Wondering what you guys recommend doing with this, or if there's even a consistently profitable approach to recommend.

The situation: by the turn or river you have a strong hand. Let's keep it simple here and say it's a straight. Your hand: KQ of clubs. The board: Jd 10h 8d 3c Ad. You led the betting pre-flop and had two callers; by the turn one had folded (as you led with a continuation bet on your open-ended straight draw). The river is the A which gives you a straight but also puts a flush possibility out there. You have position on the last remaining player. He has never yet played back at you; called the pre-flop raise, called the semi-bluff on the flop. You checked it through on the turn.

When the A lands on the river, he leads out with his first act of aggression, a big bet that almost puts him all-in. It's a ridiculous overbet if he made the flush (well, it's ridiculous no matter what he has... but I'm getting ahead of myself).

Do you call with the straight, or fold to the bet?

My thinking goes like this: the overbet indicates a likely bluff or semi-bluff. His hand is very probably not as strong as the flush he's repping with that bet. However, a smart player (and let's assume you don't know how smart he is) might do the exact same move hoping for a call when you misread it as a bluff; or a really bad fish might overbet because hey, it's a flush, exciting! It's hard to say what percentage of players are smart enough to try this as a move, or what percentage is dumb/new enough to overbet like that when completing the flush. If I knew it would be easier to calculate an approximate cost/profit of calling this bet.

Also the bet itself is a problem, precisely because it's an overbet. In this example let's say it's a pot-sized bet. He's giving you 2-1 pot odds on the call, which means it has to be a bluff or semi-bluff (no flush) 1 time out of 3 for you to break even. So you don't have to be sure it's a bluff, but you better be pretty confident that a lot of the time it will be a bluff.