A Simple Kelly Problem I Can't Solve

[Stephen H]

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)

[rjp]

OK, just wanted to clear that up.

[Stephen H]

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.