Disclaimer: this turned out pretty long...
Inspired by pghfan's thread, I thought I would make an attempt to quantify calling vs shoving in a few short stacked spots. In all situations we'll assume that button min-opens 40% (I've used 22+,A2s+,K2s+,Q6s+,J7s+,T7s+,97s+,86s+,76s,65s,A2o +,K9o+,Q9o+,J9o+,T9o,98o which is not perfect but whatever). Preflop pot is 2.5bb, so we're getting 4.5:1 to call. I'll start with a trivial stack size and go from there.
Stack - 2bb
So this is trivial, getting 4.5:1 to shove all hands have the necessary equity, we call it off.
Stack - 3bb
If we shove, we're getting 5.5:2 and so need 26.7% equity. 32o has 29.87% so we can profitably get in any 2.
What about call? We can safely assume our opponent will put the last bb in regardless. We'd be getting 6.5:1 on the flop, so we need 13% to get it in. The important part is:
if there are any flops where we can correctly fold then we're better off calling pre.
It turns out that we have <13% equity on about 30% of flops in this instance. So in the case we have 32o, if we call pre and fold the 30% of bad flops, then our equity on the remaining is ~38% (I've explained how I reach this number at the end and since it is an approximation I've rounded down).
EV(call)=.3*(-1)+.7*(.38*7.5-2)
EV(call)=.295bb
EV(shove)=.2987*7.5-2
EV(shove)=.240bb
So calling wins about .055bb more than shoving. Now I'm sure people have 2 initial reactions: 1. WTF who cares? and 2. Great now I have 1bb left 30% of the time instead of a (very slightly -EV) shot at 7.5bb. Well #1 I don't have much to say about, but re: 2 when you have 1bb you have a guaranteed profitable spot in the sb and if you win that get a whole orbit with 3-4bb where you will likely get several more. Moving right along...
Stack - 4bb
Now getting it in we're getting 6.5:3 and need 31.6%. We should now fold a bunch and the worst hand you can get in (used holdemviewer for this) is 63o with 31.8%. EV(shove)=.318*9.5-3=.021bb. Again I think the him autoshoving flop is a very good assumption, so let's look at call. Postflop we're getting 7.5:2 so need 21% to get it in.
In this case we actually can fold 45% of flops! And our equity on the flops we don't fold is about 45%. So,
EV(call)=.45*(-1)+.55*(.45*9.5-3)
EV(call)=0.25bb
Now this is actually a real gain. Shoving 63o in this spot is pretty clearly worse cEV wise than call/fold bad flops. We should also note that we can almost certainly take some hands that are slightly -cEV shoves and turn them into +cEV calls.
Stack - 5bb
For this one I thought I'd pick a decent hand that we can clearly get in and actually flops ok to see what happens (picking AA or something is stupid since there are of course no flops we can fold). So we're getting 7.5:4 and need 34.8%. We'll use 98s that has 40.75% and so EV(shove)=.4075*11.5-4=.69bb, super easy clear shove>fold.
Postflop we'll be getting 8.5:3 and need 26% to get it in.
We can fold 40% of flops. Our equity on the flops we don't fold is about 55%. (As an aside, we're still folding 40% of the time and our opponent is shoving 4 at 5.5 so cbetting any 2 will still show a profit for him and our assumption is still pretty reasonable.)
EV(call)=.4*(-1)+.6*(.55*11.5-4)
EV(call)=.995bb
and EV(call)-EV(shove)=.3bb
That's pretty damn significant, and worse hands will be able to fold more flops and should (I think) show a bigger difference.
As our stack gets deeper our assumption about our opponent always shoving gets weaker (I think up until now it is near 100% accurate) so I'm going to stop it there, but here's a summary and some of my comments:
1. If our opponent never folds preflop and always shoves flop then call/decide is better than shove with all hands where there are flops we can correctly fold. If there are no flops we can fold (eg if we have AA) then call/decide=shove.
2. Some of the gains even with extremely small stacks are non-trivial.
3. The wider our opponent opens the fewer flops there are we can correctly fold and so the smaller the call edge gets.
4. Since we're always unsure about exactly how wide our opponent opens we can just err on the side of putting it in on more flops to ensure we aren't making mistakes that makes call<shove.
5. As our stack gets a little deeper, we'll probably just shove our value hands to ensure our opponent doesn't fold postflop when he hits his worst flops, so our range is actually quite weak to defend. This gives him a pretty strong incentive to cbet and makes the "opponent never folds postflop" assumption stronger and I think will allow this model to extend to slightly deeper stacks fairly accurately.
6. When our opponent starts bet/folding some and checking back some I don't really know what will happen. Checking back seems like it will be a net positive since our range is really quite weak but I could be wrong.
How I calculated equity on the flops we don't fold:
Take the 63o example. Our equity pf is 31.81% and we're calling 55% of flops, folding 45%. If x is the avg equity we have on the flops we call and y the avg equity on flops we fold then .55x+.45y=.3181 We want to know x. Looking at the graph, from the 55% to 100% on the x-axis, equity goes in a near straight line from 20% to 10%. So our average equity on the bottom 45% of flops is .5*(20%+10%)=15%. This is y. Then we solve for x=(.3181-.45*.15)/.55 and get x=.4556. Because there's some estimating used there I rounded down to 45%.
Because the assumptions don't hold as well later (and because this is quite a bit of work) I don't want to try extend this to larger stacks right now. But I do want to note that our intuition about what is correct play with short stacks can be quite wrong, even in seemingly obvious situations. I think the T9s example from the other thread is an absurdly easy call. We can't profitably shove, but by calling and folding bad flops, and on good flops sometimes winning a cbet, sometimes getting it in with decent equity I think we'll show a substantial profit. Also, if our opponent chooses to not cbet often then I think we get the better of that. If he were to promise to check it down we'd have a hugely profitable flat (like 1+bb), I think the more of our equity we get to realize without more money going in the better.
If anyone actually finds this interesting, maybe at some point I'll try to find a decent way to approximate some slightly deeper stacks. If anyone smarter than me wants to try to do that, that would be nice
Also, I have tried to read through this and edit/check math but there are a lot of words/numbers so I may have made some (hopefully small) errors. If so, feel free to point them out and I'll try to fix them.
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