How reactive gambling can backfire: ruin probability is increasing in $p$, Hölder continuous in initial fortune

A gambler with an initial fortune $x$ starts by betting a dollar, then doubles the bet after every win and halves the bet after every loss. Let $p\in (0,1)$ be the probability of winning for each round. We show that the gambler survives with positive probability if and only if $p < 1/2$ and $x > 2$. Moreover, the ruin probability is increasing and real-analytic in $p$, but a singular, Hölder continuous function of $x$.