Inducing Bluffs by Checking

Sometimes in poker you'd like to induce your opponent to bluff – and not only when you have a strong hand. On the river with a medium-strength hand, it's often better to check and call than to bet out if your opponent will likely abandon his bluffing hands to a bet.

When there are cards to come, the situation can get a bit more difficult. Checking the turn in order to induce a bluff on the river sounds a little self-destructive. After all, it gives an opponent a free chance to outdraw you, and may not earn you the extra bet anyway. But under some circumstances, the risk may be worth it, and it's worth figuring out what those circumstances are.

A common case that's worth considering is one in which you raise preflop, the big blind elects to defend, and you flop a big hand, perhaps two pair. When your opponent check-calls the flop, and checks again on the turn, conceivably your best bet may be to check in order to make it appear that you've abandoned your bluff, and to induce your opponent to bluff on the river. On the other hand, against some opponents, the turn is your last chance to extract payment from a drawing hand, or some other hand that would call on the turn but needs to improve to call a bet on the river.

To do this right, the best approach would be to work out a range of different situations so that we can understand when checking is or isn't appropriate. But for now, I've taken the first step – working out a simple situation in which checking is marginally better than betting. To do this, I've broken the analysis down into nine possible outcomes, five of which you bet on the turn:

1. Your opponent folds.
2. Your opponent calls, then checks and calls with a loser on the river.
3. Your opponent calls, then bets a winner on the river.
4. Your opponent calls, then bets a loser on the river.
5. Your opponent calls, then checks and folds on the river.

And four of which you check on the turn:

6. Your opponent improves and you call.
7. Your opponent fails to improve and doesn't call a bet.
8. Your opponent fails to improve and does call your bet.
9. Your opponent bluffs and folds to your raise.

The details I've assumed are as follows: There are four big bets in the pot before your action on the turn. You have a hand that you know is the best right now, and know that your opponent has something weak enough to be folded on the turn. Perhaps you have A-Q with a board of A-Q-7-5, while your opponent has pocket eights. So, we're taking some liberties with your ability to read your opponent, but the numbers below can of course be adjusted for other situations. Rounding slightly, we'll call this a 4 percent chance of your being outdrawn. We'll also assume that you're smart enough not to raise the river when your opponent may have hit, but that you'll always call (effectively, we're rolling bet-call and bet-raise-fold together). Finally, your opponent will fold to your bet on the turn 80 percent of the time, but will bluff 30 percent of the time on the river if you check the turn, and half that if you bet the turn, and will call your river bet 50 percent of the time.

Now, all we have to do to calculate your expectation for betting versus folding is work out the expected value (EV) for each of these situations, weight them according to the probability of each, and see which is bigger. I've done this, and here are the tables for betting and checking, respectively. (Note that the EV is relative to your stacks before your action on the turn.)

Mission accomplished – we've identified a situation that's not completely implausible in which checking the turn may be marginally more profitable. Now, we just have to figure out what it means.

Most obviously, we've described a fairly extreme situation, and checking is only a hair better than betting out. So, the first lesson to take home is that when in doubt (and even most of the time you're not in doubt), bet the turn. The risk of dropping an opponent who might be induced to give you action on the river is rarely great enough to outweigh the cost of missing a bet that your opponent might call.

Of course, adjustments to these numbers could favor one move or the other. Opponents you deem more likely to bluff the river (we all know a few opponents who will jump on nearly every opportunity to steal once you've shown this kind of weakness) are better candidates for the check; similarly with opponents who are likely to keep you honest on the river once they know it will cost only a bet to see your cards. Of course, such opponents may rarely be as likely as 80 percent to fold on the turn anyway, which would tilt the balance back in favor of betting. Opponents who may have slightly better hands – underpairs, or flush or straight draws – in particular, hands worth calling the turn, are likewise worth betting into, because they will be less likely to fold anyway. And as the existing pot gets larger, betting out becomes more appropriate, since the risk of losing the whole pot is magnified. Finally, opponents who are likely to keep you honest once they've reached the river, calling your final bet to get to show down, are better candidates for the check, especially if they're more liable to abandon their hands on the turn.

I've found that having a few prototypical situations in mind helps in evaluating new situations, even though they'll rarely match exactly. But there are numerous ways to deepen this fairly simple analysis. We can play with a wider range of estimates for the parameters that went into this analysis, to understand a wider range of situations. We also can break this analysis down further, to include some potential uncertainty about your opponent's holdings. Suppose you have top set, and believe your opponent could have only a smaller pair or a flush draw. The former may be a clear check on the turn, while the latter may be a clear bet. So, you have two estimates for both your checking EV and your betting EV. In order to calculate the best option, you need to calculate weighted EV estimates separately for both checking and betting. That's just an example, and to save you some work, I'll tell you what comes out. The more likely the flush draw is, the more the balance tilts in favor of betting the turn. Missing your last opportunity to cash in on your opponent's missed draws is too catastrophic to overcome unless the chances of a flush draw are truly tiny.

Regular readers of my column will, I hope, recognize this analysis as yet another application of a simple arithmetic recipe. A variety of poker situations can be subjected to this kind of analysis, in which the EV's of competing decisions are compared directly, by calculating a weighted sum of each potential outcome. The appropriate calculations can be made in any spreadsheet program in minutes, and can occasionally yield surprising results. Even if you don't find the percentages I've used for this example helpful, I hope you'll find some use for the overall approach.