nash equilibrium hands are the hands that you can shove profitably when you know villain will alway play (in this case calls) optimally
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unexploitable hands are simply the probability that the hand you hold has the most pre-flop equity of all the hands left to act

this is actually a pretty simple case, BB vs SB, because ur hand really only has to be better than Q7o to be unexploitable since that is the hand where half are better and half are worse and there are only two hands left to act --

but because of bunching, you actually need different hands at different table configurations and at a nine man table u need QTo, but to compensate for position you really need QTs and this does make a difference even though ur shoving because the BB can decide whether or not to call so position is still worth a finite amount

so when we program our "perfect calling machine", we are going to call with all hands better than QTs, this gives one side of our nash optimum strategy...we are always calling with all hands better than QTs and always folding worse, and villain knows exactly which hands we are calling and which we are folding

now we find out which hands have enough equity in pot odds with antes in play and dead money to find our left side nash equilibrium optimum strategy

so what we need to do is work backward...find all the hands that have >50% equity vs a perfect calling range that gives us our nash equlibrium hands i believe 43s falls well short because our odds pre are nearly even money for all intents and purposes i believe K7o falls outside but i think our perfect calling range should be

QTs+, K9s+, A7o+,66+ but i don't think anyone has done the math on what perfect callng range hands are it's actually wierd math

Cheers Todd that explains it perfectly, it's also my understanding of unexploitable that you can turn your hand over and the shove can't be +ev. I think my problem with it's application is that if people aren't calling according to Nash then we win because we take the ante's more often but we are further behind when called but then it has to be +ev because we because they are deviating from the nash equibilirium.

However I am struggling to get my head around how a range can be unexploitable when then hands within it aren't by the definition of SC. Like I get the idea that we can't turn 43s face up because it allows villain to call with hands that they would have to fold like 54+ that crush us that they would have to fold if they knew only our range rather than our hand but given that people aren't calling nash then I think it's application can cost us a ton. But how is that if it requires them to deviate from the equilibrium.

Zeal I haven't factored in bunching but I am pretty sure that I can prove that shoving K7o here is +ev

Hand 1: 64.264% 63.78% 00.48% 2000802536 15121518.00 { 22+, A2s+, K2s+, Q8s+, J8s+, A2o+, K6o+, Q9o+, JTo }

I have almost never seen someone actually call with this range and I really think that in most cases shoving 15-20BB's BvB is going to be profitable against just about every single player. The K7o shove is fine. I am not a fan of raise/folding in these spots unless you have enough info to back up the guy is an absolute nit.

Because our opponent has to call based on our range, not our actual hand. So with 43s, our opponent can profitably call with everything other than 32. But if he does that and we shove AA as well, well clearly we are printing money.

So what this means is that if our opponent deviates from the NE calling range (in this case to call with everything other than 32), it may make particular hands in our shoving range no longer +EV, but the total EV of our total range will be higher. In other words, while shoving 43s now loses money, we now make far more money when we shove AA and other big hands, more than offsetting the loss from shoving 43s.

Furthermore, since we are no longer at an equilibrium, we can improve our EV even more by deviating from our NE shoving range (so not shoving the hands that are now -EV vs our opponents calling range).

So,
let our NE range make X profit vs the NE calling range,
let our NE range make Y vs our opponents anything but 32 range
let our adjusted to our opponents anything but 32 range (whatever that is) make Z profit,

then,

Z > Y > X

OK i've been researching this and doing some calcs a little and I'm pretty confident in saying that unexploitable shove hands in a 2 player game are this

44+/A7s+, A5s, A4s, A3s/ K9s+, K8s, K7s, K6s/QTs,QJ,Q9s,Q8s, Q7s

the suitedness of the cards and proximity to straights makes a huge difference, so i'm pretty sure this is a correct range of unexploitable HU shoves (i think A6 and even a6s is weaker than A5 coz a6 cant connect not sure)...i'm going to post later on my approximations for 9man table ...so K7o is def not unexploitable to shove but may or may not be within the the range of nash equilibria

i will comment later if anyone has any questions about the above HU unexploitable shove range...since it's the simplest scenario it's the easiest to calculate

OK i'm deriving this peicemeal so if anyone has any objection to my logic go ahead and speak up

if our unexploitable shove range is correct then, setting equity = 50% gives us an unexploitable calling range of

A7s+, 66+, KJs+ or top 14% of hands...

everyone ok with that?

this is for an idealzed HU match no bunching, stack sizes = infinity (pot odds = even money)