Breaking a Pat 8 in Lowball

Someone broke a pat 8 against me in a no-limit lowball game to win a large pot. He looked like a hero and thought he made a very clever move. I can show both by poker logic and mathematics that it was a terrible play. And this is not just sour grapes, but precise analysis many years later.

The game was no-limit lowball in the late '70s at Palo Alto's legendary Cameo Club, famed at the time for the biggest games in California. The "Big Apple," as it was called, the biggest game in the 13-table house, was usually $20-to-go. If the right players showed up, though, in addition to – or in place of – that game, the biggest game would be $80-to-go. It had three traveling blinds, starting with the dealer and moving left, of $20, $20, and $40. Sometimes, a "round from home" – in which the dealer position doubled the big blind by putting $80 in the pot – would make the game $160-to-go. That game was much too big for me, although I did sometimes play the $20-to-go.

Normally, I played $4-to-go, with its three traveling blinds of $1, $1, and $2. The bring-in (minimum opening bet) was always twice the size of the big blind, making the minimum in this game $4. If someone killed (overblinded) the pot, the bring-in was twice the size of the kill. A kill had to be at least twice the size of the big blind, and could be more, plus there could be more than one kill.

On this particular hand, I was dealt 7-6-4-2-A in early position, and opened for $4. The next two players folded, and then Ralph raised $20. Everyone else, including the blinds, folded back to me. I immediately shoved in my stack, another $74, a bit more than the size of the pot. Raising the pot made no sense, because that would leave me only $22. Ralph called.

I of course stood pat. Ralph thought for a while. Finally, he pulled a card out of his hand and showed it faceup, an 8. He then waited for me to show my cards, which I did. Ralph triumphantly showed that he had made a 7-4, and took the pot. He said, "I knew you had that 8-7 beat, so I had to break it." He implied that he had made a very clever play. I was of course disappointed, but that doesn't matter. What does is this. Had he made the right play?

Let's look at this mathematically. I would open from early position with any good draw plus any pat 8 or better. I would call a $20 raise with most of those drawing hands, planning to bet the rest of my chips if I made the hand, plus sometimes on a bluff. I would reraise with any hand 8 or better. I might even make that play with a smooth 9, intending to stand pat and hope that Ralph would break. I would not have a rough 9 here, although Ralph did not know that. But let's discount the possibility of a 9, even though that brings the figures even more on my side. Discounting the cards in Ralph's hand, there are 56 pat hands 8 or better. It is true that the joker makes some hands likelier than others. Also, the cards in Ralph's hand affect the likelihood of the other hands. The 8 in his hand made it somewhat less likely that I had a pat 8. Nonetheless, we can make a reasonable approximation from this. Ralph started with the best 8-7, 8-7-3-2-A. Now, breaking his hand actually made a difference only when I had a 7, so those are the real figures to deal with. Given the cards in Ralph's hand, the actual ratio is almost exactly 2-1 that I have him beat with a hand that he could draw to beat. (This could be worked out exactly, but I got close enough using Poker Probe. I dealt a million random hands against a pat 8-7-3-2-A and then enumerated the eights that lost, those that won, the sevens, and the ties.)

To call my raise, he was getting $126 for a $74 call, less than 2-1. He would either lose $74 or win $126. We don't count what he already put in the pot because that was gone. We're talking only of the $74 investment. In three times the situation came up, if he stood pat behind me, he would lose two times, for a loss of $148, and win once, for $126. That's an overall loss of $22.

Let's see how breaking the 8 fares. We know 10 cards in the deck, so he's drawing from a deck of 43. Any 4, 5, 6, or the joker makes him a winner. Accounting for the cards in my hand that he needs and are not available to be drawn, that's 11 cards that win and 32 that don't, nearly 3-1 against. He must have known when he called my reraise that I was pat. When he said he "knew" I had him beat, that must mean that it was his intention all along to draw a card if I stood pat. In four times the situation came up, if he drew, he would lose three times, for a loss of $222, and win once, for $126. That's an overall loss of $96.

The math here is in hindsight, based on my actual hand. Given that I could have had any hand better than his, his decision was actually considerably worse. It was actually about 2.6-1 – counting all the hands better than what he was drawing to – that I had him beat, so when he broke, maybe one-sixth of the time he was drawing dead. Even charitably, then, the bottom line is that both decisions were bad, but drawing a card was considerably worse than standing pat. The only good decision for him to have made was to fold.