Optimal Kelly Bet

assume_R

Well-Known Member
#1
2 questions:

  1. I understand that if your advantage is 2%, you should be betting 2% of your bankroll for optimal kelly betting for a 13.5% RoR. Or 0.7 of that, or 0.5 of that, or whatever your "kelly criteria" you want to play with is. But that assumes you're betting $0 at any advantage <0%. So how would you estimate your optimal bet if you're playing some -1% EV hands and -1.5% EV hands???
  2. Does variance have a factor in your optimal bet? Or does variance only affect that "13.5% RoR" number?

Thanks.
 

Nynefingers

Well-Known Member
#2
I'll answer your second question first. Yes, the variance does impact your optimal bet. I believe that the optimal Kelly bet is edge/variance, but I might be wrong on that. Variance does factor in though, so with a 2% edge and variance greater than 1, your optimal bet is less than 2%.

I also want to point out that Kelly betting does not result in a 13.5% RoR. Kelly betting has zero risk of ruin, if you can implement it perfectly (no table min/max, infinitely divisible bankroll, etc.) This is because each bet is a fraction of your current bankroll, and that fraction will never reach 100%, therefore you will never bet all of your bankroll and will never lose all of your bankroll. The number you are thinking of is the risk of ruin using an initial bet size (or ramp) calculated as the optimal Kelly betting scheme, but then never resizing your bets. Maybe it sounds like this is just a technicality, but I just wanted to be sure that we are clear on what we are talking about.

Now I'll go back to your first question. You are correct that ideally you would bet $0 on any unprofitable situation. If you are forced to bet on every hand (play all), then the optimal bet may change. If you are able to bet the table minimum on all unprofitable hands with no regard for the size of your spread, the optimal bet remains the same, as I understand it. If the ratio of your bets is constrained (can you really get away with spreading 5-500?) then your optimal bets on the profitable hands will be reduced. The size of the spread will determine how much it will be reduced. For more information on this topic, and Kelly betting in general, you may want to read "THE KELLY CRITERION IN BLACKJACK, SPORTS BETTING, AND THE STOCK MARKET", by Ed Thorpe. It can be found here (Archive copy). I don't pretend to understand it all, but there is some very interesting discussion about optimal betting and it's well worth the read.
 
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