Aunt Sophie Calls With Pairs

I walked heavily into the coffee shop of the Anaheim Casino, and sat in the booth next to my favorite aunt.

"Tsatskeleh!" gasped Aunt Sophie, "vass ist geshayn?" When excited, she sometimes lapses into the Mama-loshn, or "mother's tongue," that is, Yiddish. What happened? she wanted to know.

I winced as I shifted position in my seat. "Bicycle accident," I responded.

"You're limping," she observed; "it was serious?"

"Not really," I replied. "Mainly painful. I was macerated and lacerated, bruised, abused, and contused. And I learned a lesson."

"Ya," demanded Aunt Sophie, "and what might that lesson be? Don't ride bicycles?"

"No, ha-ha," I laughed. "The lesson is, don't peddle around a sharp corner; just coast. I hadn't realized that, but I certainly found out the hard way. I had taken that particular corner, a sharp switchback coming off a pedestrian bridge, at at least that speed before, so I couldn't understand what happened when the bike suddenly shot out from under me and I came down hard on my left hip and left forearm." I pulled up my sleeve to show her the road rash, which still had an uncanny resemblance to strawberry jam. "The contusions on my hip you'll have to take my word for, but I can bet they'll take on a lovely Technicolor cast as the days go by. Right now they're kind of blue and red."

"You're lucky," she observed, "nothing to have broken."

"There I agree with you completely," I assented. "As to the lesson. When I painfully picked myself up, I started examining the paved path. I'm always very careful to avoid gravel and to ride extremely slowly when the pavement is wet, but it was sunny and dry this morning and there were no rocks or gravel where I had ridden, nor were there any skid marks to indicate I had perhaps inadvertently hit the brakes. I was completely mystified. I had to carry the bicycle two miles back to my house, and limping as I was, that wasn't much fun. The rear brake pads were locked into the tire, which had burst, and the wheel itself was very bent and useless. When I took it to the bicycle repair shop, the repair guy listened to my story. He examined the left peddle and showed me how gouged it was from the pavement, and told me that when the bike is leaned over, the peddles don't have a lot of clearance, and if a peddle hits the ground, that can easily drop the bike – which must have been exactly what happened to me. He said that for sharp corners I should not peddle, just let it coast, and that's my lesson. You should see the rear wheel – Completely warped. Needs replacing. Brake pads are shot, too. The estimate to repair the damage is just over $100."

"Better that," she put in, "than something broken. And now can you answer me a question about lowball?"

"Of course," I said.

"In my game," she began, "one player has a funny habit that I think I can use to better make calls when he bets. When someone passes to him, under the edge of his draw card he sticks his cards he kept and flips the card so it turns over in the air. If an ace to 8 he catches, he bets. Not all the time he does this, and more often against more than one player, but many times. If a 9 or higher he catches, he just shows what he made. So what I want to know, can you tell me a calling strategy?"

"Hmm," I temporized, "before I answer, tell me, does he really bet every time he catches one of those cards? Most of the time? Or what?"

"Ah," she came back with, "I'm not sure, but it seems like about half the time he bets. Sometimes he just gives up and says, 'I paired.'"

"Well," I offered, "this is an interesting situation, and you can employ some game strategy to make your calling decisions. As you know, poker is a game of incomplete information. The more information you get from a player, the better decisions you can make. Clearly, by seeing this player's draw card, you are getting some information to which you would not otherwise have access. Essentially, the question is, what type of an edge are you gaining by looking at his card? One clear observation is that he is giving up some value bets when he catches a 9; you might call with a worse 9, a 10, or paint. If he shows the 9 and bets, of course, you can safely lay down all the worse hands with which you might otherwise have called. Now, here's an interesting speculation. He might do this turn-over-a-card trick in a situation in which he thinks a 9 is not worth betting. That is, he might turn over his card only when drawing to an 8. You'll have to pay attention to what hands he shows down when called. Apart from that, though, if he 'usually' bets, even when he pairs, you should always call if you have any no-pair hand, plus any pair lower than the card he catches. That's easy. You won't win all of the time, of course, but your calls will have very positive expectation. Look at the numbers. Discounting the exact cards you hold in your hand, of the 28 cards ace through 8 that he can catch when he doesn't hold the joker, 12 pair him and 16 don't. Don't worry about his catching the joker, because you can safely fold when he turns that over, which already shows an immediate gain because you would have paid him off at least some of the time when he catches it. It's 16-to-12 against his pairing, or 4-to-3. You're getting at least 4-to-1 (including the blinds). So, in seven times that the situation comes up, four times you lose one bet, and three times you win four bets. Subtract 4 from 12 and you get 8, the number of bets by which you profit in those seven times. Say he bets only half the times that he pairs. Now, it's 8-to-3 against his bluffing. That is, 16 times he makes a hand, and of the 12 times he pairs, 6 he bets; 16-to-6 is 8-to-3. So now, in 11 times that the situation comes up, eight times you lose one bet and three times you win four bets. Subtract 8 from 12 and you still get 4, the number of bets by which you profit in those 11 times."

She mulled it over. "Aha," she returned.

"Even if," I continued, "he is trying to play game-theoretically correct, he gives away some information. For example, suppose there are three bets in the pot. Game theory says that he should bluff exactly one-fourth of the time he value bets. So, if he has four small cards, there are 16 cards and the joker he can catch that he will value bet. As before, we discount the joker. So, there are four cards with which he should bluff. Game theory also says that if he bluffs correctly, you should call with a hand that can beat only a bluff three-fourths of the time he bets. (By the way, he probably doesn't play game-theoretically correct, not if he seems to bet half or more of the time in this situation.) In general, the closer he seems to be bluffing correctly, the more you should be willing to fold if the card he catches also pairs you. That is, say he turns over a 4 and you have one in your hand. You probably shouldn't call, even if your pair of fours hand is likely smoother than his. This is doubly true if you started with a pair of fives, throw a 5, and he catches a 5. That is, no matter how often you think he bluffs, you should never call in that precise situation. But, since you should usually call anyway, you should call with a lower pair than what he shows whenever the card he catches would not have paired you. Specifically, if you have a smaller pair – say your hand is A-A-3-4-7 – that's 16 cards (any 2, 5, 6, or 8) that can come that do not pair you and with which you can call, plus you can fold around half of the 11 cards that pair you, depending on how often he is bluffing. If you think he's bluffing correctly, fold half of those times. For example, say that hand consists of two red aces, and the 3, 4, and 7 are black. There remain six red cards that are pairs to cards in your hand; if he turns over one of those six red cards, fold; with any other card that he turns over and bets, call."

"And that's game theory?" she wondered doubtfully.

"Indeed it is," I supplied; "game theory very specific to a very limited situation. But you asked, and I answered your question."