Note: Not at the old Poker1 site. A version of this entry was first published in Card Player. This entry in the "Aunt Sophie" series covers poker.
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Michael Wiesenberg
Aunt Sophie and the blind bet
“So Dollink,” said my Aunt Sophie, “tell me when I should call blind bets.”
“This,” I temporized, “is one of those times I must demur, citing insufficient information.”
We sat poolside at a glass table beneath a large yellow and red umbrella. My attention was not fully on my Aunt, distracted as I was by a naiad named Sara who splashed happily in the pool on this unseasonably warm spring day.
“Even with words I know,” scolded Aunt Sophie, “you say things in such a way I got to think about it twice before I understand.”
Lowball
“My dear Aunt Sophie,” I responded, “you haven’t told me what game you’re playing nor any circumstances for me to respond. Although I presume you’re talking about your latest cardroom passion, midlevel lowball.”
Sara slid out of the pool to execute another dive, causing me again to marvel at the feat of engineering that kept the four tiny triangles of her string bikini where they belonged.
“Such a smottie,” Sophie returned. “Of course lowball I mean. And blind bets, you know what they are. Sometimes two cards I draw and the first player draws one and he’s an action player so before getting his card he bets. And so I want to know when I have the best of it by calling.”
“Now that,” I replied, “is one of those situations in which pure mathematics dominates. Many lowball situations are mostly automatic, while others are based more on tells and a knowledge of your opponent. What is your strategy, and how successful has it been?”
“Well,” Aunt Sophie offered, “I know that a pat nine is close to 50-50 against a one-card draw, so I figure half the time he makes a nine or better, and half the time he doesn’t, and this means if I call with a nine or better half the time I should win. So that’s what I do. Sometimes I feel lucky, and call if I make a 10, but overall I think I lose more of those than win, so I probably shouldn’t do that. And how do I do? I guess I break about even with the play, but it’s worth it for the times I make a good hand and raise, and then the guy feels potstuck and calls sometimes with a much worse hand. Of course, once in a while he raises back, and then I’m the one that gets stuck calling, but that doesn’t happen very often.”
“That, unfortunately,” I declaimed, “is not the optimal strategy.”
“And what,” she demanded, “is the optimal strategy?”
Positive expectation
I didn’t answer immediately, being distracted by a swiftly swimming siren. “Oh,” I finally put in. “The optimal strategy is to call whenever doing so gives you a positive expectation, and not otherwise. Well, that’s not quite optimal. Sometimes you raise, also when doing so gives you a positive expectation. And you’ll find, interestingly enough, that sometimes raising with what is almost certainly the worst hand has a positive expectation. But that’s advanced strategy. Let me begin with just the calling portion of the strategy.”
“Nu,” Aunt Sophie prompted, as my attention again wandered to the pool, “so begin already.”
“It’s all game theory,” I continued. “Your decisions always depend on the situation, and most of the situation is the size of the pot at the point you are faced with the decision of whether or not to call. You make money by calling with worse than a nine. For example, let’s say the pot is at minimum size in the 10-and-20 game you frequent, the middle blind opened for a raise and you, in the big blind, called. That puts $90 in the pot (including the dealer blind). The opener draws one card and bets blind, and you want to know what to call with. After his bet, there’s $130, so for your $40 call, you’re getting 3.25-to-1. All you have to do is beat less than one-quarter of his hands or more, and you profit. Let’s see why. Say you have a hand that wins a fourth of the time. So, in four times that the situation comes up, three times you lose your $40 call, for a loss of $120. one time you win the $130 that’s in the pot. So you win $10 over the four times. To break even, you would actually need to win four times in 13, and that would be your cutoff point. Any hand better than that cutoff point shows a profit overall, even though you might lose nearly three times out of four.”
“But, but,” sputtered Aunt Sophie, “it cost me $40 before the draw and $40 after. That’s $80. Times three is $240. And when I win, I just win two bets plus $10, which is only $90. Looks to me like a loss of $150 in four hands.”
To make money
“No, no,” I explained. “You can’t count what you put in the pot prior to the bet after the draw. That’s gone. You’re just trying to make a decision on that individual $40, whether or not to call with it. Don’t make the mistake so many players make. The money you put in the pot earlier is no longer yours. It doesn’t matter where that $130 in the pot came from. After making your $40 call, you either win that $130 or you lose your $40 call. That’s plus $130 versus minus $40. In four times, three decisions are losers and one is a winner, as I just showed you. Although it is possible to tie, the chance is so small, we can neglect it here. All you need to know is what that cutoff point is. You need to choose a hand that wins only one-fourth of the time to make money.”
“Let me understand this straight,” she demanded. “Only one time in four I gotta win and I win?”
“Precisely,” I laughed. “You’re getting money odds on your call, as I demonstrated. Now, let’s say you think your opponent is drawing to a joker-wheel, something like ace-deuce-trey-joker, and he discarded something like a king. He can catch any one of 48 different cards. Nine of those cards pair him, any ace, deuce, or trey. Three kings remain. Together that makes 12 cards, and those are, conveniently for this calculation, the one-fourth of his worst hands. So, if you call with a queen or better in this situation, you will win approximately one-fourth of the time. If you know that he does not have the joker — such as for example you have it in your own hand — you can call even rougher, and still have the best of it. If your opponent was drawing to ace-deuce-trey-four, for example, any one of 12 cards would pair him, and you could call with a king or worse. Now, I’m talking here only about a blind bet. If the player looks before betting, most of the time you can decide whether to call based on your knowledge of that player’s play. You would tend to call a player who bluffs a lot more often than one who does not.”
“My,” Aunt Sophie marveled, “with a king I can call and make money.”
Better the odds
“Depends on how much is in the pot,” I put in. “The better odds you’re getting on your call, the less often you have to win to make money. In fact, you make your decision based on how much actually is in the pot. Let’s say you had opened on the button for a raise, $40, in late position, and the middle blind raised that. The big blind called, and you, with your good one-card draw, also did. So now there’s $180 in the pot — three bets each. The middle blind draws a card, as do the big blind and you. Some players like to bet blind in this situation, and that’s what the first player does. You don’t have the joker, and it’s not a bad assumption that the first player has it. The second player says something to the effect of, ‘Shucks, I paired,’ and discards his hand. For your $40 call at this point, you can win $220, or 5.5-to-1. You have to win only 1 time out of 6.5, which is the same as 2 times out of 13, to profit here, which would mean you could call with a hand that would beat only 2/13 of the hands the first player could make. Multiply 2/13 by the 48 cards he can catch.”
“Why 48?” queried she. “Some of the cards in my hand he could probably use, and some from the other player.”
“True,” I agreed, “but you don’t know which ones. Some of those cards might just as easily pair him. Unless you actually see his hand, you have to figure this as if he were drawing from the 48 cards he hasn’t seen. Anyway, from before, only a bit more than seven cards form his worst hands. Any three or two, plus one ace. If you end up with a pair of aces in your hand, that accounts for two of the cards, so you can call in this situation with a pair of aces, or any better hand. Do you sometimes see players in your game make what looks like a crazy call with a small pair? That may seem like the play of a live one, but, as you see, sometimes it is the right play. On the other hand, say the middle blind opened the pot, and just for the minimum. You have a three-card draw, but you can make it, since it costs nothing to call. The opener bets blind, again, with his one-card draw. This time, however, your $40 call gains you only $50. You need a hand that beats the worst 4/9 of his hands. That’s a bit more than 21 of his worst hands. Nine cards pair him, plus there are four each of kings, queens, and jacks, totalling 21. To beat any of those hands, you need a 10 or better. So, depending on the odds you’re getting, you might call one blind bet with nothing worse than a 10, and you might call another with a small pair. Now please notice that this applies only to a particular situation. Sometimes you’ll see a player bet a two-card draw blind because the other player drew three. That’s different.”
“Uh huh,” she nodded, appearing slowly to grasp the concepts. “But when do I raise?”
“That my dear,” I supplied, “will have to come with another lesson.”
And with that, I walked to the edge of the pool for a closer look at what suddenly seemed to have become an empty pool. I leaned over to see if the siren was submerged just at the closer edge of the pool, when a brown hand suddenly snaked around my wrist and pulled me headfirst into the pool, there to forget entirely about lowball and game theory.