## For Turn and River Bluffs

This section won't be about deciding when to try the turn and river bluffs. Finding those situations correctly requires accurate hand reading more than anything else. This section is about how much to bet on the turn and river. Or, more specifically, it's about how to divide your remaining money between the turn and river. There are two important principles for dividing your remaining money:

1. Save enough on the turn for a credible bluff on the river.
2. Bet as much as you can on the turn while still retaining a credible river bluff.13

Save Enough on the Turn for a Credible River Bluff

Say the pot is \$1,000, and you have \$2,000 remaining. If you were to bet the pot on the turn, \$1,000, you would have only \$ 1,000 remaining to bet on the river if called. In that case, the pot would be \$3,000 plus your \$1,000 bet, so your river bet would offer odds of 4-to-l to your opponent. If your opponent called your pot-sized turn bet with a made hand (likely), he'll probably call again getting 4-to-l. So \$1,000 is not a credible river bluff.

Failing to leave enough for a river bluff hurts you two ways. Obviously, it limits your bluffing options. Instead of the turn and river bluffs, you are limited essentially only to a single street bluff. You can't take advantage of a scary river card if you don't have enough left to make it scary.

But it also takes the teeth out of your single street bluff. A \$1,000 turn bluff will be much scarier if you have \$3,000 or more behind than if you have \$ 1,000 or less. With little behind, your opponent just has to decide how often you are bluffing and compare that percentage to his pot odds.

With a lot left, though, he has to worry about winning a little when he's right, but losing a lot when he's wrong. Many times you won't follow through on the river, and he'll be left with a "paltry" \$1,000 win. But sometimes you will follow through, and he'll have a far tougher decision for far more money. (It's tougher for him because you'll bluff \$ 1,000 on the turn and give up on the river more often than you'll bluff \$1,000 and follow through for

13 Note that we are talking about bluffing sequences where you will eventually move all-in. If you are extremely deep, you may make big bluffs on the turn and river, yet still not be all-in. Those situations are more complex to analyze than what we will talk about here.

\$3,000 on the river. So, from his perspective, you're far more likely to have him beaten when he sees the \$3,000 bet than when he sees the \$1,000 one.)

Overall, the turn and river bluff play is much stronger when you have a credible threat left after your initial turn volley. How much is credible?

It should be significantly more than the size of the turn bet, and it should offer your opponent relatively short odds on a river call. As the odds get longer than about 2.5-to-l or so (a bet two-thirds the size of the pot), your opponent will call more and more often. For instance, in a given situation your opponent might fold 75 percent of the time against a pot-sized bet (offering 2-to-l), but 20 percent or less against a one-third pot-sized bet (offering 4-to-1).

On the turn, you may simply not have enough money to try a turn and river bluff. And, since the turn portion loses teeth without the river portion to back it up, you may not be able to bluff profitably at all. From the start of the hand make sure your river bluff will be credible before you launch the play.

Bet as Much as You Can on the Turn While Still Retaining a Credible River Bluff

The flipside to the first principle is that you should bet as much as you can on the turn while still maintaining a "credible" river bluff. Generally speaking, your opponent's chance of folding on the river will look like a logistic curve (also known as an s-curve).

LOGISTIC CURVE

Increasing Size of Bet *

For all bet sizes that are only a small fraction of the size of the pot, your opponents will fold roughly the same number of hands: perhaps only busted draws and the very weakest made hands. For all bet sizes much larger than the size of the pot, your opponents will also fold roughly the same number of hands: almost everything except the nuts and perhaps a couple of other extremely strong hands.

In the middle, usually around the half-pot to one-and-a-half pot range, will be a sharp change in the fold percentage, where your opponents fold stronger and stronger hands to bigger and bigger bets. The optimal size for the river bet is the smallest amount that keeps your opponent folding most of the time.

That is, to find the right theoretical size for your river bet, start at the right-most edge of the graph and follow it left until it begins to drop significantly. Stop there and look at the corresponding bet size. That's about how big your river bet should be.

LOGISTIC CURVE

Increasing Size of Bet *

LOGISTIC CURVE

LOGISTIC CURVE

Why is that the right size? Well, if you bet more than that, then you risk significantly more for only a small increase in your chance of success. In a \$ 1,000 pot, it makes no sense to bet \$2,000 for an 80 percent chance of folding when you can get a 75 percent chance for \$1,000. You'd lose an extra \$1,000 20 percent of the time, while making an extra \$2,000 (swinging a -\$1,000 failure to a +\$1,000 success) only 5 percent of the time.

But there's a more subtle reason that's just as important. The less you bet on the river, the more you can bet on your turn bluff. Betting more on your turn bluff serves two purposes:

1. Increases your chance of success, at least somewhat
2. Improves your potential reward on the river bluff if you get called on the turn

Say you have \$4,000 to distribute between bluffs on the turn and river. The pot on the turn is \$1,000. You are trying to choose between \$1,000 and \$3,000 bets or \$500 and \$3,500 bets.

You think a \$500 bluff will succeed on the turn about 30 percent of the time, and a \$1,000 bluff about 50 percent. If you've bluffed \$ 1,000 on the turn, then you think a \$3,000 river bluff into the now \$3,000 pot will work about 70 percent of the time. If you bluffed \$500 on the turn, then you think a \$3,500 river bluff into the now \$2,000 pot will work about 80 percent of the time.

Which series is better? When you bluff \$500 and \$3,500, you can have one of three outcomes: you can win the initial \$ 1,000 if your turn bluff succeeds, you can win \$1,500 if your turn bluff fails, but your river bluff succeeds, or you can lose \$4,000 if you get called down.

You'll win \$1,000 30 percent of the time. You'll win \$1,500 80 percent of 70 percent of the time, or (0.8)(0.7) = 56 percent of the time. You'll lose \$4,000 the other 14 percent. Thus, the EV of this sequence (assuming you have no chance to win by making the best hand) is \$580.

\$580 = (0.3X\$1,000) + (0.56)(\$1,500) + (0.14)(- \$4,000)

When you bluff \$1,000 and \$3,000, you can also have one of three outcomes: you can win the initial \$1,000, you can win \$2,000, or you can lose \$4,000.

You'll win \$1,000 50 percent of the time. You'll win \$2,000 70 percent of 50 percent of the time, or (0.7)(0.5) = 35 percent of the time, and you'll lose \$4,000 the other 15 percent of the time. (Note that we've set the numbers so that both sequences ultimately succeed roughly 85 percent of the time.) The EV of this sequence is, thus \$600.

\$600 = (0.50)(\$ 1,000) + (0.35)(\$2,000) + (0.15)(- \$4,000)

Betting somewhat more on the turn and somewhat less on the river increases your overall EV by \$20 even though, the way we set the numbers, your total chance of success drops slightly from 86 to 85 percent. It's because your turn bluff succeeds more often and because you win a bigger pot, \$2,000 versus \$1,500, when your river bluff succeeds.

You have to find the sweet spot. Dividing the bets \$2,000 and \$2,000 wouldn't work at all: It would violate the first principle by not leaving a credible bluff for the river.

When bluffing on the turn and river, maximize the size of your turn bluff while still leaving enough for a credible river bluff.