I thought the first thing I should look at are pre-flop shoves, purely for ease of calculation. There seems to be a common idea that when you’re short stacked, you should shove with any two cards.
So, let’s assume you’re in a 9-handed game, in mid position. The blinds get put in, it folds to you, you shove, there’s one caller behind you. The expected value from a pre-flop shove is as such:
EV = p (1.5b + 2s) – s, where p is the probability of your hand winning, b is the big blind and s is your stack.
This should be quite intuitive. When you hit your hand (p), you win the blinds (big blind + small blind = 1.5 big blinds), as well as double up your stack.
The shove makes mathematical sense if the expected value is more than 0. Thus you should make the shove as long as
EV > 0
p(1.5b + 2s) -s > 0
p(1.5b + 2s) > s
p > s/(2s+1.5b)
So therefore, the shove makes sense as long as your odds are larger than s/(2s+1.5b).
What does that mean, however? If you look at this Pre-Flop Odds chart from WestonPoker, you’ll see that the weakest pre-flop hand (3-2 offsuit) still has a 31.2% chance of winning against a random hand. Oh before I go further, an interesting side note. That chart shows 3-2 offsuit to be the weakest hand, not 7-2 offsuit as commonly said. That’s something I should look at eventually, but I just thought it was interesting to point out.
Back to preflop shoves. Considering a 3-2 offsuit preflop hand, we have
0.312 > s/(2s+1.5b)
0.312 (2s+1.5b) > s
0.624s + 0.468b > s
0.468b > 0.376s
s < 1.24b
This basically means that you should shove with any hand so long as you have less than 1.5 big blinds left. That’s unlikely to happen in a real game, however.
Let’s say you are in a situation where you are down below 10 big blinds. This is a relatively realistic situation, and is commonly accepted to be a short stack. Given s = 10b, we then have
p > 10b / (20b + 1.5b)
p > 10b / 21.5b
p > 10 / 21.5 = 46.5%
So therefore, once you go under 10 big blinds, it’s logical to shove once you have a probability of 46.5% or more.
What hands are those? Well, based on the chart, that’s the following:
- any Ace, any King, Any Queen, Any Jack other than J2 offsuit
- T5 suited or higher, T7 offsuite or higher
- 96 suited or higher, 97 offsuit or higher
- 87 suited
- Any pocket pair
That’s not exactly any two cards, but it is still a wide range to shove. Of course, this is still a very surface exploration of it, basically assuming that you’ll have one caller with any two cards. In a real game, you’ll be able to steal the blinds a portion of the time shoving; also, when you get called it’s usually against an upper range of hands, not just any random hand.
But I think that’s enough for now, I’ll go into the more detailed analysis of that in the next post.