Kelly betting on multiple outcomes

[lij45o6]

So, lets say that you have three outcomes: x_1, x_2, and x_3. Each as a probability and an expectation: e_1->p_1...e_3->p_3.

Now, I would like to know how to find the appropriate kelly bet relative to bankroll B. One way that is given is find the mean of the system:

mu = SIGMA e_i*p_1 //from k=1 to i.

This will find out mean, or "expected value"

Next, find variance:

(sigma)^2 = SIGMA (e_i*p_i - mu)^2 * p_i //from k=1 to i.

Once variance has been solved, use the following equation:

K = B*(mu)/((sigma)^2)

Which would be out kelly bet relative to our bankroll.

Is there another way of finding kelly on multiple outcomes, or is this it?

[DSchles]

Hope some of this helps.

http://math.stackexchange.com/questi...n-two-outcomes
http://math.stackexchange.com/questi...for-3-outcomes
https://en.wikipedia.org/wiki/Kelly_criterion (scroll down to "Multiple Horses")
http://www.albionresearch.com/kelly/

There's a lot more on the Internet. Just Google "Kelly wagers on multiple outcomes."

Don

[DSchles]

I found another really good one, for Kelly in general:

http://www.elem.com/~btilly/kelly-criterion/Don

[lij45o6]

Thanks Don!

[Gronbog]

I found another really good one, for Kelly in general:

http://www.elem.com/~btilly/kelly-criterion/Don

I get "404 Not found" for this link

[DSchles]

You might do better without the "Don" at the end! :-) Sorry.

Don

[lij45o6]

Don (or to whom this subject is most familiar);

With respect to risk of ruin, assuming a fixed bankroll, we can assume a risk of ruin of 0.135 without adjusting our betting structure.

For example: A CC'er possesses a 10,000 USD bankroll and plans to use a count that offers close to 2% or 0.02. The SD of typical blackjack is 1.17. His average max bet (for 2% advantage) should be:

b_kelly = (10000*0.02)/(1.17^.5) = 145 USD

Regardless if this counter changes their bet unit size (lets say it is 25 USD), their risk of ruin should be around 13.5%, correct?

Here is another example.

Say your buddy offers you a 50:50 odds coin toss. However, this coin is biased towards heads. The probability for heads and tails is:

H: 0.5025
T: 0.4975

You find the expectation of heads and tails as:

H: +1
T: -1

for a mean of 0.005

The variance of the coin is:

0.24999375

The proper kelly bet of units relative to a whole, fixed bank is:

b_kelly = (B*EV)/(var) = 0.02 or 2% of your bank,

NOW. What is the risk of ruin with a bank of 1000 USD and units of 25 USD?

Our bank in units is:

40

We have our variance of 0.24999375.

Our risk of ruin can be computed as:

r_ruin = e^[(-2*B_units*EV)/(var)] which equals 20.13% approximately.

Would the above equation in bold be the correct way to establish a risk of ruin for any game with any given edge, bankroll, and variance? I found it in BJA pg 113. 3rd ed.

[DSchles]

Looks correct to me.

Don