Standing on a Jack in Lowball

You're on the button in a limit lowball game. Everyone passes to you. Your hand is J-8-7-3-2. In a double-limit game (such as those of Northern California), you just open; in a single-limit game (such as those of Southern California), you open for a raise. So far, so good. Only the big blind calls, and that player takes two cards. What should you do? This is one that most longtime lowball players get wrong. Someone could play for decades and if he tried to figure out his playing strategy on the fly, the situation might not come up enough for him to work it out just from observation. Those who learned lowball in the '60s, '70s, and '80s automatically break the jack in this situation. (Sadly, most lowball players are from that era. That's a shame.) It turns out that in almost every other situation, breaking a jack is the correct play. But if you start with any J-8-7 or J-7-6, standing pat on the jack against a big blind's two-card draw, as opposed to drawing a card, is by far the better play.

So, here's another lowball situation that involves mathematics, although it's not quite straightforward to work out the figures. In fact, easiest is to use Mike Caro's wonderful Poker Probe, a program that compares hands, and then use the resultant figures to make your best decision.

I compared the specific hand J-8-7-3-2 against the two-card draw 6-5-joker. The pat jack won 59.42-40.58, or very close to 3-2.

Then, I compared the one-card draw 8-7-3-2 to the two-card draw 6-5-joker. The one-card draw won 55.17-44.83, a much-reduced 5-4.

You cannot assume that whenever the big blind draws two cards, that player's hand contains the joker. So, then I compared the specific hand J-8-7-3-2 to the two-card draw 6-5-4. The pat jack won 67.87-32.13, something better than 2-1.

Then, I compared the one-card draw 8-7-3-2 to the two-card draw 6-5-4. The one-card draw won 61.79-38.21, about 13-8.

A random hand should get the joker less than one time in 5, so if you take a weighted average, the pat jack should beat the two-card draw about 66.18-33.82, while the one-card draw wins 60.47-39.53. Here's where the 66.18 comes from: (4 × 67.87 + 59.42)÷5; and here's the 60.47: (4 × 61.79 + 55.17)÷5. The exact figures would need to be worked out by comparing all possible J-8-7 hands to all possible two-card draws, and all possible 8-7 draws to all possible two-card draws, but the average would be close to what I assume here. Let's round the figures off somewhat conservatively and say that, on average, a pat jack beats a two-card draw approximately 66-34, while a one-card draw to an 8 beats a two-card draw approximately 60-40.

Your argument might be: But when you draw to an 8-7, you can bet it after the other player passes, or you can call if he bets, and you win most of those. The problem is that you don't make an 8 most of the time, and, furthermore, you can never raise with the hand you make. When the other player bets, you will call sometimes with kings and queens, but, on average, your most frequent calling hands will be in the range of 8 through jack. Also, when your opponent passes, you will probably bet with a 10 or better, and maybe some of the times that you pair, so your average betting hand also will be in the range of 8 through jack. So, on average, the hands that will be bet will be those with which you make a jack or better. When you draw one card, you make a jack or better approximately 28 times out of 48 (a bit more than 58 times out of 100); you win approximately three-fifths of those contests. (That came from the last sentence of the previous paragraph.)

So, now let's see what happens in 100 matchups of each situation in a $20 bet pot. Either you opened for a raise on the button in a $10-$20 double-limit game or you opened in a single-limit game; the result is the same. We can neglect the bet after the draw here, because not many players bet into a pat hand with a bluff. You can probably safely fold almost every time your opponent bets. You can call just often enough to keep your opponent from thinking that you always fold in this situation, thus encouraging him to bluff.

The pot contains $45: your open, the big blind's $10 plus his $10 call, and the dealer blind. You either win $25 or lose $20. In 100 matchups, you win 66 x $25, or $1,650, and lose 34 x $20, or $680, for a net of $970. Divide by 100 and your profit is $9.70 per hand.

Now, let's try 100 drawing matchups. There is no bet 42 times, and you win three-fifths of those times, 42 × 3/5 x $25, or $630, and lose 42 × 2/5 x $20, or $336, for a net of $294. When there is a bet, you either win $45 or lose $40. In the other 58 times, you win 58 × 3/5 x $45, or $1,566, and lose 58 × 2/5 x $40, or $928, for a net of $638. Altogether, you win $932 ($294 + $638). Divide by 100 and your profit is $9.32 per hand.

Will you win 60 percent of the hands that involve a bet after the draw? That depends partially on how likely your opponent is to bluff, but if he bluffs correctly, he probably bets whenever he makes a 9 or better, and he also bluffs whenever he ends up with a pair higher than what he started with (by catching a cold pair) or worse. You should call with approximately a queen or better when he bets, and you should lose approximately two-thirds of the time. (Why this is so involves game theory.) But of the times he passes, you should bet whenever you make a 10 or better plus whenever you pair your top card, and he should call with a king or better. You should win more than two-thirds of those matchups. Overall, you probably win 60 percent of the time (because you can bet more often than the two-card draw can), and the hands that get bet involve the times you make a jack or better.

These figures are shaded somewhat conservatively. If you match better pat jacks (J-8-7-2-A as the best and J-8-7-6-5 as the worst) to worse two-card draws, the pat jack wins more. But let's just say that the difference is $38 over 100 hands, or 38 cents per hand, in favor of standing on the jack. Notice that you won more and risked considerably less by not drawing. You always bet exactly $20 in the pat hand scenario; your average bet is more than $30 when you draw a card. (It's approximately $31.60.) (I know. Sometimes you have to call a bet with the jack, but if you call exactly correctly, that should be a wash, and you can neglect the bet after the draw. Figuring in that bet needlessly complicates the math here.)

Now, if breaking a J-8-7 is wrong, you can see that breaking a J-9 is even worse. Yet, you see many old-timers do exactly that all the time. They know that it is correct most of the time to open on the button, when everyone has passed, with a 9 to draw to, a one-card draw to a 9 being better than the random cards remaining for two players. The value of this hand is further enhanced by having position – that is, by getting to see what the other two players do both before and after the draw. The best thing that can happen to you when you open on the button in lowball with a J-8-7 or J-9 is that both the blinds fold. That should happen as often as half the time. The next most likely thing to happen is that the little blind folds and the big blind draws two cards. Of course, both of them can call, or the big blind can draw one card. Or, one of them can raise. Whether to stand is then a different situation, but not the premise of this column. The premise here is what to do with a pat J-8-7 after your one opponent has taken one card.

It turns out that the cutoff on what jack to break is approximately J-8-5. I won't go into details here nor report all the details, but, using the previous reasoning, you should stand on J-8-6 but discard the jack and draw one card if you have J-8-5.

Again, I won't give all the details, but you should stand on J-7-6 and break J-7-5. You can break J-7-6-2-A because, even though standing is better than breaking, it's close enough to improve the betting after the draw. Also, against some players, you can raise with 7-6-3-2-A or so. But you definitely should stand on J-7-6-5-4. And you should break any J-7-5 or better.

Also, if the joker is in your hand, you should draw to your J-8-7. On the other hand, if you know the other player does not have the joker, you should be more willing to stand on your pat jack. How do you know this? Many players flash the joker when they fold, show it to their neighbors, make some comment about what a shame it is to "get this card and not be able to use it," or sit and admire the card before finally reluctantly mucking their hands.

So, bottom line, here's a play that goes counter to intuitive wisdom and how many old-time lowball players automatically play. Against a two-card draw, most of the time it is more profitable (and less risky) to stand pat with a J-8-7 or a rough J-7-6 than it is to break the hand and draw one card.