My rule of thumb in this situation is, you think your opponent has a "fair" chance or better of matching your max lead off bet with his own, then bet the max.
By "fair" I mean 1/3 or better.
I think that with what we know about the probabilitiess of winning, pushing and swinging hands, we can put some actual numbers on Monkeysystem's rule of thumb. Just to be clear, this is the situation where you lead your opponent by less than 1/2 a max bet and are acting first.
If we bet small, as discussed earlier in this thread, we advance if our opponent loses or pushes his hand and our probability of advancing is approximately:
L = 0.56
If we bet big, and our opponent takes the low, we advance if we win our hand or if we push while our opponent loses or pushes his hand. The probability of winning our hand is about 0.44, the probability of pushing while our opponent loses is about 0.051 and the probability of pushing while our opponent pushes is about .012. So our overall probability of advancing would be approximately:
H = 0.44 + 0.05 + 0.01 = 0.50
If we bet big and our opponent foolishly matches our bet, we advance as long as there is no 1/2 swing. Let's call this probability C for now.
If we then let
B = the probability that our opponent will match our big bet
then our actual probability of advancing with a big bet is:
BC + (1 - B)H
==>BC + (1 - B)*0.50
In order to prefer making the big bet, we therefore need
BC + (1 - B)*0.50 > L
==>BC + (1 - B) * 0.50 > 0.56
==>BC + 0.50 - 0.50*B > 0.56
==>B*(C - 0.50) + 0.50 > 0.56
==>B*(C - 0.50) > 0.057
==>B > 0.057 / (C - 0.50)
Now, C depends on the skill level of our opponent. The two extremes are that he is a basic strategy opponent or he plays optimally.
For a basic stratey opponent without enough chips to double/split,
C = 0.81 (How To Win Even More Blackjack Tournaments, Volume II, chapter 16, and verified using my simulator)
so we would need
B > 0.07 / (0.81 - 0.50)
==> B > 0.23
before we would prefer betting big over taking the low.
For a basic strategy opponent with enough chips to double/split to 4 hands,
C = 0.78 (obtained using my simulator)
so we would need
B > 0.07 / (0.78 - 0.50)
==> B > 0.25
before we would prefer betting big over taking the low.
For an optimal strategy opponent without enough chips to double/split,
C = 0.76 (obtained using my simulator)
so we would need
B > 0.07 / (0.76 - 0.50)
==> B > 0.27
before we would prefer betting big over taking the low.
For an optimal strategy opponent with enough chips to double/split to 4 hands,
C = 0.63 (obtained using my simulator)
so we would need
B > 0.07 / (0.63 - 0.50)
==> B > 0.54
before we would prefer betting big over taking the low.
So, depending on the skill level of our opponent, we're looking for something between a 23% and 54% chance that he will match our big bet before we would favour it over taking the low. Now, the 54% scenario is unlikely because we not not expect someone who matched our big bet to know the optimal strategy for overcoming that bet. In reality, such a player would double or split more often than called for by basic strategy, but would do so far from optimally. Also, when he does not have enough chips to double or split, the range is a more manageable 23%-27%.
For a long time, I have resisted
ever giving up the low on the final hand. However, this is one aspect of my game that really needs to improve and, from what I've learned in this thread, I will definitely be doing it a lot more in some of the events that I play.
