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.
Aunt Sophie raises the blind bet
“Nu, Dollink,” said my Aunt Sophie, “you said when I should raise the blind bets you would tell me.”
We were back poolside after I took a brief visit back to my apartment to change into dry clothes, occasioned by Sara having pulled me into the pool when I foolishly peered over the edge to ascertain her whereabouts. Aunt Sophie had spent the time making three lattes on my Krups espresso machine. Sara dripped and sipped demurely.
“Ah yes,” I said, trying not to inhale the cocoa on the foam, “I explained that when to call a blind bet from a one-card draw in lowball depends on the size of the pot. When to raise, though, depends on other factors. Those are the constitution of your hand, and how your opponent plays, as well as the size of the pot.”
“You mean,” asked Aunt Sophie, pushing me a square of the Rhubarb Paradise she had made before coming over, “that you don’t just with the good hands raise?”
“You’re referring to,” I queried rhetorically, “what your opponents call ‘raising hands’? And where is the cutoff point? Is any seven a raising hand? How about any six?”
“Well of course,” she retorted, “with any six I would raise.”
“It’s not as cut-and-dried as that,” I remarked. “I could show you situations in which you would not even raise with some sixes, and others in which you would raise with any eight or better.” I tapped a few figures into my notebook computer.
“What?” she demanded. “Did you hear that, Sara? The genius says not with a six I would raise? When would that be?”
Sara, realizing she was not required to supply any answers here, continued sipping quietly.
“The situation is somewhat artificial,” I admitted. “Nonetheless, let me posit one in which you would not raise with a six. You open in first position; a solid player behind you raises. All fold till the middle blind, who puts in one more bet. You and the first raiser call. The first player says, ‘The free look wasn’t there,’ and throws a king face up on the table. (Never mind that the king came from the middle of his hand, meaning he must somehow have contrived to look at the cards in positions 1, 2, 4, and 5 in his hand and somehow specifically missed the one in the middle.)”
“Even I can tell you’re getting sidetracked,” whispered Sara.
“You draw one,” I continued, “and, without hesitation, the player behind you pitches his discard. This lack of hesitation indicates that he probably was drawing all the time, which means that he’s drawing to a killer hand. Your hand is 2-3-4-5-K. When he sees that all are drawing, the first player says, ‘I made mine’ before receiving his card, and tosses in a bet. On the draw, you catch a six. If you were to raise, you would be making a big mistake. Here’s why. Since you don’t have the joker, it’s almost guaranteed that one of the other two does. To make the blind bet so confidently into two one-card draws, there’s a very good chance that the first player to draw has it, and, furthermore, is likely drawing to a wheel. To have raised on the come like that, the player behind you is likely drawing to either a 6-4 or a wheel. Can’t specify for sure the exact hands, of course, but let’s say it’s something like A-joker-2-3 for the first and A-2-3-6 for the second, who, for the sake of argument, also discarded a king. Now, based on an examination with Mike Caro’s Poker Probe, in this exact situation, you have considerably the worst of it. The first hand wins more than 46% of the time, while the 6-4 draw wins a bit over 27%, and you come in third at about 26.5% In other situations, you could be even worse off, by the way.
“(Parenthetically,” I said, parenthetically, “sometimes you could be better off. Here, it’s unusual that the draw to the six fares better than the 5-4-3-2. It’s because two aces and a six are out, cards that are needed by the non-joker wheel draw, but the 6-3-2-A is only hurt by one four and one five. This 3:2 penalty ratio makes the 6-3-2-A slightly more likely to win than 5-4-3-2, although it often wouldn’t be. That, however, is just a quibble. Because even when ahead of the draw to a 6, as, for example, a matchup among A-joker-2-4, A-3-4-5, and A-2-3-6, that wheel draw without the joker still fares no better than about 45:29:26. The hand needs to win a third of the time just to have neither positive nor negative expectation.)
Most of the time
“Anyway, let’s see what happens when you raise after catching a six in this situation. If it’s a 10-and-20 game, and you originally came in for a raise, the pot contains $135 before the draw: your $20 open plus two $10 raises three ways, plus $15 contributed by the dealer and big blind. Now there’s a $20 blind bet. You raise. Let us further posit that the first player will not call your raise with anything worse than a seven, and that the player behind you will not call two bets cold unless he makes his hand. That is, he will fold if he catches a seven. Most of the time you make the six, you win; it works out to about 65-to-35 that the six wins.
“How often does your raise get called, though? When the first player makes a seven or better and when the second makes a six or better. The only time you win when you raise and get called is when the first player catches exactly a seven, and the other player does not beat your hand. The two events combined occur approximately 9.38% of the time. (This is done by an exact calculation. The first player catches a seven 4 times out of 37, and the second player catches a non-winning card 30 times out of 36. Multiply 4/37 by 30/36 to get 0.09009.) How much is that worth to you? This is figured by multiplying $175 times 0.09009, to get $15.77. To restate what might not be obvious, a bit more than 9% of the time you make $175; 9.09% of $175 is $15.77. That is what this part of the situation is worth.”
I paused for a sip of latte. “Tell me,” I requested, “what do you do if your raise is raised?”
“What do I do?” she snorted. “My teeth I grit and call, and usually I’m beat, but still I have to call.”
“And if they both raise?” I qualified.
“Both?” she echoed. “If both, then I make a good laydown. Surely one has a straight six beat.”
“Okay. In that case, you lose $40 when both players beat you, because they both raise, and you don’t call two raises. The cases in which this happens are the first player catches a four, five, or six, and the second player catches a four or five, which is 6/37 x 5/36 + 2/37 x 6/36 = 0.0315315. Multiply by $40 and you get a loss of $1.26 per hand.”
“And,” I went on, “You lose $60 when only one player beats you, because you call one raise. I don’t think it’s an unwarranted assumption to say that if either player makes his hand, he raises your raise. So, you lose $60 if the first player catches a four, five, or six, and the second player catches a nonwinning card, or the first player catches a non-winning card and the second player catches a four or five, which is 6/37 x 31/36 + 2/37 x 30/36 + 31/37 x 6/36 = 0.3243242. Multiply by $60 and you get a loss of $19.46 per hand.”
“Of course,” I expostulated, “You win $155 when neither calls the raise. This happens only when the first player catches an ace, two, three, or an eight through king, and the second player catches an ace, two, three, six, or a seven through king, which is 4/37 [first player catches ace, two, or three] x 30/36 [second player catches non-winnner] + 21/37 [first player catches eight through king; three kings are gone] x 30/36 [second player catches non-winnner] = 0.5630629. To find out how much that is worth, multiply by $155 and you get $87.27.”
“Using the previous figures, then,” I proceeded, “Your net is $15.77 – $1.26 – $19.46 + $87.27, for a total positive expectation of $82.32.
“Now let’s say you flat call instead of raise. This time let’s say that the player behind you will call with a 9 or better, a not unreasonable assumption, since he will assume that the money odds are causing you to call with almost anything. We can further state that he raises only if he makes his hand, and the first player reraises with a six or wheel. As before, you call one raise, but not two.
“So, you win $155 when the first player catches a losing card and the second player catches a 10 or worse. I won’t bore you with the details, but this comes up about 40.09% of the time. To find out how much that is worth, multiply by $155 and you get $62.14.”
“You win $175 when the first player catches a losing card and the second player catches a seven, eight, or nine. Times $175, for a profit of $44.14 per hand.
“You lose exactly $20 when the first player catches a winning card and the second player catches anything, because one of two things happens: either there is no raise, or there’s a raise and a reraise, and you fold; but you’re beat in both cases. That works out to 8/37, or 0.2162162. Multiply by $20 for a loss of $4.32.
“You lose $40 when the first player catches a losing card and the second player catches a winning card (because you call one raise). This happens 13.06% of the time. Multiply by $40 to get a net loss of $5.23. We can assume the first player flat calls a raise when he makes a 7-4. If he reraises, your actual loss is even less.
“You never lose $60, because you don’t call two raises.
“This time your net, without having raised with that six you made, is $62.14 + $44.14 – $4.32 – $5.23, for a total positive expectation of $96.73. Recall that when you raised, your net was $82.32. So not only do you win nearly $14 more by not raising, you risk considerably less. To belabor the obvious, you risked $20, which sometimes, rarely, became $40, to win nearly $100, almost a 5-to-1 return, versus risking $40, which sometimes became $60, to win a bit more than $80, barely a 2-to-1 return.”
Aunt Sophie looked at me, light appearing to dawn behind her eyes. “Is that,” she offered, “why Mike Caro’s lowball strategy sometimes just has you call with a six after the draw, instead of the automatic raise that many players make?”
“Ah,” I returned, “the light dawns.” I looked over to Sara’s chair, to discover only a puddle and fading damp footsteps leading to the pool, wherein a siren slowly paddled.