Hey folks
Imagine NLHE, River head's up game where players have option to check or shove. SPR=1 and the ranges are 100% vs 100%, so there aren't polarity advantage or other bs, so raw equity is even.
a) KdQd8s 7s 2c
b) Th9d8c 5s 2d
c) Ah8h2h Qc 4c
Peek whatever texture you like, you can even create your own for this purpouse. How do you construct open shove range for OOP? What says humans poker theory?
Enjoy

First of all, the IP player will have an advantage in this game on any board.

OOP player should be checking a lot and shoving pretty polarized (mix bluff hands, so that IP can't exploit with blockers).

IP player should then fold exactly 50% vs the shove vs hands that have no showdown value. Vs the bluff hands that have some showdown value, he should be folding a little bit over 50%.

When OOP checks, the IP should then jam anything that has more than 50% equity vs OOP calling range and balance it with bluffs (mainly the bottom of his range). He should be betting pretty much the same % that he would be calling vs shove (around 45-50%) to make the OOP value hands indifferent.

OOP should then fold exactly 50% vs no showdown value hands (unless those hands are betting 100% of the time) and a bit more vs bluffs with SDV.

It is unlikely that there would be many weird value-bluff merge plays in these 1 SPR spots, but they aren't impossible. Blockers are very important in these spots.

After some thinking I solved how often OOP should be betting too. It should be around 22%:

I made this with my leet paint skills:


We know that IP bets ~50% of the time.

When OOP bets 22.5% of the time (like in the pic), IP should be value betting vs check top ~34,5% (pink) and therefore bluffing ~17,25%. This adds up to ~52%, which is around how much IP is betting.


I believe that these solutions should work for any possible board.

I solved this spot with monker also and was v. curious, because there is a lot of checking OOP with top range, and my thinking was to create polarized range, and monker did put a lot 2p and sets into checking range, and didn't pick up top hands linearly (i am not sure how it corelates with real game scenario, cause this is theoretical model 100p vs 100p).
Can you guys recommand me some papers where this is explained with some math behind?

It probably puts 2p/sets into the checking range because they don't block any of the opponent's bluffs.

I'd expect the same thing from top pair top kicker type of hands, since opponent probably won't ever be bluffing A high hands vs check.

As I've said, blockers are very important in these spots.

In my experience for the OOP player the most nutted hands get checked and the tier right below that gets donked. For example check the nut and 2nd nut flush and donk the 3rd and 4rd nut flush. This is if we assume OOP is on the defensive.