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Buttons play looks like AJ,KQs(except KQc),QQ+,KQo imo. You are 50% vs this range. If you rank them for likelyhood I would say KQo,AJo are a little less likely because of the preflop action. So you arent looking good vs this range(if you remove KQo you have 25% equity). Now what hands will villain raise? Imo he wont raise any of them. I believe our equity is around 35% vs buttons range. Now how often will we beat BBs hand and how often will he call? Imo we need BB to call every time with a worse hand for it to be +EV (0.35*.1=>1/3). So this isnt a valuebet. Check call.

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The problem with this math is that you ignore the fact that when you check, the button will bet with most of the hands that have you beat so you lose that bet anyway.

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Well in my analysis you dont beat any hand anyway(tie with a few) so you might as well let him bluff and if BB raises you can fold.