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This situation/question really breaks down to subranges. And not only our subranges, but the subranges of our villain, and how he is likely to play his subranges.
Preflop villain should have subranges, which will allow us to make some assumptions about what range we are actually facing postflop. That is, we expect him to 3bet his premiums for value. Then we expect him to call all the hands he feel is profitable to play a pot with IP (SCs, Suited Broadays, AJ, etc type hands). Then we expect him to 3bet bluff some % of the time, and those hands would come from the hands that he couldn't profitably call with (suited gappers, K9,Q8, etc type hands). Of course we can't be sure he is playing in this manner, and he cold just be flatting anything, even premiums.
However, postflop his range, and ours, are going to become divided even more. On this flop, we are going to want to bet our strong hands to extract value from his many worse hands that can call. Remember in order to value bet we need to have >50% equity against his ranve (to truly be getting value when called). So, we would bet things like 99+, AT, KT, etc} because we expect worse to call often enough.
Likewise, we are going to want to bet a good % of the hands that didn't connect (assuming he doesn't float often, or call too lightly). That's stuff like KQ, KJ, etc. We aren't betting because we have >50% equity, but we are betting because we aren't likely to win a showdown if we check down, some better hands will fold, and our fold equity in combination with pot equity (having gutshots + overs, etc) will make this profitable.
Now in between we are going to have a range that we can't bet for value, as we don't expect worse to call often enough, but we feel that if we check we are ahead of his betting range. This will form our check/calling range and will incorporate hands like 9x, weaker Tx, and depending on his betting frequency when checked to, check/calling hands like 77-88 will likely be profitable. Profitable on that street that is, but you might be setting yourself up to a fair amount of tricky turn/river spots that you are going to be prone to making mistakes in.
Then obviously we are going to have a check/folding range, and that's hands we either can't bet for value or don't need to bet, yet we don't feel is ahead of his betting range when we check. So we check/fold, and that's where 88 fits in this spot assuming villain isn't going to play incredibly straight forward and toss everything missed on this flop and call everything that hit (aka.. If villain can float, bluff, etc I would check/fold 88 here).
SiltStrider hit the nail on the head with what I was referring to with cbetting 22 and checking 88. With 22, there are going to be alot of more hands that have decent equity against us, and a fair amount more hands that beat us. Any pp 44-88 now has us crushed, yet those hands are likely to fold to a cbet. Any overcards have at least 24% equity or better, and because we hold 22 villain will obviously have overcards. So as a whole, 22 is not going to win showdown as often, and therefore profits more from turning it into a bluff and picking up on our flop cbet fold equity.
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