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

Paper

Abstract

A gambler with an initial fortune

starts by betting a dollar, then doubles the bet after every win and halves the bet after every loss. Let

be the probability of winning for each round. We show that the gambler survives with positive probability if and only if

and

. Moreover, the ruin probability is increasing and real-analytic in

, but a singular, Hölder continuous function of