How reactive gambling can backfire: ruin probability is increasing in $p$, H\"older continuous in initial fortune

TL;DR

This paper analyzes a gambling strategy where the ruin probability depends on the win probability and initial fortune, showing that survival chances are positive only under specific conditions and that ruin probability varies smoothly with p but irregularly with initial fortune.

Contribution

It establishes the precise conditions for positive survival probability and characterizes the ruin probability's dependence on p and initial fortune, highlighting its analytic and H"older continuity properties.

Findings

Survival probability is positive iff p<1/2 and x>2.

Ruin probability increases with p and is real-analytic in p.

Ruin probability is H"older continuous in initial fortune x.

Abstract

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, H\"older continuous function of $x$.