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1st off... There is no such thing as a 2-bet. The initial BB post counts as the first bet, so MP made the 2nd bet (referred to as simply a raise), and you 3bet (re-raised).
But yes, in general, when in a multiway pot most villains will play their hand more straightforward. There are more villains, which means more hand combos, which means a greater chance of a strong holding. It is because of this that your betting/calling/raising/etc range should be altered and weighted more towards stronger holdings.
In a 3bet pot, this should come into play even more. The pot being larger means the SPR is generally smaller. This leads to more straightforward play, because you have less room to maneuver.
Anyways, my logic on the flop is this... Their ranges are likely pretty strong.. Obviously real reads would be better here than just stats, but either way, BTNs stats don't look like someone who likes to Cold Call 3bets lightly. If we look at our hand and anaylze it against their possible ranges, we can pretty much assume that we likely don't have >50% equity against their calling range. So I don't feel we can cbet the flop for value. However, we do have decent Showdown value, so I also don't see a need to cbet in hopes of folds.
Basically to explain the above paragraph, I don't expect worse to call all that often, and I don't expect better to fold. Given we have good SD value, I'm checking here looking to pot control a bit, and evaluate on what the BTN and other villain do. On an ABCD theorem standpoint, we have B range hand.
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