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Obviously if you put him on ranges 95% of the time where 6 out of 7 times you lose and 1 out of 7 you split, you should fold, which is what I think you did. With little to no reads, though, I couldn't pinpoint a range on anyone that specifically or that confidently, but that's me. If you have more stats and notes to share, tell us. If you're basing it on how most villains play at those stakes, maybe that's valid, and while I do tend to give an unknown player more credit than he may deserve until he proves otherwise, I also try not to give him too much either.
But, to play devil's advocate:
Pre-flop action: 4x raise from MP makes sense with AK, JJ, KK, and AA, not so much 22 or KJ - other hands that would make sense AQ, TT, QQ, suited connectors, and maybe a few gappers or pocket pairs if he was loose or attempting to balance his range.
Flop action: 68% re-raise of the pot makes sense with AK, AA, QQ and with position on the bettor, lots of drawing hands like AQ or AT. JJ, KK, and 22 seem less likely for a raise of that amount - I would expect more with a set.
Turn action: Eliminating the sets JJ, KK, 22, and 77 seems even more likely. A smart villain would have thought, "how do I get the rest of my $576 in the pot"? He would want to get the money in the pot. How do you make money WITHOUT putting money in the pot? You've called him twice. Why would he put the brakes on now? At this juncture, after two streets of betting, he has only put in 15% of his remaining stack after the flop. If villain regularly plays made sets this way, then you WANT to play pots with him because he is not maximizing his value - not even close. AK, QQ, and drawing hands seem MORE likely now plus a backdoor flush may have come into the picture.
River action: The fact that there are two 2's on board almost eliminates the possibility to me that he has pocket 2's just based on statistics and even more so how the hand played out. It's worth considering that AK and AA are less likely villain holdings to a degree simply because you have an ace and a king. Like I've said, you could definitely be beat. He could easily show down with hands like AA or maybe he was playing a set in a way I'd consider tricky or sub-optimal. I can tell you if I flopped a set, money would be getting in the pot or you would be folding. But, with few reads, playing this hand out 100 times, I'd guess against a bad, average, good, and possibly very good player, he doesn't have you beat more than 70% of the time.
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