How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843)

by Ellenberg, Jordan

  But what the Pensées are most famous for is thought 233, which Pascal titled “Infinite-rien” (“Infinity-nothing”) but which is universally known as “Pascal’s wager.”

  As we’ve mentioned, Pascal held the question of God’s existence to be one that logic couldn’t touch: “‘God is, or He is not.’ But to which side shall we incline? Reason can decide nothing here.” But Pascal doesn’t stop there. What is the question of belief, he asks, if not a kind of gamble, a game with the highest possible stakes, a game you have no choice but to play? And the analysis of wagers, the distinction between the smart play and the foolish one, was a subject Pascal understood better than almost anyone on earth. He had not quite left his mathematical work behind him after all.

  How does Pascal compute the expected value of the game of faith? The key is already present in his mystic revelation:

  Everlasting joy in return for one day’s effort on earth.

  What is this, but a reckoning of the costs and benefits of adopting faith? Even in the middle of ecstatic communion with his savior, Pascal was still doing math! I love this about him.

  To compute Pascal’s expected value, we still need the probability that God exists; say for a moment we are pretty fervent doubters and assign this hypothesis a probability of only 5%. If we believe in God, and we turn out to be right, then our reward is “everlasting joy,” or, in the economists’ terms, infinitely many utils.* If we believe in God and we turn out to be wrong—an outcome we are 95% sure will be the case—then we pay a price; maybe more than the “one day’s effort” that Pascal suggests, since we have to count not only the time spent in worship but the opportunity cost of all the libertine pleasures we forwent in our quest for salvation. Still, it’s a certain fixed sum, let’s say a hundred utils.

  Then the expected value of belief is

  (5%) × infinity + (95%)(−100)

  Now, 5% is a small number. But infinite joy is a lot of joy; 5% of it is still infinite. So it swamps whatever finite cost imposed on us by adopting religion.

  We’ve already discussed the perils of trying to assign a numerical probability to a proposition like “God exists.” It is not clear any such assignment makes sense. But Pascal doesn’t make any such dodgy numerical move. He doesn’t need to. Because it doesn’t matter whether that number is 5% or something else. One percent of infinite bliss is still infinite bliss, and outweighs whatever finite costs attach to a life of piety. The same goes for 0.1% or 0.000001%. All that matters is that the probability God exists is not zero. Don’t you have to concede that point? That the existence of the Deity is at least possible? If so, then the expected value computation seems unequivocal: it is worth it to believe. The expected value of that choice is not only positive, but infinitely positive.

  Pascal’s argument has serious flaws. The gravest is that it suffers from the Cat in the Hat problem we saw in chapter 10, failing to consider all possible hypotheses. In Pascal’s setup, there are only two options: that the God of Christianity is real and will reward that particular sector of the faithful, or that God doesn’t exist. But what if there’s a God who damns Christians eternally? Such a God is surely possible too, and this possibility alone suffices to kill the argument: now, by adopting Christianity, we are wagering on a chance of infinite joy but also taking on the risk of infinite torment, with no principled way to weight the relative odds of the two options. We’re back to our starting point, where reason can decide nothing.

  Voltaire raised a different objection. You might have expected him to be sympathetic to Pascal’s wager—as we’ve already seen, he had no objection to gambling. And he admired mathematics; his attitude toward Newton approached worship (he once called him “the god to whom I sacrifice”) and he was romantically entangled for many years with the mathematician Émilie du Châtelet. But Pascal was not quite Voltaire’s sort of thinker. The two men stood at odds across a gulf as much temperamental as philosophical. Voltaire’s generally chipper outlook had no room for Pascal’s dark, introspective, mystical emissions. Voltaire dubbed Pascal “the sublime misanthrope” and devoted a long essay to knocking down the gloomy Pensées piece by piece. His attitude toward Pascal is that of the popular smart kid toward the bitter and nonconforming nerd.

  As for the wager, Voltaire said it was “a little indecent and puerile: the idea of a game, and of loss and gain, does not befit the gravity of the subject.” More substantively: “The interest I have to believe a thing is no proof that such a thing exists.” Voltaire himself, typically sunny, leans toward an informal argument by design: look at the world, look how amazing it is, God is real, QED!

  Voltaire has missed the point. Pascal’s wager is curiously modern, so much so that Voltaire has not caught up to it. Voltaire is right that, unlike Witztum and the Bible coders, or Arbuthnot, or the contemporary advocates of intelligent design, Pascal is not offering evidence for God’s existence at all. He is indeed proposing a reason to believe, but the reason has to do with the utility of believing, not the justifiability of believing. In a way, he anticipates the austere stance of Neyman and Pearson we saw in chapter 9. Just like them, he was skeptical that the evidence we encounter will provide a reliable means of determining what is true. Nonetheless, we have no choice but to decide what to do. Pascal is not trying to convince you God exists; he is trying to convince you that it would be to your benefit to believe so, and thus that your best course of action is to hang out with Christians and obey the forms of piety, until, just by force of propinquity, you start to truly believe. Can I put Pascal’s argument in modern terms better than David Foster Wallace did in Infinite Jest? I cannot.

  The desperate, newly sober White Flaggers are always encouraged to invoke and pay empty lip-service to slogans they don’t yet understand or believe—e.g. “Easy Does It!” and “Turn It Over!” and “One Day at a Time!” It’s called “Fake It Till You Make It,” itself an oft-invoked slogan. Everyone on a Commitment who gets up publicly to speak starts out saying he’s an alcoholic, says it whether he believes he is yet or not; then everybody up there says how Grateful he is to be sober today and how great it is to be Active and out on a Commitment with his Group, even if he’s not grateful or pleased about it at all. You’re encouraged to keep saying stuff like this until you start to believe it, just like if you ask somebody with serious sober time how long you’ll have to keep schlepping to all these goddamn meetings he’ll smile that infuriating smile and tell you just until you start to want to go to all these goddamn meetings.

  ST. PETERSBURG AND ELLSBERG

  Utils are useful when making decisions about items that don’t have well-defined dollar values, like wasted time or unpleasant meals. But you also need to talk about utility when dealing with items that do have well-defined dollar values—like dollars.

  This realization arrived very early in the development of probability theory. Like many important ideas, it entered the conversation in the form of a puzzle. Daniel Bernoulli famously described the conundrum in his 1738 paper “Exposition on a New Theory of the Measurement of Risk”: “Peter tosses a coin and continues to do so until it should land ‘heads’ when it comes to the ground. He agrees to give Paul one ducat if he gets ‘heads’ on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled.”

  This is obviously a rather attractive scenario for Paul, a game he should be willing to ante up some entrance fee to play. But how much? The natural answer, given our experience with lotteries, is to compute the expected value of the amount of money Paul gets from Peter. There’s a 50/50 chance that the first throw of the coin lands heads, in which case Paul gets one ducat. If the first throw is tails and the second is heads, an event which happens 1/4 of the time, Paul gets two ducats. To get four, the first three throws have to fall tails, tails, heads, which happens with probability 1/8. Carrying on and addin
g up, Paul’s expected profit is

  (1/2) × 1 + (1/4) × 2 + (1/8) × 4 + (1/16) × 8 + (1/32) × 16 + . . .

  or

  1/2 + 1/2 + 1/2 + 1/2 + . . .

  That sum is not a number. It’s divergent; the more terms you add, the bigger the sum gets, growing without bound past any finite threshold.* This seems to suggest that Paul should be willing to spend any number of ducats for the right to play this game.

  That sounds nuts. And it is! But when the math tells us something that sounds nuts, mathematicians don’t just shrug and walk away. We go hunting for the kink in the tracks where either the math or our intuition has gone off the rails. The condundrum, known as the St. Petersburg paradox, had been devised by Nicolas Bernoulli, Daniel’s cousin, some thirty years before, and many of the probabilists of the time had puzzled over it without coming to any satistfying conclusion. The younger Bernoulli’s beautiful untwisting of the paradox is a landmark result, and one that has formed the foundation of economic thinking about uncertain values ever since. The mistake, Bernoulli said, is to say that a ducat is a ducat is a ducat. A ducat in the hand of a rich man is not worth the same as a ducat in the hand of a peasant, as is plainly visible from the different levels of care with which the two men treat their cash. In particular, having two thousand ducats isn’t twice as good as having one thousand; it is less than twice as good, because a thousand ducats is worth less to a person who already has a thousand ducats than it is to the person who has none. Twice as many ducats doesn’t translate into twice as many utils; not all curves are lines, and the relation between money and utility is governed by one of those nonlinear curves.

  Bernoulli thought that utility grew like the logarithm, so that the kth prize of 2k ducats was worth just k utils. Remember, we can think of the logarithm as more or less the number of digits: so in dollar terms, Bernoulli’s theory is saying that rich people measure the value of their pile by the number of digits after the dollar sign—a billionaire is as much richer than a hundred-millionaire as the hundred-millionaire is richer than the ten-millionaire.

  In Bernoulli’s formulation, the expected utility of the St. Petersburg game is the sum

  (1/2) × 1 + (1/4) × 2 + (1/8) × 3 + (1/16) × 4 + . . .

  This tames the paradox; this sum, it turns out, is no longer infinite, or even very large. In fact, there’s a beautiful trick that allows us to compute it exactly:

  The sum of the first row, (1/2) + (1/4) + (1/8) + . . . , is 1; this is the very infinite series that Zeno encountered in chapter 2. The second row is the same as the first, but with every entry divided by 2; so its sum must be half the sum of the first row, or 1/2. By the same reasoning, the third row, which is just the second row with each term halved, must have half the sum of the second row; so 1/4. Now the sum of all the numbers in the triangle is 1 + 1/2 + 1/4 + 1/8 + . . . ; just one more than Zeno’s sum, which is to say, 2.

  But what if we sum down the columns first instead of the rows? Just as with the holes in my parents’ stereo set, it can’t matter whether we start counting vertically or horizontally; the sum is what the sum is.* In the first column there is just a single 1/2; in the second, there are two copies of 1/4, making (1/4) × 2; in the third, three copies of 1/8, making (1/8) × 3, and so on. The series formed by the column sums is none other than the sum Bernoulli set up to study the St. Petersburg problem. And its sum is the sum of all the numbers in the infinite triangle, which is to say: 2. So the amount Paul should pay is the number of ducats his personal utility curve tells him 2 utils is worth.*

  The shape of the utility curve, beyond the bare fact that it tends to bend downward as the money increases, is impossible to pin down precisely,* though contemporary economists and psychologists are constantly devising ever-more-intricate experiments to refine our understanding of its properties. (“Now just get your head settled comfortably at the center of the fMRI, if you don’t mind, and I’m going to ask you to rank the following six poker strategies in order from most enticing to least enticing, and after that, if you wouldn’t mind just holding still while my postdoc takes this cheek swab . . . ?”)

  We know, at least, that there is no universal curve; different people in different contexts assign different utilities to money. This fact is important. It gives us pause, or it ought to, when we start making generalizations about economic behavior. Greg Mankiw, the Harvard economist whom we last saw in chapter 1 faintly praising Reaganomics, wrote a widely circulated blog post in 2008 explaining that increased income taxes proposed by presidential candidate Barack Obama would lead him to slack off at work. After all, Mankiw was already at an equilibrium, where the utility of the dollars he’d earn from another hour of work would be exactly canceled by the negative utility imposed by the loss of an hour with his kids. Diminish the number of dollars Mankiw makes per hour, and that trade stops being worth it; he cuts back on work until he drops to the income level where an hour with his kids is worth the same to him as an hour spent working for his Obama-diminished pay. He agrees with Reagan’s view of the economy as seen from the standpoint of a cowboy-movie star; when the tax rate goes up, you make fewer cowboy movies.

  But not everybody is Greg Mankiw. In particular, not everybody has the same utility curve he has. The comic essayist Fran Lebowitz tells a story about her youth in Manhattan, driving a cab. She started driving at the beginning of the month, she said, and kept driving every day until she’d made enough money to pay for rent and food. Then she stopped driving and wrote for the rest of the month. For Lebowitz, all money above a certain threshold contributes essentially zero further utility; she has a different-looking curve than Mankiw does. Hers goes flat once her rent is paid. What happens to Fran Lebowitz if income taxes go up? She works more, not less, to bring herself back up to the threshold.*

  Bernoulli was not the only mathematician to arrive at the idea of utility and its nonlinear relation with money. He’d been anticipated by at least two other researchers. One was Gabriel Cramer of Geneva; the other was a young correspondent of Cramer’s, none other than the needle thrower Georges-Louis LeClerc, Comte de Buffon. Buffon’s interest in probability was not restricted to parlor games. Late in life, he reminisced about his encounter with the vexing St. Petersburg paradox: “I dreamed about this problem some time without finding the knot; I could not see that it was possible to make mathematical calculations agree with common sense without introducing some moral considerations; and having expressed my ideas to Mr. Cramer, he told me that I was right, and that he had also resolved this question by a similar approach.”

  Buffon’s conclusion mirrored Bernoulli’s, and he perceives the nonlinearity especially clearly:

  Money must not be estimated by its numerical quantity: if the metal, that is merely the sign of wealth, was wealth itself, that is, if the happiness or the benefits that result from wealth were proportional to the quantity of money, men would have reason to estimate it numerically and by its quantity, but it is barely necessary that the benefits that one derives from money are in just proportion with its quantity; a rich man of one hundred thousand ecus income is not ten times happier than the man of only ten thousand ecus; there is more than that what money is, as soon as one passes certain limits it has almost no real value, and cannot increase the well-being of its possessor; a man that discovered a mountain of gold would not be richer than the one that found only one cubic fathom.

  The doctrine of expected utility is appealingly straightforward and simple: presented with a set of choices, pick the one with the highest expected utility. It is perhaps the closest thing we have to a simple mathematical theory of individual decision making. And it captures many features of the way humans make choices, which is why it remains a central part of the quantitative social scientist’s tool kit. Pierre-Simon Laplace, on the last page of his 1814 treatise A Philosophical Essay on Probabilities, writes, “We see, in this Essay, that the theory of probabilities is, in the end, only common sense boiled down to
‘calculus’; it points out in a precise way what rational minds understand by means of a sort of instinct, without necessarily being aware of it. It leaves nothing to doubt, in the choice of opinions and decisions; by its use one can always determine the most advantageous choice.”

  Again we see it: mathematics is the extension of common sense by other means.

  But expected utility doesn’t get at everything. Once again, the troubling complications enter in the form of a puzzle. This time, the puzzle-bearer was Daniel Ellsberg, who later became famous as the whistle-blower who leaked the Pentagon Papers to the civilian press. (In mathematical circles, which can be parochial at times, it would not be outlandish to hear it said of Ellsberg, “You know, before he got involved in politics, he did some really important work.”)

  In 1961, a decade before his explosion into public view, Ellsberg was a brilliant young analyst at the RAND Corporation, consulting with the U.S. government on strategic matters surrounding nuclear war—how it could be prevented, or, barring that, effectively conducted. At the same time, he was working toward a Harvard PhD in economics. On both tracks, he was thinking deeply about the process by which human beings made decisions in the face of the unknown. At the time, the theory of expected utility held a supreme position in the mathematical analysis of decisions. Von Neumann and Morgenstern,* in their foundational book The Theory of Games and Economic Behavior, had proven that all people who obeyed a certain set of behavior rules, or axioms, had to act as if their choices were governed by the drive to maximize some utility function. These axioms—later refined by Leonard Jimmie Savage, a member of the wartime Statistical Research Group with Abraham Wald—were the standard model of behavior under uncertainty at the time.

  Game theory and expected utility theory still play a great role in the study of negotiations among people and states, but never more so than at RAND at the height of the Cold War, where the writings of von Neumann and Morgenstern were the subject of Pentateuchal levels of reverence and analysis. The researchers at RAND were studying something fundamental to human life: the process of choice and competition. And the games they studied, like Pascal’s wager, were played for very high stakes.