I think we often find ourselves in this spot. Everybody knows that when you get down to 10 big blinds, you should shove or fold, and the question is how good a hand do I need to shove, or should I be patient and wait for a better hand or later position. So, I hope this analysis will be useful to others. Obviously, I could not do this full analysis in real time while I was playing, but I will now explore this the way I would have if I had been allowed to leave the table for a while, go to my computer, and then come back. Doing this type of post mortem will hopefully help me develop an intuition to use when I’m at the table.

The question when contemplating a shove is what is your expected number of big blinds (EBB) at the conclusion of the hand. If you fold, your EBB is 10. If you shove and everybody folds, your EBB is 11.5, a 15% increase to your stack. So, we need to know how likely it is that everyone will fold and how often you will win when called. In general, your EBB when shoving 10 big blinds is:

EBB = p1 * 11.5 + p2 * 21.5

where

p1 = probability that everyone folds

p2 = probability that you win when called

If EBB is greater than 10, then it was a good play, otherwise, the expected number of big blinds is smaller, in which case, it was a bad play. This analysis assumes that everyone has you covered, and ignores several other meta-game factors.

Calculating p1:

First, let me describe a calling range for each of my opponents, based on my knowledge of their play, their image of my shoving range here, and their stack sizes. (Real names have been replaced with pseudonyms for privacy.)

Mike (immediate left) 9k stack: 77+, AK, AQ, AJ, KQs, which is a 7.5% range

Mark (button) 20k stack: 66+, AT+, KQ, KJs, a 10% range

Lisa (SB) 8k stack: 22+, AT+, KJ+, a 13% range

Julie (BB) 15k stack: 77+, AK, AQ, a 6% range

So, having assigned them a range, it’s easy to calculate the probability that a shove will get through. It’s the combined probability that each of them folds (for purposes of simplicity for the entire write-up, I’m going to limit the discussion to the case where exactly one person calls and ignore multiple callers, which would have a negligible effect on the analysis.):

.925 * .9 * .87 * .94 = 68%

So, independent of my hand, when I shove in this situation, I will get a fold 68% of the time. However, my deep stacked opponents should open up their ranges significantly against me in this spot. In particular, the two big stacks, Mark and Lisa, should be much more willing to call. Let’s say that these guys adjust their range to: any pair, any 2 broadways, 9Ts, AT+ and A5s+, which is a 20% range. Then, the probability that my shove gets through is

.925 * .8 * .87 * .8 = 52%

So, against players who adjust their range, my shove gets through about half the time.

So, the values I will use for p1 are 86% and 52%.

Calculating p2:

Now, let me look at my equity against these players with my A8. First let’s look at the actual ranges I assign my poker buddies, and then we’ll look at the adjusted ranges.

Plugging A8 against their ranges into Flopzilla, I get the following equity for my hand against these players:

Mike: 32%

Mark: 36%

Lisa: 38%

Julie: 28%

So, my combined equity is:

.075 * .32 + .1 * .36 + .13 * .38 + .06*.28 = 12.6%

Now, with the wider calling ranges from the big stacks, the equity values are:

Mike: 32%

Mark: 44%

Lisa: 38%

Julie: 44%

And the combined equity of the adjusted ranges is

.075 * .32 + .2 * .44 + .13 * .38 + .2*.44 = 25%

Interesting to note that when my opponents widen their range, I get called more often, but my A8 does a lot better because they are now calling with many more hands that are behind me.

So, the values I will use for p2 are 12.6% and 25%.

Calculating EBB:

Now, let’s plug into the formula for EBB. In the first case, against my friends, as I expect them to play,

EBB = .68 * 11.5 + .13 * 21.5 = 10.6

Against the wider range, the result is

EBB = .52 * 11.5 + .25 * 21.5 = 11.4

I found this enlightening. Shoving A8 in this spot is just about a break even play. Given the likely margin of error in my range calculations and other simplifying assumptions, I would say that A8o is a wash, and that meta-game considerations (e.g. How much better do I think I am than the other players? How much do I care about outlasting a few other players?) should probably dominate.

Interesting to note that against stronger play, where the others are adjusting and widening their range, the expected number of big blinds is actually higher and almost identical to the EBB when everyone folds. So, you really don’t care if people call. A 15% increase in your stack is non-trivial, so I conclude that if you perceive your opponents as calling with a wider than normal range, then a shove is worthwhile in this spot with A8o, but otherwise, it is probably not worth the risk. Right on the margin. AT+ is a definite shove, and A6 is a fold.

The other night, I found myself right on the margin.

*Acknowledgement: Thanks to Brian Woods for helpful comments and suggestions.*