let's say there are 9 players each with 3.5BB stacks and two blinds posted, SB and BB. The only bet allowed is either all-in or fold (or call the all-in)

how would you solve for the solution of this game?

I can point you to programs that will compute the solution, or do you actually want to know how they compute it?

which programs?

Mathematics of Poker is a good book that solves several of these types of toy games. If you have a decent mathematical or statistical background it's a pretty easy read. If not then no big deal, the GTO stuff is all pretty straightforward calculations that you're probably already comfortable with.

Only downside to the book is the introduction is written by Chris Ferguson, a decidedly non-GTO choice.

ICMizer and Holdem Resources Calculator (HRC) can find Nash Ranges for push or fold games with the click of a few buttons.

those seem to solve for tournament situations/payouts?

They can do chipEV too. The ICM aspects are an extra factor on top of the 'basic' maths.

I solved it in my head years ago when I was twice as ******ed as I am now. You could take a computer's opinion as a base strategy and theorize (based on a situation) what you should shove.

The near optimal play will look something like this

early position: AA, KK, QQ, JJ, AK (def folding TT and AQ even though on some tables it might be +EV) --->
middle position: AA, KK, QQ, JJ, TT, AK, (situationally 9s / 8s / AQ)
late position: AA, KK, QQ, JJ, TT, AK, AQ, TT (99, 88 & if no prior action) (situationally all pairs & AJ suited)

Source: been short stacking my whole life, you meant 35 BB though.. right?

(disclaimer: incoming short stack mob mentality superiority complex results consistent swing oriented slow roll over bet)

the computer is basically calculating your range vs your opponents range .. I've done the calculations with a pen and paper for fun. I realized how effective an all-in/fold playstyle is against deeper stacks when short stacking. In the situation you provided, shoving hands like AQ and TT is -EV and therefore should be shoved by short stack strategists only.

Shoving JJ in early position is (debatably) +EV on an average table, anyone disagree with this statement?
I'm also not too sure about TT in Middle position.. any opinions

^ 5/5

FWIW, I ran OP's original situation (3.5bb deep, shove or fold) thru ICMizer (ChipEV mode) and it computed the Nash/GTO solution. (Hopefully I have set up the software correctly).

For 9-max (no antes), UTG jams 20% and you can see the calling percentages (going diagonally) are 10%, 11%, 11%, 12%, 12%, 13%, 23%, 51% (BB). (The first column, and others labelled 'O', is for overcall frequencies).



The opening frequencies for the other positions are:
UTG+1: 21%
MP: 25%
MP1: 28%
HJ: 31%
CO: 36%
BTN: 42%
SB: 77%

Naturally the calling ranges are wider when the open shove comes from later positions. The BB obviously calls at very high frequencies due to the amazing pot odds. For instance, when it's folded round to the SB and he jams 77%, the BB should call (even wider) with 83%.

If stacks are deeper, the risk/reward ratio is worse, so in most spots the shoving ranges will be tighter, as will the calling ranges. An additional nuance is that a more profitable strategy for deeper stacks would include raising not all in. Tools like the one I used work best for pure shove or fold solutions.

Other calculation tools are available from your nearest poker software supplier.

^ very cool post. I'm assuming this doesn't factor in any ICM considerations?

No. It's in ChipEV mode (like it's a cash game).
The ICM factors would produce entirely different results for a 1 table SNG, an MTT, or a satty that had X amount of tickets. The program allows you to enter the payout structure for those things.

Whosnext can give you the exact answer if the betting or folding is simultaneous. Only God can presently give you the exact answer if its not.

If everyone has 3.5 blinds mcflys solution is valid. In practice though when you have 3.5 blinds most other players behind have much larger stacks, and that forces them to tighten up their call ranges based on how many players are left to act behind them due to the added cost when one of the remaining players raise. You also gain a lot in scenarios where two or more people call and one is forced to fold out their equity postflop. That means in normal situations you should be shoving wider than the numbers suggest with a 3.5bb stack. Figuring out by how much is beyond the scope of existing solvers though unless you constrain larger stacks to playing a similar push/fold game which isn't realistic.

^ For the purposes of calculating strategies for a 3.5bb short stack, you could probably setup decent enough abstractions for the larger stacks with existing tools.

Allow the larger stacks to play 3-bet / 4-bet raise-or-fold, that should work fine for this scenario where you are mostly interested about the short stack strategy.

(Side note: For 3.5bb it looks like the ranges are tight enough so that it isn't much of a problem, but for even shorter stacks the limitation to max. 3 active players will become quite significant. Just something to keep in mind about the mentioned tools.)