A Simple Kelly Problem I Can't Solve

[hedgie43]

Sillyarms,

The goal is to maximize bankroll growth, not EV.

[psuasskicker]

[ 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 -

[rjp]

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.).

[kyc12]

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

[MCS]

[ QUOTE ]
[ QUOTE ]
You people are hilarious.

[/ QUOTE ]

Thanks for adding so much...

[/ QUOTE ]

Pretty standard.

[Thremp]

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)

[rjp]

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.

[Stephen H]

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.

[rjp]

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]

[03 Z4]

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