Starting to become more serious

I've been playing blackjack on a recreational basis for many years using the system described in Stanford Wong's book, mostly at $10 and, more recently, $25 tables. As I begin to invest more time and energy, I'd like to get a better understanding of betting strategy. Stanford seems to outline a linear betting scheme (which I use), with betting increasing in proportion to the player's edge on the house, whereas many of the postings on this site describe a non-linear or exponential approach (e.g., 1x, 2x, 4x, 9x, etc.). Assuming a bankroll of $10,000, what might be an optimal betting strategy for me? A related question is, I've started to keep a side count of aces in the two deck games, but I'm not entirely sure how to incorporate this into my betting. Right now I just shade up or down my count-based bet depending on whether the remaining cards are ace-rich or ace-poor. Any thoughts appreciated.

People seem to throw around the term "optimal betting strategy" a lot. Unfortunately, this term isn't really well-defined... exactly what is it that you are trying to optimize?

To optimize your EV (ie, the expected value of your bankroll after the bet) you bet the minimum when the house has the edge, and your entire bankroll when you have the edge. The problem with this is that your risk of ruin is 100%, which is, ummm.... problematic.

The much-used (and much-misused) Kelly Criterion optimizes the EV of the log of your bankroll after the bet. It's an interesting idea because the log of 0 is negative infinity (sort of) so a TRUE Kelly bettor would have a risk of ruin of 0, since he would never bet his entire bankroll on anything but a sure thing, since a non-zero probability of losing a bet of your entire bankroll would bring the EV of the log of your bankroll down to negative infinity... the Kelly Bettor would prefer not to make this bet.

Now, it has been determined that for a game WITH EVEN MONEY PAYOFFS (which blackjack is not), the correct "Kelly-sized" bet is equivalent to the fraction of your bankroll which matches your edge. For example, in Kelly's original paper (if I remember it correctly... I have it kicking around somewhere) he uses the example of being able to bet on the result of a baseball game after it has been played, but before the result is "common knowledge". You have a data line, on which you receive a stream of data, with each peice of data representing the result of a particular game. However, there are errors on this line, and each "result" only has a probability p of being corect. So if there is a 25% error rate on your line (so p=75%) and you were able to make even-money bets, your edge here is 50%, and so betting 50% of your bankroll would maximize the EV of the log of your bankroll.

The problem with applying this to blackjack is that a blackjack bet is not an even money bet. Even when you have the edge, you still usually lose more hands than you win. You edge comes from the fact that a blackjack pays 3:2, and you can increase your bet (by splitting/doubling) in many favourable situations. I think I read somewhere (very possibly on this web site... I don't remember) that a "true Kelly bet" for blackjack (ie a bet that maximizes the EV of the log of your bankroll) is something around 75% of your edge multiplied times your bankroll. So, with a 2% edge, it would be about 1.5% of your bankroll.

Another problem with applying Kelly Betting to any blackjack (or pretty much any real-world situation) is that for this whole "risk of ruin = 0" thing you need the ability to make arbitrarily small bets if necessary, which, of course, you can't do, because of things like table minimums. I mean, if your bankroll is down to $100 and the table minimums are $5, you aren't technically "ruined" since your bankroll is non-zero, but you also can't make any more bets if you want to stick to the Kelly Criterion (unless it's an even money bet where you edge is 5% or more)

Anyway, I've babbled enough, and haven't really given you an answer. My point here is that when someone talks about "optimal betting" or an "optimal betting ramp", your first question should be, what is it that's being optimized?

Keith

10 grand bank, DAS, S17, 8 decks, 7/8 pen:

TC =< 0 don't play
TC = 1 $10
TC = 2 $30
TC = 3 $50
TC >= 4 $70 or two hands of $50

your ROR will be a safe 0.35% and you will win about $17 per 100 hands (using I18). If you play in AC or connecticut like I do, this strategy is not practical, especially if you play weekends, which is nearly impossible because of crowded conditions and higher table mins (and neither is the pen :) ).

So, a more "gutsy" strategy with above rules that's easier to implement, I'd do something like:
TC < 0 don't play
TC = 0 $25 (only way to get into a table)
TC = 1 $25
TC = 2 $50
TC = 3 $75
TC >= 4 $100 or two hands of $72 (round it to $70)

ROR will be 9.4% and you will win 20.5 dollars per 100 hands.

Stanford seems to outline a linear betting scheme (which I use), with betting increasing in proportion to the player's edge on the house, whereas many of the postings on this site describe a non-linear or exponential approach (e.g., 1x, 2x, 4x, 9x, etc.).

I don't know who is suggesting 2x something, but if you bet, say, $20x the TC you're betting approximately in proportion to the edge, as Wong and everyone else recommends. It's not exponential. And if someone tells you to bet 2x Kelly (or anything over 1x Kelly), you should hold tight to your bankroll and run fast!

Optimal Betting Theory has been very well defined for a few years. For example, CVDATA/CVX have to generate optimal bets before they generate SCOREs. It does it automatically. Or see the series of articles on Richard Reid's web site below.

ETF

I could give you an optimal betting strategy based on what CVCX spits out but I'd need to know how much your bankroll is? Actually, if you are getting more serious you should probably get the software for yourself. You will encounter many different games so it is important to analyze each one individually. Usually, this is what I do:

TC......units bet (at $25 table)

0.........1 or none
1.........2
2.........6
3.........9
4.........12

My 1 to 30 spread at a $10 table looks like this

TC........units bet
0..........1 unit (2 red chips)
1..........5 unit
2..........15 units
3..........22.5 units (9 greens)
4..........30 units (12 green)

It is basically the same as my 1 to 12 green spread except I bet $10 when I don't have the advantage. You might last an hour and maybe in optimal conditions, 2 hours playing this way before they tell you to leave.

This is a good all-around bet ramp. I used this on 6 deck and 2 deck games. Of course, I get backed off a lot but I play a winning game and that is what's important to me. If you never get backed off, you probably aren't playing with a good EV score, in my opinion unless you got some Mad crazy ace sequencing skills.