It is known that when OOP, you should be betting when your range is stronger, and checking when your range is weaker.

Before, I thought that generally when your range is stronger, to always bet with no check/raises. When your range is weaker, you checkraise hands that can likely get 3 streets of value vs opponent, as long as you can bluffcatch enough by the river (and still keeping a few strong hands in your range). I feel like when range is weaker, that is definitely correct, though how much you check/raise and cap your raise is debatable and based on how much weaker your range is. If it's a 3bet pot 100bb in PLO and the flop is A44, you have no check/raises really.

This is all based on the completely polarized toy game, where IP has no incentive to bet. However in practice, the more similar the ranges are, and the more protection IP needs, the more incentive he has to bet. Most boards, IP will have some hands in his range that can bet 3 streets for value.

I thought this it would look more like this, when OOP for different range strengths. Betting frequencies are approximate, and assume proper bluff/value ratios. This is also factoring in position, so an even range advantage below would be a slight edge if positions were removed.

edit - check/call and check/raise %s are from your checking range.

Super Strong - Bet 75%, check/call 15% (below min def but not exploitable I think since betting so much). No check/raise.
Strong - Bet 60%, check/call 30%. 0-5% is ok if you can balance it.
Even - Bet 50%, check/call 40%. Check/raise 10%
Weaker - Bet 0%, check/call 50%. Check/raise 7%
Super Weak - Bet 0%, check/call 50%. Check/raise 0%.
Super super super weak - I remember seeing an example where you'd have to defend less than MDF if you are super capped. This I'm really not sure about. On one hand, most people don't turn enough hands into bluffs in this spot and you can safely fold. On the other hand, what happens vs someone who does turn enough hands into bluffs?

As for which hands to check/raise when your range is even to stronger, probably your very strongest hands.

This also assumes fairly uniform ranges though. There are a lot of spots where one player will have more nutted hands, but the other player will have a lot more stronger marginal hands. I suppose in this situation, the player with more nutted hands would just be betting more polar with a wider check/call range.

Be interested to know any of your thoughts. I've come to these conclusions just through play, logic, and an NL range frequency thing I made with excel macros. Curious of how it compares to sims.