Re: Utility functions and optimal f

From Stanford Wong's BJ21

Posted by ML on 14 Dec 1997, at 1:39 p.m., in response to Utility functions and optimal f , posted by Karel Janecek on 14 Dec 1997, at 10:29 a.m.

Hi and thanks for the response. Thanks to Steve too.

I am going to have to print it out and study some before I understand all you are saying but let me address one point I think I do understand.

Some background so you will know better the extent (or lack) of my knowledge. I know the equation B(n)= ((1+f)^W)* ((1-f)^L)*B(0) tells the bankroll after n plays for any specific mix of wins and losses which might occur in a simple game with bets being resized instantly as the same fraction of bankroll. I was using p and 1-p to show I was working with the one outcome which is the expected outcome.

For example, for a simple game and a one percentage advantage, we all know the fraction should be .01.

So, arbitrarily taking 1000 as n we can write a short program.

10 input f

20 x=((1+f)^505)*((1-f)^495)

30 print f

I one plugs in different values for f, .01 will give the biggest value.

If one takes a complex game, Opie with Griffin betting for example and takes 8000 for n and puts in expected values of outcomes,

8000*.6*.51=2448 *.49=2352

8000*.3*.51=1224 *.49=1176

8000*.1*.53=424 *.47=376

20 x=((1+f)^23.52)*((1-f)^24.48)+((1+2*f)^12.24)* ((1-2*f)^11.76+((1+6*f)^4.24)*((1-6*f)^3.76

x is largest at what I call the Yamashita value.

When the middle bet multiplier is changed to 3.5 (the optimal spread), x is maximum at the Yamashita value.

Is what I am observing something trivial which makes sense to everyone else, is it a coincidence, or is there some reason? Whatever it is I do not see how it ties into anything to do with logarithmic increase.

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