The Minds of Others

The Minds of Others

I’m an optimist in the sense that I believe humans are noble and honorable, and some of them are really smart.… I have a somewhat more pessimistic view of people in groups.

—STEVE JOBS

An investor sells a stock to another, one convinced it’s headed down and the other convinced it’s going up; I think I know what you think but have no idea what you think I think; an economic bubble bursts; a prospective lover offers a gift that says neither “I want to be more than friends” nor “I don’t want to be more than friends”; a table of diners squabbles over who should treat whom and why; someone trying to be helpful unintentionally offends; someone trying hard to be cool draws snickers; someone trying to break from the herd finds, dismayingly, the herd following his lead. “I love you,” says one lover to another; “I love you, too,” the other replies; and both wonder what exactly the other means by that.

What does computer science have to say about all this?

Schoolchildren are taught to conceive of literary plots as belonging to one of several categories: man vs. nature, man vs. self, man vs. man, man vs. society. Thus far in this book we have considered primarily cases in the first two categories—that is to say, computer science has thus far been our guide to problems created by the fundamental structure of the world, and by our limited capacities for processing information. Optimal stopping problems spring from the irreversibility and irrevocability of time; the explore/exploit dilemma, from time’s limited supply. Relaxation and randomization emerge as vital and necessary strategies for dealing with the ineluctable complexity of challenges like trip planning and vaccinations.

In this chapter we shift the focus and consider the remaining two genres—that is, man vs. man and man vs. society: in effect, the problems that we pose and cause each other. Our best guide to this terrain comes from a branch of mathematics known as game theory, a field that in its classical incarnation had an enormous impact on the twentieth century. In the past couple of decades, cross-pollination between game theory and computer science has produced the field of algorithmic game theory—which has already begun to have an impact on the twenty-first.

Recursion

Now, a clever man would put the poison into his own goblet because he would know that only a great fool would reach for what he was given. I am not a great fool, so I can clearly not choose the wine in front of you. But you must have known I was not a great fool—you would have counted on it—so I can clearly not choose the wine in front of me.

—THE PRINCESS BRIDE

Arguably the most influential economist of the twentieth century, John Maynard Keynes, once said that “successful investing is anticipating the anticipations of others.” For a share of stock to be sold at, say, $60, the buyer must believe he can sell it later for $70—to someone who believes he can sell it for $80 to someone who believes he can sell it for $90 to someone who believes he can sell it for $100 to someone else. In this way, the value of a stock isn’t what people think it’s worth but what people think people think it’s worth. In fact, even that’s not going far enough. As Keynes put it, making a crucial distinction between beauty and popularity:

Professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one’s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe who practice the fourth, fifth, and higher degrees.

Computer science illustrates the fundamental limitations of this kind of reasoning with what’s called the “halting problem.” As Alan Turing proved in 1936, a computer program can never tell you for sure whether another program might end up calculating forever without end—except by simulating the operation of that program and thus potentially going off the deep end itself. (Accordingly, programmers will never have automated tools that can tell them whether their software will freeze.) This is one of the foundational results in all of computer science, on which many other proofs hang.* Simply put, any time a system—be it a machine or a mind—simulates the workings of something as complex as itself, it finds its resources totally maxed out, more or less by definition. Computer scientists have a term for this potentially endless journey into the hall of mirrors, minds simulating minds simulating minds: “recursion.”

“In poker, you never play your hand,” James Bond says in Casino Royale; “you play the man across from you.” In fact, what you really play is a theoretically infinite recursion. There’s your own hand and the hand you believe your opponent to have; then the hand you believe your opponent believes you have, and the hand you believe your opponent believes you to believe he has … and on it goes. “I don’t know if this is an actual game-theory term,” says the world’s top-rated poker player, Dan Smith, “but poker players call it ‘leveling.’ Level one is ‘I know.’ Two is ‘you know that I know.’ Three, ‘I know that you know that I know.’ There are situations where it just comes up where you are like, ‘Wow, this is a really silly spot to bluff but if he knows that it is a silly spot to bluff then he won’t call me and that’s where it’s the clever spot to bluff.’ Those things happen.”

One of the most memorable bluffs in high-level poker occurred when Tom Dwan wagered $479,500 on Texas Hold ’Em’s absolute worst possible hand, the 2–7—while literally telling his opponent, Sammy George, that he was holding it. “You don’t have deuce-seven,” George replied. “You don’t have deuce-seven.” George folded, and Dwan—with, yes, deuce-seven—took the pot.

In poker, recursion is a dangerous game. You don’t want to get caught one step behind your opponent, of course—but there’s also an imperative not to get too far ahead of them either. “There’s a rule that you really only want to play one level above your opponent,” explains poker professional Vanessa Rousso. “If you play too far above your opponent, you’re going to think they have information that they don’t actually have—[and] they won’t be able to glean the information that you want them to glean from your actions.” Sometimes poker pros will deliberately bait their opponent into a convoluted recursion, meanwhile playing completely by-the-book, unpsychological poker themselves. This is known as luring them into “a leveling war against themselves.”

(Luring an opponent into fruitless recursion can be an effective strategy in other games, too. One of the most colorful, bizarre, and fascinating episodes in the history of man-vs.-machine chess came in a 2008 blitz showdown between American grandmaster Hikaru Nakamura and leading computer chess program Rybka. In a game where each side got just three minutes on the clock to play all of their moves or automatically lose, the advantage surely seemed to be on the side of the computer—capable of evaluating millions of positions every second, and of making its move without twitching a muscle. But Nakamura immediately gridlocked the board, and proceeded to make repetitive, meaningless moves as fast as he could click. Meanwhile, the computer wasted precious moments fruitlessly searching for winning variations that didn’t exist and doggedly trying to anticipate all the possible future moves by Nakamura, who himself was simply doing the chess equivalent of twiddling his thumbs. When the computer had nearly exhausted its time and began flailing so as not to lose by the clock, Nakamura finally opened the position and crashed through.)

Given recursion’s dangers, how do poker professionals break out of it? They use game theory. “Sometimes you can come up with reasons to make exploitive [leveling] plays, but a lot of the time you are just making inferior plays for reasons that are really just noise,” Dan Smith explains. “I try really hard to have a base level of theory understanding in most situations.… I always start by knowing or trying to know what Nash is.”

So what is Nash?