Survival probability in reactive gambling: monotonicity, singularity, and Hölder continuity

Abstract

A gambler starts with fortune $x$ and unit bet, then aims to double the bet after every win and halve it after every loss, while the actual bet is capped by the current fortune. Each round is won independently with probability $p\in(0,1)$. We study $f(x,p)$, the probability that the gambler is never ruined. We prove that $f(x,p)>0$ exactly when $x>2$ and $p<1/2$. For fixed $x>2$, $f(x,\cdot)$ is strictly decreasing and real analytic on $(0,1/2)$, and vanishes on $[1/2,1)$: playing a more favourable game lowers the chance of surviving it. For fixed $p\in(0,1/2)$, $f(\cdot,p)$ is strictly increasing and singular continuous on $[2,\infty)$. An elementary two-step argument gives an explicit global Hölder exponent, with constant one, which is strictly larger than $\log_2(1+p)$ and tends to $\frac12\log_2(111/46)=0.635426\ldots$ as $p\uparrow1/2$; we place it in a hierarchy of computable exponents and show, by an exact finite computation, that the optimal exponent is strictly smaller than the pointwise exponent at $x=2$ once $p$ is close to $1/2$. We determine that boundary exponent, and give the complete phase diagram when wins and losses use unequal multipliers.