Re: The Mathematics of poker by Bill Chen & some dude...
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From what I have seen of if so far it is defintely a book that is very daunting for the non mathematically minded. I have a mathematics and statistics degree - from a long time ago now and I am rusty, but even for someone like me I think it will need very careful reading and involve some relearning to grasp the equations. At the minute I'm still trying to work out the RoRU equation. I wish Chen had given some examples of it in this book [img]/images/graemlins/frown.gif[/img]
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Example:
Suppose you have played 22,500 hands of 9-handed limit holdem, and you've won 337.5 bets, with a std deviation of
17 bets/100. So your win rate w is 1.5 bb/100, and your standard deviation s is 17.
Traditional risk of ruin says, for a 300 bet bankroll:
ror = exp(-2*w*b/s^2) = exp(-2*1.5*300/289) = 4.44%
This is the risk of ruin if your TRUE win rate is 1.5 bb/100 and your TRUE standard deviation is 17 bb/100.
But you don't really know your true win rate, and you don't have enough of a sample to conclude that it's really super close to those numbers. Instead, we can approximate your win rate by another normal distribution. Suppose your win rate is a random variable with mean w and standard deviation that we can calculate:
s_w = 17/(sqrt(225)) = 1.1333
If we assume your win rate is distributed normally (it isn't, but it's a better approximation than assuming your win rate is exact), then we can do better by calculating your RoR for all the points in that distribution and summing them by their weights. To do this we use calculus, as in our book, and arrive at the RoRU formula:
roru = r(w,b)*(exp(2b^2(s_w^2/s^4)))(phi(w - 2b(s_w^2/s^2)) + phi(-w/s_w)
which if I've done the math right, comes out to 17.89%.
This reflects the uncertainty about our win rate -- the base RoR is our risk of ruin if we are 100% accurate on our win rate and standard deviation. But here we might have just overperformed our expectation - we're only about 1.5 standard deviations away from zero. And at w=zero, risk of ruin is 100%, while at w = 3.0 (the other side), risk of ruin only goes down by, say, 3%.
-- some dude
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