progressive black jack - math question

[108suited]

This is sort of funny becuase I never fully accepted the progressive betting blacjack system. The only system that I'm aware of to beat blackjack is cardcounting, and I realize that card counting may be difficult to do due to the rules in most casinos.

Back to the progressive blackjack scenario. The idea of this is that one progressively increases his bets during a win streak of hands.
Example: Min bet is 10. Every time you win a 2 consecutive bets at a certain limit, you increase your bet by a predetermined amount. The following is an example schedule:
10 10
20 20
30 30
50 50
As soon as you lose a hand, you go back to the minimum bet of 10.

The name of the book that I read on this is Progressive BlackJack by Donald Dahl (in case anyone is interested). I've been suspicious all along of this idea, but intuitively it seemed like the system might have some merit. Assume that the house has a 49/50 advantage or that it's 50/50. The progressive betting system is suppose to somehow make up for the overall disadantage of being an uderdog on your bets.

Now someone please give a math formula to disprove progressive betting

Commentary and questions also welcome.

[gaming_mouse]

The "math" to disprove is the simple fact that the house has an edge on every dollar that you bet (before a hand is dealt).

So it doesn't matter how you change the sizes of your bets -- in the long run you are going to lose (total amount bet)*(house advantage).

HTH,
gm

[PairTheBoard]

I think this system might actually work slightly to your Disadvantage. If the last hand dealt had a lot of high cards or Aces in it you would have been more likely to win the hand. If the last hand dealt had a lot of small cards in it you would have been more likely to lose the hand. It seems reasonable then to think that if you won the last hand it is more likely the hand was rich in high cards and Aces. If that's the case you would want to reduce your next bet, not increase it.

PairTheBoard

[AaronBrown]

It might help a little or hurt a little, depending on your assumptions. It won't matter much. However, counting is easy to learn and you can disguise your counting as progressive betting.

For example, consider the simplified model that your probability of winning in blackjack starts out at 49% and moves up or down 0.5% at each hand. 20 hands are dealt between reshuffles. With fixed betting, you expect to lose 2% of each bet. So if you bet $25 per hand, you expect to lose $0.50.

With progressive betting I started with a $20 bet so the average bet size was the same in both schemes. That had a loss of $0.47 per hand.

I know this isn't realistic about blackjack, I'm just trying to demonstrate that you can gain a slight advantage from progressive betting if the winning percentage is a random walk.

[Siegmund]

If the house's edge is constant, varying the size of your bets makes no difference at all to your rate of return, as others have pointed out.

On the other hand, whether you won or lost the last few hands does give you a bit (a very little bit) of information about whether the deck is in your favour or not.

To oversimplify the situation and pretend we are talking about only even money bets: if the count is positive, your chance of winning two in a row might be .51*.51=.2601; if the count is negative your chance of winning two in a row might be .49*.49=.2401. If you are not a card counter, you have no prior information about what's in the deck - but you can now conclude that you are more likely (by a margin of 52%-48%) to be facing a positive deck than a negative one.

How much of an advantage is this? Not much of one. A card counter knows for a certainty whether the deck is running positive or negative, and by varying his betting, he gets himself an edge of say 2% over the long run. You are in possession of about 1/50th as much information about whether the deck is positive or negative as the card counter is. So, to a first approximation, I'd guess you are going to do something around .04% better than if you placed flat bets.

That's a very very tiny gain in EV, in exchange for a large increase in your variance. Following this strategy certainly increases your risk of ruin / expected time to profit / bankroll needed to play.

A roulette analogy is perhaps useful. Seeing the ball land in the same quadrant a few times in a row is a reason why betting on that quadrant next time is better than betting somewhere else on the wheel: IF it's a perfectly balanced wheel, any bet is as good as any other; if it happens to be unbalanced, you have some weak evidence of what direction it's most likely leaning. But that's still no excuse to play roulette. If you are making -5.1% bets while the other poor suckers are making -5.3% bets... you still all lose in the end.

[Girchuck]

Except ofcourse in BJ your chance of winning a hand is not even close to 50%. It closer to 45%. Thats why they pay 3:2 on blackjack, and thats why the games that pay 6:5 for blackjack are completely unbeatable by any counting system.

[TomCollins]

Let's just change it to a generic win loss game with 47.5% chance of winning.

There are 9 scenerios:
Scenerio Frequency Value
L (2.5% -10
WL 24.9% 0
WWL 11.8% 0
WWWL 5.63% 20
WWWWL 2.67% 30
WWWWWL 1.27% 60
WWWWWWL .603% 70
WWWWWWWL .286% 120
WWWWWWWW .259% 220

So adding up those, you get an EV of -$1.225 per "round".
You could change the win % of the game very easily. Even with a 49.9% chance of winning, you get an EV of -$.05.

[AaronBrown]

I agree. My calculation is meant to mimic the edge changes in Blackjack, not the whole game. For most purposes over a large number of hands it doesn't matter much how the payoffs are distributed, just what the house edge is.

[AaronBrown]

That's true, but if the odds don't change, there is no advantage to progressive betting. It could only help if the wins came in streaks.

For example, if you bet on a team in the world series every time it had won the previous two games, you would be +57 bets since 1903.

[TomCollins]

Teams that win 2 games are more likely to win in general.