Aunt Sophie Wonders About Game Theory

"Nu, tsatskeleh," said Aunt Sophie, "you remember before about game theory you told me."

I piloted my new Volvo C70 coupe through heavy traffic on Interstate 5, taking it easy. I would much rather have been on my more maneuverable Harley, but Aunt Sophie, adventurous though she was, would not have wanted me to convey her in that fashion to an evening of lowball. We'd get there when we got there, I knew, and conversation with my favorite aunt could make any drive pleasant. The Anaheim Club wasn't going anywhere.

"Yes," I responded. "Didn't you follow what I said?"

"Of course," she returned. "I'm no dummy."

I deftly avoided hitting an Excursion that cut in front of me and slammed on its brakes to avoid rear-ending the decelerating ML55 in front of it, given that there had been only three car lengths - car lengths, not bloated SUV lengths - between me and the Mercedes to begin with.

"Then, what is the problem?" I wondered.

"Problem?" she demanded. "No problem. It's just a variable on the bluffing I'd like to do, but I wanted to run it over you first."

I winced at the malapropisms, but knew attempts at education in that department were ill-advised and had little chance of success. Better, as she would say, I should concentrate on the education of a poker player.

"So, run me over," I urged.

"Against some players," she began, "when I bet I should use game theory for bluffing. I should bet, you said, so that the ratio of the good hands to the bluffs is the same as what the caller gets."

"Nicely put," I offered.

"Thank you," she replied. "So, an example you gave. In a typical pot, I open for a raise and the big blind calls. The big blind draws one card, and so do I. He checks and I'm going to bet anything from a 10 on down. Say I'm drawing to 4-3-2-A. I don't know what cards he has, so how they affect my hand, that I don't know, either. Anyway, I bet with any 5 through 10."

The Excursion started to jump over into a perceived two-car hole in the lane to our right, saw at the last second that a TT had already moved into the space, and quickly swerved back in front of me. I had seen the TT and known what would happen, and had left plenty of room for the Excursion to get back in. I was just glad that the behemoth had not hit the diminutive German roadster. The resultant collision would have been harder to avoid, although there had been an opening of a few car lengths in the carpool lane, and I could have jumped the double yellow to get there. I should have been in that lane already were our exit not coming up soon.

"So far, so good," I said.

"That's 24 cards," Aunt Sophie remarked. "OK, four and a half small bets has the pot. My bet after the draw is the same like two small bets, so, if I bet, he gets a little better than 3-to-1 to call. This means the ratio of good hands to bluffs I make should be about 3-to-1."

"I'm glad," I inserted, "you say 'about,' because some of the time you bet an 8, 9, or 10, and he has passed a better hand."

"Yes," she agreed, "so the ratio I don't use of three and a half to one I don't use to take care of those times, but just instead only 3-to-1, because mostly this player would bet his eights and nines and sometimes tens."

"Did you figure that out on your own?" I asked in awe. "That 'fudge factor,' I mean."

"Yes," she proudly announced. "Didn't I say I was no dummy?"

"Yes," I temporized, "I knew that. Go on."

"OK," she went on, "so to get 3-to-1 from 24 cards, that means I pick eight bluffing cards that also I bet with the good cards. Now, like you said, that would be the six worst cards, so I would bet if it came a 4 or a 3, plus two deuces, that's eight cards. It could be if I had a red deuce, I bet if I catch either of the black deuces."

"Very good," I commended. "What's the problem?"

"So this," she mused, "I been thinking. He's gotta call two-thirds of the time, the other side of the fraction."

"Well," I suggested, "he doesn't really have to call that fraction of the time. It actually doesn't matter what he does, because he can do nothing to counter proper betting."

"Yah, yah," she shot back. "This I know, but if I see he isn't calling properly, then my betting frequency I can adjust. If he never after passing calls, then I always bet; if he calls too much, then I never bluff."

That Excursion was determined to make my driving life difficult. Our exit was a few miles ahead, and I wanted to ease over to the right. But whenever even the smallest break in traffic seemed to be sliding in the next lane back toward us, the Excursion started to creep toward it. I might not be able to get over before that whale found his own hole to jam into.

"And how," I put in, "would you know if he was calling too much?"

"Aha!" she exclaimed. "Two-thirds of the time he's supposed to call. That would be any card that doesn't pair him. If he usually bets eights and nines, that leaves in the deck 40 cards he could maybe call with. Five cards from his hand plus eight eights and nines, take away from 53, that leaves 40. Approximately 26 cards he can call. Any 10 through king he should call, that's 16 cards, plus another nine or 10 cards. If his bottom cards are small and he pairs, that's any of nine more cards. So, if I see he's calling with big pairs, he's calling too much and I bluff less, maybe not at all."

"Right," I assented.

"So," Aunt Sophie resumed, "I notice that a player who correctly calls, with pairs a lot of the time he calls. So maybe, I'm wondering, instead of bluffing only with my worst hands, sometimes I should be bluffing with the best bluffing hands. That is, if I bet the kings and queens and he calls a lot of the time with pairs, I win even though I was bluffing. What is wrong with that?"

"Hmm," I answered, "that's a good question, but I think I can give you a good answer. Most of the time you're not drawing to a wheel. You should be betting your worst hands as a bluff in the proper ratio, because these are hands that probably cannot win in a showdown. If you bet kings and queens, then to be betting in the proper ratio, you have to give up on some of the larger pairs. That is, you bet, as you correctly identified, all the tens and better, plus all the cards that pair your top two cards, plus two more cards that pair your next higher card. But if you bet all of those plus a number of other cards, you're bluffing too often. Proper strategy against you would then be to call almost every time."

"But I don't mean," she extrapolated, "to bet all of the top pairs. I mean to bet randomly. Sometimes you bet big pairs, sometimes kings and queens you bet. Then when he calls with pairs, some of the times you're bluffing you win anyway."

"But that's not good," I explained. "As I said, most of the time you're not drawing to a wheel. You're drawing to a 6, 7, or 8. If you pair tops and don't bet, you automatically lose all of those pots, including a certain number of times that you would have won by betting. If you pair eights and bet, your opponent can hardly call with a pair of sevens. But if you just show down, he gets a gift, a pot he would not have won had you bet. Sure, two-thirds of the time you bet that hand, you lose, but you still profit the times he doesn't call, whereas, had you bet, you would have won one-third of the time. By not betting your worst hands, you give up those opportunities, and he'll still win approximately the same fraction of the time on the other bets. In fact, if he adjusts properly, he'll still do better than if you bet those worst hands of yours. He would just not call as often when he saw that your average betting hand when you were bluffing was somewhere in the range of a king, rather than a pair. Betting kings, queens, and jacks against someone who calls with almost all pairs is not bluffing; it's betting for value. When someone sees you doing that, his proper response is to check and call with all eights, nines, and tens. You don't want that, because it means value betting suddenly becomes less profitable. No, the proper employment of game theory for bluffing when someone has passed to you in lowball is to bet your best hands plus the proper fraction of your worst hands. You show down the rest of the hands and hope for the best. Why waste a bluff on a hand like a king that has a reasonable chance of winning in a showdown?"

The Excursion driver appeared to be momentarily distracted. He stuck a cell phone to his ear and didn't see the hole that opened to his right, which I quickly yet cautiously slid into. Negotiating two more lanes was easy, and I was over to the right in plenty of time to take our exit to the Anaheim Club. I drove to the front door, left the car with valet parking, and escorted my aunt into the cardroom, where I hoped she would properly use her newfound understanding of the theories of John Nash as applied to the relatively simple - at least in mathematical terms - paradigm of poker.