So player has 10bb on the button and nobody has opened and i assume its antes time also so 2.625bb are at blinds and antes usually.
Absolutely any pair will do here of course.
First of all try 22 and lets say they call you with any range you can imagine and see what happens. Try 20% first.
You will get called 36% of the time and have 43.7% equity for a result of 9.56*0.36+12.625*0.64=11.52 so +1.52 for pushing with 22.
Try calling ranges of 10% (real tight) and this goes to 8.7*0.19+12.625*.81=11.88 for +1.88bb gain.
Try 30% ranges and its 9.95*0.51+12.625*0.49=11.26 or +1.26bb
You cant go wider than 30% because its bs for the SB to do such a call and then face a push by the BB with some 10% hand because it will not have any decent advantage vs pusher's range anyway but it will also endure a disaster when BB over shoves or calls.
This indicates an avg 30% range for callers is not far from the final result.
So most definitely all pairs are in the range basically because the chance of an over pair is small 11% and all other hands are not ahead of 22 in any material way if no behind a bit typically. Plus they cant call wider than 30% anyway because it gets ugly for them especially the sb.
Now with same logic you can try A2o and still it will pass ok plus it has some removal effect.
so all AX will do.
Lets try to find the worse equity you need to have anyway when called if they are near 30% ranges. You probably need at least 0.3bb edge for ICM reasons so look for them if they call around 30% range to leave you with equity near 37% when called.
So you create the following list that passes this threshold;
ie 45% push range roughly
This then for confirmation forces BB at near 35% and the SB at around 25%. Of course it is a lot tighter for the BB if SB calls, say 10% or less then.
To verify this use a Nash equilibrium numerical approach like;
Which indeed shows without above explanation that it is more like ;
45.4%, 22+ Ax+ K2s+ K5o+ Q4s+ Q9o+ J6s+ J9o+ T6s+ T9o 96s+ 98o 85s+ 75s+ 65s 54s for the 10bb pusher
and SB calls/overshoves 19.5%, 44+ A2s+ A7o+ K9s+ KTo+ QTs+ with BB then calling only 7.1%, 88+ ATs+ AJo+
otherwise if SB folded BB calls with;
35.4%, 22+ Ax+ K2s+ K7o+ Q7s+ Q9o+ J8s+ JTo T9s
So overall i suppose going for
45.4%, 22+ Ax+ K2s+ K5o+ Q4s+ Q9o+ J6s+ J9o+ T6s+ T9o 96s+ 98o 85s+ 75s+ 65s 54s
seems reasonable especially since if the image of pusher is not very loose from prior history the blinds will definitely not call as optimally as above but rather a bit tighter if not a lot tighter.
The above assumed the blinds are ~3 times bigger stacks or better.
If the BB is a small stack too it may get a bit tighter for the pusher and the BB, say 40% push vs 30% call for BB 25% for SB etc.