Why should we be more likely to barrel if we have equity when called?

By mersenneary

Two HUSNG.com users, “Moarskatesvidz” and “Tunaisbusto” are arguing about a hand. In the hand, We raise 43s preflop, and the board comes Kc8d5h. We continuation bet and get called. The turn is the nine of diamonds, and our opponent checks. The question: Should we be more likely to bluff the turn if we are suited in diamonds, or suited in spades?

Moarskatesvidz: It's gotta be diamonds. At least, then, we have some chance of winning the pot, even if we're called.

Tunaisbusto: I don't think so. I think we should be more likely to take the free card when we have a flush draw. We should bet when we have no chance of winning the hand otherwise.

Moarskatesvidz: No, that's the exact opposite of what's right. We should bluff when even when we're called, we still have a chance.

Tunaisbusto: But if you bet, you can get raised, and have no chance of hitting your draw.

Moarskatesvidz: But if you bet, you can win a huge pot if he calls and you hit the river.

Which user is right? Well, both are sometimes. Against some opponents at some stack depths, the answer is that you should be more likely to bluff with the 43 of diamonds. Against other opponents and at other stack depths, the answer is that you should be more likely to bluff with 43 of spades. So which are which?

It's important to remember we always have to compare the equity from bluffing against the equity from checking behind. Let's say there's t400 in the pot on this turn. If we give up without the flush draw, we'll lose 200 chips from the start of the hand. So for betting to be better than checking, we need at least -200 in equity from betting from the start of the hand. Let's say we plan to bet t300. If we get called x% of the time, betting has the same expectation as checking if (1-x)(200) + (x)(-500) = -200. That simplifies to x = 57.1%, or 4/7. Any calling frequency less than 57.1% makes bluffing better than checking behind.

For the flush draw, let's make a couple generous assumptions. For checking behind the turn, let's say that if we hit the river, which happens 18% of the time, we win 300 more chips on average, for +500 from the start of the hand from our opponent. The other 82% of the time, we miss and give up, and lose 200 chips. (0.82)(-200) + (0.18)(500) = -74. So, the expectation from checking behind with the flush draw is -t74 from the start of the hand.

If we bet the turn, let's say we never get jammed on, we always give up on the river unless we hit, we bet t700 into t1000 when we hit the river, and we always get called. I'll spare you the math (feel free to work through it if you'd like), but in this case, any calling frequency lower than 69.3% means that bluffing is better than taking the free card and checking behind.

So given these conditions, we need a fold 30.7% of the time when we have the flush draw for bluffing to be better than checking behind, and 42.9% of the time when we don't have a flush draw. Thus, when you're getting a fold from 30.7%-42.9% of the time, it's best to barrel with your flush draw but check behind without. Below 30.7% or the time, it's best to check behind with both hands. When getting a fold 43% or more, it's best to bluff with both hands.

A less mathematically intensive way to put it is that in betting, we're not really risking 300 extra chips. When we're called, we're going to get those chips back 18% of the time, plus a much bigger premium when we hit compared against how much we get paid off when we check behind. When we bet without equity, we are risking the full 300 with no additional opportunities.

Obviously, though, these were extremely favorable conditions for the flush draw. We've said we're always getting paid off on the river, and we're never getting raised on the turn. There are other situations where it's better to bet without the flush draw and check behind with it. For example, if stacks are short, and betting doesn't leave much behind, then we'll get raised much more often, and even if called, we'll be getting less of a premium on our river bet cashing in on the flush.

The other major detail to think about is how much of your opponent's turn calling range is strong enough to call a river bet when you hit your flush. On some boards, the flush completing makes it so there's really nothing left that you represent as a bluff, and you'll have a hard time getting paid off. On other boards, a backdoor flush isn't that scary with plenty of missed straight draws and air hands to pick off on the river.

On the example hand, if there were 400 in the pot with 680 effective behind, I'd be more likely to bet without the flush draw than with one. Too often, our opponent will either jam or fold to a bet of 250-300. We can bet bigger and call a shove, but sometimes betting bigger just so that we're able to bet/call doesn't lead to the best expectation for a betting size. We're stuck in an awkward stack size and it can be best to make sure we get our share of the pot and check behind. With a much worse expectation from checking behind, 43 of spades can sometimes afford to make the bluff here instead, feigning a bet/call hoping to get a fold enough of the time.