brh

Originally Posted by brh

Originally Posted by brh

Thanks Gramazeka, I need these results for another thread:

I am interested in the case where the Kelly bettor sets their goal at Infinity.

1. Prob [Bank reaches "b", before loosing "a"](k) = [1-a^(1-2/k)]/[b^(1-2/k) - a^(1-2/k)]

If lim b-> infinity (k<2),
Prob [Inf,a](k) = [1-a^(1-2/k)]/[ 0 - a^(1-2/k) ]

= - [1-a^(1-2/k)]/[ a^(1-2/k) ]

= 1 - a^(2/k-1)

If k = 1 (2/k-1) = 2 - 1 = 1.

So Prob[Inf,a](1.0) = 1 - a => P(Inf,0.5](1.0) = 0.5 = 50%

If k = 1/2 (2/k-1) = 4 - 1 = 3.

So Prob[Inf,a](0.5) = 1 - a^3 => P(Inf,0.5)(0.5) = 1 - (0.5)^3 = 0.875 = 87.5%

Turning this around:
2. Prob` [Bank looses "a", before reaching "b"](k) = 1 - [1-a^(1-2/k)]/[b^(1-2/k) - a^(1-2/k)]

If lim b-> infinity (k<2),

Prob` [Inf,a](k) = a^(2/k-1)

So Prob`[Inf,a](1.0) = a => P`(Inf,0.5](1.0) = 0.5 = 50%

So Prob`[Inf,a](0.5) = a^3 => P`(Inf,0.5)(0.5) = (0.5)^3 = 0.125 = 12.5%

To summarise, a Kelly bettor who has a goal of Infinity has:

a 50% chance of being at half the starting banrkoll at some stage for k=1,

a 12.5% chance of being at half the starting banrkoll at some stage for k=1/2.

I needed to paste this quote from here :
http://www.blackjacktheforum.com/sho...ed=1#post18058

Thanks to Gramazeka, I can fill in the final gaps in the above post.

Firstly the paper I was referring to was here: http://www.blackjackforumonline.com/...eads-howto.pdf

This is where I got to before falling asleep:

....
But here we have the BIG proviso that if W(n) = $5000 [ can't tell you <n>, Wiener functions are funny like that ],

And the table minimum is $5 the Kelly bettor is in deep do-do (Aussie for shi#), because the Bankroll is now
$5000, $B = $5, resizing lower is not an option, any they may go to ruin after only another N0 = 20000 rounds.
So it IS possible for a Kelly bettor to go to ruin due to table minimums.

So there we have it - the fixed bettor has a 1.83% ROR after about 80000 rounds.

The Kelly bettor has an unquantifiable but finite (without resorting to Stochastic Differential Equations) ROR of
somewhat less than 1.83% ROR in this case due to a table minimum of $5, probably about the same
order of 80000 rounds.
...

Well thanks to Gramazeka the risk for the Kelly bettor IS quantifiable:

...
To summarise, a Kelly bettor who has a goal of Infinity has:
a 50% chance of being at half the starting banrkoll at some stage for k=1,
a 12.5% chance of being at half the starting banrkoll at some stage for k=1/2.

...
But here we have the BIG proviso that if W(n) = $5000 [ can't tell you <n>, Wiener functions are funny like that ]
...
We can't put an estimate on the number of rounds, but we now have the result that for a Kelly bettor with a starting bankroll,
of $10000, with a goal of infinity, with k=1/2, has a 12.5% chance of at some stage halving that bank ie W(n) = $5000.

Now the Kelly bettor has resized to a spread of $5 - $40 with a bankroll of $5000 = EKB and now becomes a fixed bettor
with an ROR of 13.6% since the table minimum prevents any further lowering of $B by resizing.

Since these probabilities are disjoint, this means that the Kelly bettor HAS a total ROR of 12.5% x 13.6% = 1.7%.

This is not appreciably different than the 1.86% for the fixed bettor !!

So after being unlucky enough (12.5%) for the bankroll to halve to $5000, unless the Kelly bettor replentishes
the Banrkoll from other sources - they are basically rooted.

If they persist spreading $5 - $40 with a $5000 bank, unless they are lucky (and advantage players can't rely on this),
they are way overbetting and may go broke after N0 = 20000 rounds.

So the other option is to reduce the spread to say $5-$10 or $5-$20. In a game like this a 1-2 or 1-4 spread will blow
out your N0 to somewhere > 100,000 rounds. So you may not lose your precious $5000 bankroll, but you will be
playing red chip for the rest of your natural life.

Chew on this .