aligned} P(x^n || y^{n-d}) \equiv \prod _{i=1}^n P(x_i | x^{i-1}, y^{i-d}). \end{aligned}$$

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We use a convention that, if $i-d \le 0$, $y^{i-d}$ is set to null. Mostly, the cases with $d=0,1$ are used:

$$\begin{aligned} P\big (x^n || y^{n}\big ) = \prod _{i=1}^n P\big (x_i | x^{i-1}, y^{i}\big ), \end{aligned}$$

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$$\begin{aligned} P\big (x^n || y^{n-1}\big ) = \prod _{i=1}^n P\big (x_i | x^{i-1}, y^{i-1}\big ). \end{aligned}$$

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The joint probability of $x^n$ and $y^n$ is decomposed as

$$\begin{aligned} P\big (x^n , y^n\big ) = P\big (x^n || y^n\big ) P\big (y^n || x^{n-1}\big ) . \end{aligned}$$

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$$\begin{aligned} \begin{array}{ll} \because P(x^n , y^n) &{}= \prod \limits _i P(x_i, y_i | x^{i-1}, y^{i-1}) \\ &{}= \prod \limits _i P\big (x_i | x^{i-1}, y^i\big ) P\big (y_i | x^{i-1}, y^{i-1}\big ) \\ &{}= P\big (x^n || y^n\big ) P\big (y^n || x^{n-1}\big ). \end{array} \end{aligned}$$

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The causally conditional entropy is defined as

$$\begin{aligned} S(X^n || Y^n) \equiv - \langle \ln P(x^n || y^n)\rangle = \sum _{i=1}^n S\big (X_i | X^{i-1}, Y^i\big ) . \end{aligned}$$

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The directed information, introduced by Massey [33], is defined as

$$\begin{aligned} I_\mathrm{dr}(Y^n \rightarrow X^n) \equiv S(X^n) - S(X^n || Y^n) . \end{aligned}$$

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It can be explicitly written as

$$\begin{aligned} I_\mathrm{dr}(Y^n \rightarrow X^n) = \left\langle \ln \frac{P(x^n || y^n)}{P(x^n)} \right\rangle = \sum _i \left\langle \ln \frac{P\big (x_{i+1} | x^{i}, y^{i+1}\big )}{P\big (x_{i+1}|x^{i}\big )} \right\rangle . \end{aligned}$$

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Appendix 2: Markovian Coin Tossing and 1D Ising Model

We here show the equivalence of the Markovian coin tossing discussed in Sect. 4.1 with the 1D Ising model. Without loss of generality, we can parametrize the conditional probability $P(y_{i+1}|y_i)$ as

$$\begin{aligned} P(y_{i+1}|y_i) = \frac{\exp \left[ \beta J \ y_{i+1} y_i \right] }{2 \cosh \beta J} . \end{aligned}$$

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One can see $0 < P(y_{i+1}|y_i) < 1$ and the normalization condition $\sum _{y_{i+1}}P(y_{i+1}|y_i)=1$ is satisfied. The new parameter J can be related to the flipping rate $\epsilon$ as

$$\begin{aligned} \beta J = \frac{1}{2} \ln \frac{\bar{\epsilon }}{\epsilon } . \end{aligned}$$

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By rewriting the normalization condition of $P(y^n)$ in the following way, the correspondence to the Ising model is evident:

$$\begin{aligned} \begin{array}{ll} 1 &{}= \sum \limits _{y^n} P(y^n) \\ &{}= \sum \limits _{y^n} \exp \left[ \sum \limits _i \ln P(y_{i+1}|y_i) \right] \\ &{}= \frac{1}{\left( 2 \cosh \beta J\right) ^n} \sum \limits _{y^n} \exp \left[ \sum \limits _i \beta J \ y_{i+1}y_i \right] \\ &{}\equiv \frac{\mathrm{tr}\left[ e^{- \beta H} \right] }{Z} . \end{array} \end{aligned}$$

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Thus, the numerator is the definition of the partition function of the Ising model without external fields. The average of the exponential of g is written as

$$\begin{aligned} \begin{array}{ll} \langle \exp \left[ n g_n \right] \rangle _{y^n} &{}= \sum \limits _{y^n} P(y^n) \prod \limits _i (1 + f_i y_i) \\ &{}= \frac{1}{\left( 2 \cosh \beta J\right) ^n} \sum \limits _{y^n} \exp \left[ \sum \limits _i \beta J \ y_{i+1}y_i + \sum \limits _i \ln (1 + f_i y_i) \right] . \end{array} \end{aligned}$$

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When $f_i(y_i|y^{i-1})$ is independent of $y^{i-1}$, the numerator of RHS is the partition function of the Ising model in a weird form of magnetic field. In one dimension, the symmetry breaking never occurs in the Ising model at finite temperature. In the context of the Markovian coin tossing, the absence of symmetry breaking corresponds to the fact that, at finite values of $\epsilon$, the coin flips after finite number of trials, and the “magnetization” always vanishes,

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{1}{n}\sum _i \langle y_i \rangle = 0 . \end{aligned}$$

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