The Kelly Criterion And Utility Function Optimisation For Stochastic Binary Games: Submartingale And Supermartingale Regimes

TL;DR

The paper reformulates the Kelly criterion for a memoryless binary Bernoulli game, showing that the optimal fractional stake $\mathcal{F}_{K}=p-q$ maximizes a log-utility $\mathsf{U}(\mathcal{F},p)$ and relates it to Shannon entropy via $\mathsf{U}_{K}=\log(2)-\mathscr{H}(p,q)$. It characterizes wealth dynamics as a (sub/super)martingale depending on the sign of $\mathsf{U}(\mathcal{F},p)$ and derives Doob-based results, variance, and volatility estimates, highlighting the exponential growth of wealth under the Kelly regime for large $N$. The work also extends to fractional Kelly fractions $f\mathcal{F}_{K}$ to balance growth against risk, illustrating the trade-off with concrete numerical examples. Overall, the study provides a rigorous, information-theoretic foundation for optimizing growth rates in stochastic binary games and clarifies the regimes in which wealth accumulates or depletes. The results have implications for risk-controlled betting and investment strategies where outcomes are binary and memoryless.

Abstract

A reformulation of the Kelly Criterion is presented. Let $\mathfrak{G}$ be a generic stochastic Bernoulli binary game with outcomes $\mathscr{Z}(I)\in\lbrace -1,1\rbrace$ of N trials for $I=1...N$. The binomial probabilities are $\mathsf{P}(\mathscr{Z}(I)=1)=p$ and ${\mathsf{P}}(\mathscr{Z}(I)=-1)=q$ with $p+q=1$. For a fair game $p=q=\tfrac{1}{2}$ and for a biased game $p>q$. If $\mathscr{W}(0)$ is the initial wealth then at the $I^{th}$ trial one bets a fraction $\mathcal{F}$ so that the bet is $B(I)=\mathcal{F}\mathscr{W}(I-1)$. If one wagers $B(I)$ and wins one recovers the original wager plus $B(I)$ if $\mathscr{Z}(I)=+1$, or a loss of $B(I)$ if $\mathscr{Z}(I)=-1$. The wealth at the $N^{th}$ trial/bet for large $N$ is the random walk $\mathscr{W}(N)=\mathscr{W} (0)+\sum_{I=1}^{N}B(I)\mathscr{Z}(I)=\mathscr{W}(0)\prod_{I=1}^{N}(1+\mathcal{F}\mathscr{Z}(I))$ with expectation $\mathsf{E}[\mathscr{W}(N)]$. Defining a 'utility function' $\mathsf{U}(\mathcal{F},p)=\mathsf{E}[\log(\mathscr{W}(N)/\mathscr{W}(0))^{1/N}]$ then $\mathsf{U}(\mathcal{F},p)$ is optimised by the Kelly fraction $\mathcal{F}=\mathcal{F}_{K}=p-q=2p-1$, which is essentially a critical point of $\mathsf{U}(\mathcal{F},p)$. Also $\mathsf{U}(\mathcal{F}_{K},p)$ can be related to the Shannon entropy. If $[0,1]=[0,\mathcal{F}_{*})\bigcup [\mathcal{F}_{*}]\bigcup (\mathcal{F}_{*},1]$ with $\mathsf{U}(\mathcal{F}_{*},p)=0$ then $\mathsf{U}(\mathcal{F},p)>0, \forall\mathcal{F}\in[0,\mathcal{F}_{*})$ and $\mathscr{W}(N)$ is a submartingale for $p>1/2$; also $\mathsf{U}(\mathcal{F},p)<0,\forall \mathcal{F}\in(\mathcal{F}_{*},1]$, and $\mathscr{W}(\mathcal{F},p)$ is a supermartingale. Estimates are derived for variance and volatility $\mathsf{VAR}(\mathscr{W}(N))$ and $σ(\mathscr{W}(N))=\sqrt{\mathsf{VAR}(\mathscr{W}(N)})$. For large $N$ and $\mathcal{F}=\mathcal{F}_{K}$, $\mathsf{E}[\mathscr{W}(N)]$ grows exponentially.

Figures (6)

• Figure 1: Utility function $\bm{\mathsf{U}}(\mathcal{F},p)$ versus $\mathcal{F}$ with a peak at Kelly fraction $\mathcal{F}_{K}=0.4$
• Figure 2: Utility function curves $\bm{\mathsf{U}}(\mathcal{F},p)$ for p=0.52, p=0.515, p=0.51
• Figure 3: Plots of $\bm{\mathsf{E}}[\mathscr{W}(N)]=\mathscr{W}(0)(1+\mathcal{F}_{K}(2p-1)-p(1-p)\mathcal{F}^{2})^{N}$ vs $N$ for $p=0.51, p=0.52, p=0.53$ for the Kelly fraction $\mathcal{F}_{K}=2p-1$
• Figure 4: Plots of $\bm{\mathsf{E}}[\mathscr{W}(N)]=\mathscr{W}(0)(1+\mathcal{F}(2p-1)-p(1-p)\mathcal{F}^{2}))^{N}$ vs $N$ for $p=0.51, p=0.52, p=0.53$ for $\mathcal{F}=0.2\gg\mathcal{F}_{*})$
• Figure 5: Plots of $\bm{\mathsf{E}}[\mathscr{W}(N)]$ vs N for $p=0.52$ and $\mathcal{F}_{K}=0.04$ and $F_{K}=f\mathcal{F}_{K}=\tfrac{2}{3}\mathcal{F}_{K}=\tfrac{2}{75}$

Theorems & Definitions (61)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Definition 2.6
• Lemma 2.7