A Simple Kelly Problem I Can\'t Solve
 

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.

Re: A Simple Kelly Problem I Can\'t Solve
 

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

Re: A Simple Kelly Problem I Can\'t Solve
 

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.

Re: A Simple Kelly Problem I Can\'t Solve
 

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

Re: A Simple Kelly Problem I Can\'t Solve
 

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.

Re: A Simple Kelly Problem I Can\'t Solve
 

[ 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.

Re: A Simple Kelly Problem I Can\'t Solve
 

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]

Re: A Simple Kelly Problem I Can\'t Solve
 

You people are hilarious.

Re: A Simple Kelly Problem I Can\'t Solve
 

[ QUOTE ]
You people are hilarious.

[/ QUOTE ]

Thanks for adding so much...

Re: A Simple Kelly Problem I Can\'t Solve
 

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

The bet has higher EV. Take the bet.

Re: A Simple Kelly Problem I Can\'t Solve
 

Sillyarms,

The goal is to maximize bankroll growth, not EV.

Re: A Simple Kelly Problem I Can\'t Solve
 

[ QUOTE ]
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.

[/ QUOTE ]

You have to understand that I'm a noob sports bettor and do not follow Kelly very closely so I'm sorta talking out my ass here...

Mathematically, doesn't this drop the win from $42 to $22? Kelly I thought would determine bankroll size based on doing nothing, but doing nothing here isn't $0, it's $20.

I'm rereading your post and I'm seeing that you're saying this already of course. So I guess I'm agreeing with what you said there. It becomes sort of a tricky question then cause the line will move pretty constantly. $42 drops to $22, but when you hedge your win isn't $20 it's actually whatever you hedge for, so the win moves again.

I dunno...I'm fuzzy this morning and not willing/able to think about it more critically than that. But from a pure math perspective it looks to me like max br growth by Kelly would say you should be hedging at least a portion (about half?) of this bet.

- C -

Re: A Simple Kelly Problem I Can\'t Solve
 

Now that my brain isn't fried (and assuming I understand your original post correctly) then the expected growth of your 51.92% bet at +100 odds is 0.074%.

The arb, however, means you will lose nothing 51.92% of the time and will win 0.15% of your bankroll 48.08% of the time, for an expected growth of 0.072%.

Doesn't look like much of a difference either way (with the obvious exception that there is no risk in the arb), but your orginal post doesn't clearly define the parameters that are necessary to calculate this properly (bankroll size, if you're betting a fractional Kelly, etc.).

Re: A Simple Kelly Problem I Can\'t Solve
 

How about using the difference between the the bet and the arb for your edge in Kelly's input? Should be close enough

Re: A Simple Kelly Problem I Can\'t Solve
 

[ QUOTE ]
[ QUOTE ]
You people are hilarious.

[/ QUOTE ]

Thanks for adding so much...

[/ QUOTE ]

Pretty standard.

Re: A Simple Kelly Problem I Can\'t Solve
 

Utah,

I can't do the math and I'm not expert. But I think here you don't arb. If the line moves in your favor you would begin to arb out some of the position (IIRC this is optimal for all bets)

Re: A Simple Kelly Problem I Can\'t Solve
 

Where's Ganchrow when you need him? [img]/images/graemlins/smile.gif[/img]

Looks to me the arb would be best, as you're getting roughly the same bankroll growth with no risk.

Re: A Simple Kelly Problem I Can\'t Solve
 

Okay, I'm a complete noob at this and can't actually solve the math, but I found this problem facinating and perhaps I can add a little here. rjp seems to be on the right track, except he doesn't account for if you should arb only part of the bet; he's just comparing plain bet vs full arb. I've done a bit of reading of the Kelly and arbs for this so I'd appreciate any help if I made an obvious mistake.

Normally with a Kelly problem you're just trying to max f, the percentage of your bankroll that you risk, given a fixed value for win percentage, loss percentage, and bet return. In cases where f* gets capped by the max bet size with the site, you can use the cap if your calc would have you bet over it.
But in this case, you have the ability to play around with the win/loss percentage and the bet return amounts. You could take the full arb and make it a 100% bet with a return of 1%, you could not arb at all and have a 52/48 chance on an even money bet, or something in between.

So the normal growth function for Kelly is:
G(f)=p*log(1+bf)+q*log(1-f)
where:
f=fraction of bankroll bet
p=chance of winning bet
q=chance of losing bet
b=return on bet
and the familiar Kelly formula is just the maximum of that function in terms of f.

For this problem, I think we can restate the growth function as:
G(f1,f2)=p*log(1+r1)+q*log(1+r2)
where:
f1= % of bankroll bet on side 1 of the arb/bet
f2= % of bankroll bet on side 2 of the arb
r1= net return if side 1 wins
r2= net return if side 2 wins (one of these may be negative if the arb is unbalanced)
p,q= odds of each side winning

The basic idea is the growth function is the chance of that change in bankroll happening, times the log of the new bankroll size (in terms of a % of old bankroll).
I'll admit I can't solve this math into an easy formula, but I popped it into an Excel spreadsheet and let solver do its magic - maximizing the growth function value by adjusting the % of bankroll bet under the following constraints:
bet on side 1 <= 1 (or 100% of bankroll)
bet on side 2 <= 1 - bet on side 1 (or bet1 +bet2 can't be bigger than 100% of bankroll)

The result? It said to bet the farm in a weighted arb position - put 51.92% (Look familiar? - at a guess this is a function of the soft line being an even money bet) of your bankroll on the soft line, and arb with the remaining 48.08%. If the profitable side wins, you win 3.84% of bankroll; if it loses, you drop 1.917% of bankroll.

I have absolutely NO IDEA how to account for a capped bet amount at this time, but it obviously would need to be expressed in terms of % of bankroll. For now I would just throw it into the solver constraints.

Re: A Simple Kelly Problem I Can\'t Solve
 

The only thing I'd change for:

G(f1,f2)=p*log(1+r1)+q*log(1+r2)

is

G(f1,f2)=p*log(1+ (f1*dec odds - 1) )+q*log(1+ (f2*dec odds -1))

Not sure if this is what you meant by r1 or r2... but throwing that into an optimizer should get you what you want. A good optimizer would let you specify that f1 and f2 must be >= 0 and < some fraction, as we run into limits here...

Similar computations have been a pain in the ass for me in the past... which is why when I get my PS3 this weekend I'm going to be making this much easier for myself. [img]/images/graemlins/cool.gif[/img]

Re: A Simple Kelly Problem I Can\'t Solve
 

Boy do I feel dumb trying to understand the math. :-))) And I'm an electrical engineer.

Re: A Simple Kelly Problem I Can\'t Solve
 

Yea, it seems I didn't explicitly give the relation of r1 and r2 to f1 and f2 and the whole function declaration looks dumb.

I think I really meant something like:

G(f1,f2)=p*log(1+(f1*b1)-f2)+q*log(1+(f2*b2)-f1)

As you actually have to account for the fact that you lose the f2 bet when you win the f1 bet, and you lose the f1 bet when you win the f2 bet. That way, if you bet 0 on the arb, it simplifies back to:

G(f1,0)=p*log(1+f1*b1)+q*log(1-f1)

which is the initial formula. (b1 & b2 = dec odds -1 for respective bets, just easier to write in the formula)

Re: A Simple Kelly Problem I Can\'t Solve
 

OK, just wanted to clear that up.

Re: A Simple Kelly Problem I Can\'t Solve
 

After playing around with some numbers and different bankroll sizes and bet limits, it appears there are ranges where kelly would suggest you bet the limit if you look at just the +EV bet, but if you add in the arb opportunity, you should be arbing some of the bet as well to maximize your BR growth rate.
For example, assuming you have a BR of $30k in the above example, you would bet $1K on the soft line, then arb with $427.11 at the other book.
There is, though, a threshold where the risk involved in the bet isn't worth arbing simply because your BR is so big. If your BR is $100K, you'd just bet the soft line and be done with it.

Bottom line is, if this situation comes up a lot for you, it's probably worth putting together a small spreadsheet and using the solver to figure out your bet sizes. But, depending on your bankroll/max bet ratio, you might find that you're never doing the arb.