Effect of proportional betting on trip risk

From Stanford Wong's BJ21

Posted by Kim Lee on 18 Mar 1998, 3:56 p.m.

A "Kelly" proportional bettor needs to carry a big fraction of total bankroll to avoid short-term ruin on a lengthy trip. Choosing bets in proportion to total bankroll will reduce the probability of a trip ruin as bets shrink with losses. Don's formula for trip ruin with constant betting is

r = N((B-mu*t)/sigma/Sqrt(t))+exp(2*mu*B/sigma^2)*N((B+mu*t)/sigma/Sqrt(t))

where

r = probability of ruin,
N(.) = cumulative normal distribution,
B = negative of trip bankroll (units),
mu = hourly mean (units),
sigma = hourly standard deviation (units),
t = playing time (hours).

The "Kelly" optimal bankroll is (approximately) W = sigma^2/mu. The chance of trip ruin for a Kelly bettor is given by Don's formula with B replaced by -W*ln(W/(W+B))and mu replaced by mu/2. The formulas give similar results for short trips or small bankrolls. But the Kelly bettor has much less risk on an extended trip with a large bankroll. For example, on a t = 36 hour double deck trip (mu = 2.5, sigma = 27) Don's constant bettor has a 10% risk of ruin with a trip bankroll of -B = 200 units. A Kelly bettor only needs W = 291.6 units of total bankroll. The Kelly bettor has only a 2% risk of trip ruin with 200 units.

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