JL also says you don’t actually need to perform the calculation; you only need to understand how it works in order for it to be useful.
Basically, if there’s a lot of dead money in the pot, it’s worth considering to shove and try to steal it. The incentive goes up if there are fewer players behind you who can call, and the stronger your starting hand is.
If you don’t need to actually perform the calculation, you don’t need to know how to do it.
What I said above does explain how to do it, but I still don’t expect you to actually do it at the table. It’s too much to calculate, and it’s a guess what your opponent’s calling range is in any case.
If you have the software, you can plug in your shove range and a guess of your opponent’s call range, and the software will tell you the outcome probabilities exactly, and you can put that information back into the equation.
(% of times your opponents fold * blinds) + (% of the time your opponent calls) * (% times you get called and win * pot size when they call) - (% times you get called and lose * pot size when they call)
Let’s say you figure your opponent will call a shove with a fairly tight range of hands, say the top 10% of starting hands.
So that means they’ll fold to a shove 90% of the time. That means you win 90% of the blinds+antes when you do shove, and in some hands you may even get a few extra chips from limpers or early opens that also fold to your shove.
As you can see from there, that’s already a lot of variables that can change quite a bit from one hand to the next. So is it worth doing that work for every hand, or is it only necessary to visualize a large pile of chips sliding your way 90% of the time if you’re playing against someone who will only call with their top 10% of hands?
So the remaining 10% of the time you’re getting called. If they’re calling 10% of the time, with the top 10% of hands, then you need to work out how good your shove range is against the top 10% of hands.
Say you’re shoving with 50% of your hands. You you get the small pot 90% of the time, and you get called 10% of the time. Of that 10%, the opponent has a range advantage over you. Depending on exactly how you construct those ranges, it can affect the calculation, but we can loosely estimate that you’re going to lose to the 10% best hands about 60-70% of the time, perhaps more, but not much more than 80%.
The size of the big pot is the same whether you win it or lose it, and it is equal to the (size of the smaller stack * number of callers) + whatever dead chips came from other players who put chips in the middle before folding.
Whatever that amount is, you lose it some % of the time, probably somewhere between 60-80%, and you win it the balance of the time. I didn’t take the time to plug ranges into a solver to figure this out, it’s just a rough estimate.
90% * (small pot) + (10% * (20-40% * large pot) - (60-80% * large pot))
Say that you’re employing this strategy at 10BB effective stacks. Small pot is 1.5BB (for simplicity, we’ll consider a game where there is no ante, and situations where there are no limps and no opens ahead of your shove).
Additionally, we’ll split the range of estimated wins/losses and use a 30-70% split of times you win vs. times you lose when you get called 10% of the time.
=0.9 * 1.5BB + 0.1 * ((0.3% * 21.5BB) - (0.7 * 21.5BB))
= 1.35BB + 0.1 * (6.45BB - 15.05BB)
= 1.35BB + (0.1 * -8.6BB)
= 1.35BB - 0.86BB
So if the numbers we plugged into the equation are accurate, then you’re profitable shoving, making on average about a half a big blind overall.
But keep in mind, 90% of the time you’re taking 1.5BB with no risk. With effective stacks at 10BB, that’s huge. If you are able to get away with this 2-3 times in an orbit, you’re taking a lot of chips (relative to your stack size) away from your opponents with little risk (there’s always some risk you’ll get called, but otherwise you’re taking the pot).
Does that mean you should shove a 90% range of hands, then? Well, no. Just because your opponent’s calling range is 10% of their starting hands, doesn’t mean that will hold true if you’re shoving 90% of the time. Because if your opponent see you shoving 90% of your hands, they’re going to quickly realize that you’re shoving a lot of junk, and they’ll adjust their calling range accordingly, meaning that the equation will need to be re-calculated. So if they start calling wider, you’ll need to tighten up and shove less often with better hands.
You’ll also need to use the concepts that this equation illuminates to determine when you should be calling when your opponent shoves.