A Simple Kelly Problem I Can't Solve

[Utah]

Here is a Kelly situation I encounter often. I am not sure of the answer and I won't cloud anything up with my logic yet.

In simple terms, I am faced with a max bet that is equal to or below my Kelly value where I can arb the bet with a book that is overfunded (ie, if I don't arb it the money will go unused in the second book). Intuitively, the problem seems incredibly easy - you are below Kelly and any arbing has negative expectation. Don't arb. But, by some twisted logic and math I can make a strong case for arbing a portion of the bet.

Do I arb here? Is there ever a case where you arb when the soft side is below Kelly value?

Book A: overfunded
Book B: Overfunded

Arb: Book A: +104 Book B: +100 (book B is soft)
Non-vig line approx -108
Softline value = 4.2%
Chance of win = 48%
Value of arb = 1%
My basic Kelly calculator says bet $1000
Max Limit For Bet for Book B = $1000

I bet the $1000 max at Book B. Do I arb any portion of the bet with Book A?

note: I am very ill and not thinking 100% clearly. So, this may be a very easy problem. But, I don't think so.

[ImStillBen]

Umm, you're leaving out a key piece of information here, genius.

[Thremp]

I'm almost positive that you should still hedge. Actually no, I'm not. I don't know at all. Yes, I have no [censored] clue. I will publicly say I do not have any idea on the answer.

[internetdonk]

Here is the math answer.... Flip a coin to decide if you should hedge... Then flip a coin to decide which decide to take..

[rjp]

The answer is easy: which option maximizes your expected growth?

If you don't want the risk then you obviously arb, but you want to know what the Kelly answer is, so there you have it. [img]/images/graemlins/smile.gif[/img]

I'll leave you to do the math, but I imagine taking on the risk will give you a higher expected growth.

[Utah]

[ QUOTE ]
The answer is easy: which option maximizes your expected growth?

[/ QUOTE ]That doesn't seem to be an easy question to answer because it is dependent on the +ev of the bet with the expected winrate of the bet. For the bet in question, the edge is 4.2% and Kelly says the optimal bet is $1000. So, I expect to make $42 dollars on the bet. But, that doesn't seem right because I can instantly arb for $20 with zero risk.

So, in order to make the softside bet only, I need to give back the $20 I can get for basically doing nothing. In essence, I am being charged $20 for the bet upfront and thus in lowers the rate to something under 4.2%. If it is lower than 4.2% then the kelly value drops below $1000 and I am betting too much.

I believe kelly compares between betting and not betting. But, I don't think it accounts for the alternative of, "don't take any risk and you get X amount for free".

To summarize:
false - take bet and expect $42. Don't take bet and expect nothing
True - take bet and expect $42. Don't take bet and get free $20.

Something seems amiss. Again, I could be missing something completely obvious and my logic could be 100% wrong.

[rjp]

Well it should be easy to calculate. Obviously the arb is the best if you can bet as much as possible, but in this case it depends on all the other variables as there are limits in place.

Lets say you have a 50/50 chance of winning (substitute the true odds / payoffs as necessary), then the growth function for the arb is:

0.5*ln(1+%Profit from Bet1)+0.5*ln(1+%Profit from Bet2)

Compare this with the bet that takes on the risk and you have your answer. Oh, and make sure you specify %Profit from the bets as % of your bankrolls...

I could probably make this clearer, but hopefully it makes sense. [img]/images/graemlins/wink.gif[/img]

[ImStillBen]

You people are hilarious.

[rjp]

[ QUOTE ]
You people are hilarious.

[/ QUOTE ]

Thanks for adding so much...

[sillyarms]

Your Ev is $42 on the bet.
Your Ev is $20 on the arb.

The bet has higher EV. Take the bet.