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