Eight

How to Outguess Card Games

Card games are exercises in keeping secrets. That’s why cards have plain backs and we hold them close to the chest. Above all, players struggle to keep their next move unpredictable. Do they succeed?

Barry O’Neill tried to answer that in a 1982 experiment at Northwestern University. He invented a simple game with two players, both having the same hand of four cards: ace, 2, 3, and joker. Each player was to choose a single card and put it facedown on the table. Then the cards were compared. Player A won if both cards were jokers or both were number cards that didn’t match (like a 2 and a 3). Player B won otherwise. The loser of each hand paid the winner five cents.

By design, this game wasn’t much like any familiar game. O’Neill wanted to see how well players would strategize a completely unfamiliar game. They did pretty well. The optimal strategy would be to randomly play the joker 40 percent of the time and each of the number cards 20 percent. The players put down the joker 39.4 percent of the time. That’s amazing when you consider that they were playing by gut instinct. They were not given an opportunity to calculate the best strategy and may not have known how.

There were two notable errors. One was that they overplayed aces, picking them 22.3 percent of the time. The other was that the players alternated too much. A joker should be followed by a number card 60 percent of the time. But actually it was more than that. Players did not want to play the joker twice in a row, or three times in a row. They were especially likely to switch after the card they played won.

Look beneath the surface, and O’Neill’s game says much about strategizing in poker, bridge, and other games. The most fundamental decision in poker is whether to bluff (to make a bet on a weak hand). When the bluffer is lucky, no one will be willing to match the bet, and he’ll win the pot without a showdown.

There is a mathematical formula for bluffing. The probability of your bluffing should be raise/pot. Here, raise is how much you would raise if you bluff. This may be constrained by table rules, social norms, and how much money is in your wallet. Pot is the amount that would be in the pot after your raise, and after your opponent(s) sees that raise.

Example: The pot is $100 right now. You’re thinking of aggressively raising a full $100. Only one other player is in the game. If you add $100 and so does the opponent, the pot will have $300. The formula says you should bluff with a probability of $100/$300, or 1 in 3. This guarantees that a player who sees your raise has a 1/3 chance of winning $300 (worth $100 on average). But since it costs $100 to see the raise, the opponent is only breaking even. He cannot expect to make a profit (a profit that would have to come from your pocket!).

Good players know all this, but even the best can be poor at inventing a suitably weighted random sequence of bluffs and folds. The usual problems of randomization are complicated by the fact that the bluff-or-fold decision arises only when a player has a weak hand. Also, the ability to outguess opponents is limited by the game’s incomplete information. When opponents win without a showdown, there is no way of telling whether they were bluffing or not.

In general, card players avoid streaks of the same choice. Poker novices are embarrassed at being caught in a bluff and rarely try it twice in a row. Better players occasionally throw in two or three bluffs in a row. But after two in a row, most players feel a pressure to fold on the next weak hand, especially if it comes immediately after an exposed bluff.

A player on a winning streak—without showdowns—may have been bluffing for some of those wins. When players take a risky move (bluff) and win, they are likely to try something different the next time. They don’t want to be “greedy” and “keep going back to the well.” When a player on a streak raises yet again, it’s more likely than usual that he’s not bluffing and has a strong hand.

It’s a good idea to get nonhuman help in randomizing your bluffing. I’ve mentioned the possibility of using a watch. Needless to say, looking at your watch only when you’ve got a bad hand would be a most fatal tell. You’ve got to glance at your watch (inconspicuously) with every hand.

Look to see where the second hand is, percentage-wise, on its sweep from twelve to twelve. Say you intend to bluff with 33 percent probability on this hand. If the second hand is in the first 33 percent of the cycle (between twelve and four), you bluff.

An alternative is to use the colors or values of your cards as a randomizing device. You might use the leftmost number card in your hand. This will be a value from 1 (ace) to 10. Multiply by 10 (getting a value from 10 to 100). You bluff only if the multiplied number is less than the intended bluff percentage. With a 33 percent bluff percentage, you would bluff on an ace, 2, or 3. This is less precise but much less likely to be exploited as a tell.

One classic poker tell is important to any outguesser at cards: pupil dilation. The pupil is the black, central part of the eye, within the colored iris. When a player draws a desired card, the pupils grow larger (dilate). This is no urban legend but a lab-tested effect that’s reliable enough to be used in mentalism. There’s a card trick where the performer shows an audience member the queen of hearts and says it represents money or sex (either works). The volunteer then draws cards from the shuffled deck one at a time. A good pupil reader can spot when the audience member draws the queen of hearts from the increase in pupil size.

Psychologist Eckhard Hess was one of the pioneers of pupil reading. Around 1960 he did an experiment with photographs, all landscapes except for one pinup picture of an attractive woman. Hess shuffled the photos and showed them, one at a time, to his male lab assistant. At the seventh photo, the assistant’s pupils suddenly enlarged. It was the pinup picture.

The pupils may speak truth even when the lips lie. With the 1964 presidential campaign in full swing, Hess showed University of Chicago students and faculty photos of Democratic president Lyndon Johnson and Republican challenger Barry Goldwater. Everyone said they favored liberal Johnson over hard-right Goldwater. But Hess found that about a third of the people had a more positive pupil response to Goldwater than to Johnson. Hess proposed the “interesting possibility… that in the liberal atmosphere of the university these people found it difficult to utter any pro-Goldwater sentiment.”

It’s not hard to read pupils, but you have to be looking directly at the person’s eyes when the change occurs. The change is fast enough to see, and you are looking for that change rather than the pupils’ absolute size (which varies with room lighting and drug use). A typical positive reaction is a 10 percent increase in the diameter of both pupils (a 20 percent increase in area). A bad turn of luck may cause the pupils to shrink. It looks like this, roughly actual size:

The change takes place in the half second or so after the player sees a fateful card. Not surprisingly, some serious players wear dark glasses as a countermeasure.

Recap: How to Outguess Card Games

• When card players make strategic random decisions, they avoid streaks of the same choice. A player who bluffs this time is less likely to bluff on the next weak hand.

• You might want to use a watch to randomize your bluffing, but be sure to glance at the watch with every hand.

• A good pupil reader can tell whether you drew the card you needed—if he can see your eyes.