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 learns about game theory
“Ay, tsatskeleh,” Aunt Sophie exclaimed, “Such a dream I had. So scary. A regular nightcrawler.”
“Go on,” I hazarded, already preprogrammed to ignore malapropisms.
We occupied our accustomed station, a booth in the Anaheim Club’s coffee shop, sipping caffeinic potables while taking a break from our respective games, 100-200 hold’em in my case, 20-40 lowball in hers.
“In the future,” she began, “I somehow got transported to, the year 2009. I’m in this club, and I look around, and all the same people are playing all the same games.”
“So how,” I asked, “did you know it was the future?”
“A poker magazine,” she responded, “I just happened to grab, and there was the date on the cover, and the issue was all about the tenth annual Tournament of Champions. But that’s not what scared me. I looked inside and found installment 285 of ‘World Series of Murder,’ and, as usual, at the end was ‘To be continued.’”
“And did you see,” I wondered, “who won the TOC of 2009? I could probably get a bet down in 10 years.”
“Listen, Dollink,” she returned, “the real future I might not have been dreaming. Maybe one of those alternating universes, so your money you shouldn’t risk. If we’re even alive, kenanhora. But I really had a practical reason for grabbing you for a bit of schmussing from your game. I read recently about how to bet using game theory, but I’d like to know to defend against that type of betting.”
“Okay,” I assented, “as long as you understand one thing. And that is that you can nearly always do better than game theory by knowing your opponent. As an extreme example, if you know a particular player always bluffs when he misses a hand, then you always call with all your good hands plus most of the hands that can beat a bluff. For example, in lowball, you draw ahead of this frequent bluffer who also drew, and catch a 10 or worse on the draw. You pass and call when he bets. You might even be willing to call in such case with a pair of aces. Let’s say you do this. Assume $100 in the pot before the draw in a 20-40 game, two bets from each of you, plus the blinds. You draw first and make a pair of aces. You check. Your opponent, who drew one card, bets $40. The exact results depend on the specific cards in each hand, so let’s talk about you drawing to A-joker-2-3 and catching an ace, and your opponent drawing to 4-5-6-7. Assume further that you each discarded a king. Of the 43 unseen cards, you win when you call with aces whenever he catches a 4, 5, 6, or 7. Twelve of those remain in the deck. He bets no matter what he catches, which is actually the worst thing for you; you’d be better off if he showed down his 10s and jacks. Anyway, the odds are 31-to-12 against his pairing. Let’s see what happens if you call every time he bets. If you lose, you’re out $40. If you win, you profit by $140. Thirty-one times you lose $40, for a loss of $1240. Twelve times you win $140, for a win of $1680. Your net over the 43 times is $440, a bit more than $10 a hand. Thus, in this situation your expectation, sometimes called positive EV, is about $10.23. If he doesn’t bet the 10s and jacks, that’s eight times you don’t lose $40, lowering your loss to $920, with a net of $760. However, the number of times he bets lessens by eight also, so your EV per bet is $760 divided by 35, which is a nice EV of $21.71 for every call you make. Showing down 10s and jacks, and even queens, is much more typical of a player who bluffs every time he pairs.”
“Aha,” offered Aunt Sophie, “I see. Even though I lose most of the time I call, I still profit overall by calling every time a player who bluffs every time he misses his hand.”
“And that,” I continued, “is the very worst situation. If you have a king, you win the pot 13 times out of 43, and lose $40 at most 30 times. If he doesn’t bet his jacks and 10s, you lose 22 times. In the first case, you have a net profit of $620, or a positive EV of $14.42 per call. In the case of his not betting jacks and 10s, you net $940, for a very healthy $26.86 EV.”
“Mm-hm,” she mused. “This is good.”
“Now,” I went on, “let’s look at the other extreme. Your opponent never bluffs. Likely such a player also shows down anything worse than a 9. If you called a bet with anything worse than a 10, your EV would be minus $40 for each call. So, in effect, by not calling, you profit by $40 every time he bets.”
“Yah,” she mused, “but I’m interested in how to call against the player who bluffs the correct amount of times.”
“Yes,” I interpolated, “you’re interested in a proper application of game theory. You would use this only against someone you had never played against before or someone you could not read. If you have no idea whether an opponent bluffs too much or too little, a good assumption is that he bluffs the proper amount. Okay, a person who bluffs the correct amount of time must bluff in such a way that if you call perfectly he has no net gain. Obviously if you call too much, he should rarely bluff, and if you don’t call enough, he should bluff frequently. If you call correctly, he should bluff such that the proportion of his legitimate bets to his bluffs is the same as the pot odds offered his opponent by calling him for a bluff. What does that mean?”
I paused for a moment to allow the waitress to top off our coffees, and bring Aunt Sophie a second bowl of tiramisu.
“Let’s,” I picked up, “go back to the earlier example. Say he bets whenever he makes an 8 or better, and half the time that he makes a 9, that is, drawing to 4-5-6-7 with you holding five cards he needs, he makes a betting hand 14 out of 43 times. You’re contemplating a call with a pair of aces. Two aces remain in the portion of the deck from which he drew, three 2s, three 3s, four 8s, and two 9s. Actually, four 9s remain, but he bets on only two of them, so this is the equivalent of two remaining.”
“And why,” Aunt Sophie queried, “would he bet half his 9s?”
“Ah,” I countered, “for two reasons. One is that he wouldn’t want to bet every 9 because as soon as you figured that out, you would pass almost all the 9s and an 8 once in awhile. The other, better, reason, though, is to simplify the math in this discussion.”
“Aha,” she acknowledged.
“Remember,” I inserted, “when he bets, the pot offers you $140 for your $40 call, which is 140-to-40 or 7-to-2. A convenient method a lot of games theory players use to decide when to bluff is to let the deck decide for them. They form a ratio between the number of cards that make their hand and the number of cards with which they bluff. Fourteen cards make him a legitimate betting hand, so he should bluff in a ratio of 14 to some figure that works out to the same ratio as 7-to-2; 14-to-4 is that ratio, so he should bluff with four of the cards he catches in this situation. He can just decide that if he misses his hand, most of the time he will just show it down, but four out of the 43 cards that he catches — and he decides which four before he sees the card he draws — he bluffs. For example, if he has a red 7 in his hand, he could decide that if he catches the other red 7 or any 6, he bets. The undealt cards contain one red 7 and three 6s. This is all predicated on his not knowing your calling frequency, and thus assuming that you will call the correct percentage of the time.”
“Uh, just a minute,” Aunt Sophie temporized. “He makes a betting hand with 14 cards, so 14 you put to 4, and 14-to-4 is equal to 7-to-2. Yes, I see.”
“Uh huh,” I nodded. “Out of 43 draws, 25 times he shows down and the best hand wins, and that’s usually his, but it doesn’t matter, because you can’t do anything about it, since there’s no bet. The other 18 times, though, you apparently have a decision. I say ‘apparently,’ because for any one decision, it doesn’t matter what you do, because if he bluffs correctly, no calling strategy can do better than any other.”
“Really?” she questioned. “You mean it doesn’t matter what I do?”
“Not exactly,” I supplied. “It matters inasmuch as it affects his strategy. Remember that I said you can usually do better than game theory; game theory is good only for those situations in which you can’t do better, and that’s against an unknown player or a player you can’t read.”
“Can you show this,” she requested, “in maybe figures?”
“Of course,” I said. “Let’s look at this in terms of your opponent’s expectation. His ratio of legitimate bets to bluffs in this situation is 7-to-2. That is, in nine times he bets, seven bets are on legitimate hands and two are bluffs. When you call, he either wins $140 or loses $40; when you fold, he wins $100.”
The waitress freshened my coffee and I had a small bite of my blueberry pie.
“Let’s say you call every hand,” I extemporized. “Seven times he wins $140, totalling $980. Two times he loses $40, totalling $80. Subtract and you get $900. That is, he wins $900 over the seven hands.
“Let’s say instead you fold every time,” I posited. Well, that’s easy, he just wins nine pots in a row, totalling $900.”
“But can’t I call somewhere in between?” she speculated.
“Try it,” I replied. “Say you call half the time. Seven of his bets are legitimate, and you call half of them, which gains him 3.5 times $140, or $490. You fold half of those hands, which gives him 3.5 times $100, or $350. Add them to get $840. The other two hands are bluffs. You call once, costing him $40, and fold once, winning him $100, for a net of $60. Add that to $840 and you again get $900. So you see, if he bets properly, no matter what you do, he profits by $900 in nine bets.”
“So then why,” she demanded, “does it matter what I do?”
“Ah,” I retorted. “If he’s good enough to employ game theory for his betting strategy, he’s also good enough to be observant. If he sees you calling more often than you ought, he bluffs less often; contrariwise, if he sees you calling less often than you ought, he bluffs more. The only way you can prevent an adjusment to his strategy is by calling properly. This is a good way to keep a player you can’t read or who seems to be able to read you from outplaying you.”
“Okay,” she acknowledged, “but the only thing that article didn’t say is what proper calling strategy is in this situation.”
“You must call in such a way,” I explained, “that his bluffing bets exactly break even. Since you don’t know when he’s bluffing, you have to call all his bets in the same ratio. And that ratio is the complement to his, and it depends on the odds you get by calling. In the example we have been dealing with, when he bets $40 into a $100 pot, you get odds of $140 to $40 by calling, or 7-to-2. Your opponent knows that if he can get away with a bluff more often than two times in nine, he profits; if he cannot get away with a bluff that often, he does not. His bluff breaks even if he gets away with it exactly two times in nine. You must call seven times out of nine to be calling exactly right. Let’s look at just his bluffs. Out of nine bluffing bets, he gets called seven. He loses $40 seven times and wins $140 twice. That’s exactly break-even.”
“So how do I call exactly seven times out of nine?” she asked.
“You can do it the same way he does,” I proposed. “You could decide ahead of time 9 bad cards and then call with 7 of them. For example, if you don’t know the constitution of the other player’s hand, you are drawing from a deck consisting of 48 cards. When you draw to that ace-joker wheel, approximately 24 of those cards make you a 10 or worse. And 7/9 of 24 is about 18 or 19; you don’t have to be precise here. You can just choose 18 cards with which you will call, say all the 10s, jacks, queens, and kings, plus any ace. You threw away one king, so to get up to 18, you also count all the aces. Another way to make your decision would be if you have a digital watch, right when it comes time to make your call, take a quick glance at it; 47 is about 7/9 of 60, so if the seconds on your watch read 46 or less, call. Just one thing about that, though. Don’t let your opponent see you look at your watch. If he knows you do that every time, he might be able to correlate your calls and folds to the time on his own watch.”
“Tsk,” she sighed. “Such a lot to remember.”
“Not really,” I concluded. “You don’t have to use this much. Against most players, your knowledge tells you how often to call, because they either bluff too much or too little. The same with bluffing into them; they either call too frequently or too seldom. Use game theory only against players you can’t read or who you think can read you too well.”