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

by Ellenberg, Jordan

  Just over six hundred people in all were executed in those years. Stevens offers a figure of 1% for the prevalence of mental retardation in the U.S. population. So if mentally retarded prisoners were executed at exactly the same rate as the general population, you’d expect about six or seven members of that population to have been put to death. Viewed in this light, as Scalia points out, the evidence shows no particular disinclination toward executing the mentally retarded. No Greek Orthodox bishop has ever been executed in Texas, but can you doubt Texas would kill a bishop if the necessity arose?

  Scalia’s real concern in Atkins is not so much the precise question before the court, which both sides agree affects a tiny segment of capital cases. Rather, he is worried about what he calls the “incremental abolition” of capital punishment by judicial decree. He quotes his own earlier opinion in Harmelin v. Michigan: “The Eighth Amendment is not a ratchet, whereby a temporary consensus on leniency for a particular crime fixes a permanent constitutional maximum, disabling the States from giving effect to altered beliefs and responding to changed social conditions.”

  Scalia is right to be troubled by a system in which the whims of one generation of Americans end up constitutionally binding our descendants. But it’s clear his objection is more than legal; his concern is an America that loses the habit of punishment through enforced disuse, an America that is not only legally barred from killing mentally retarded murderers but that, by virtue of the court’s lenient ratchet, has forgotten that it wants to. Scalia—much like Samuel Livermore two hundred years earlier—foresees and deplores a world in which the populace loses by inches its ability to impose effective punishments on wrongdoers. I can’t manage to share their worry. The immense ingenuity of the human species in devising ways to punish people rivals our abilities in art, philosophy, and science. Punishment is a renewable resource; there is no danger we’ll run out.

  FLORIDA 2000, THE SLIME MOLD, AND HOW TO CHOOSE A WINGMAN

  The slime mold Physarum polycephalum is a charming little organism. It spends much of its life as a tiny single cell, roughly related to the amoeba. But, under the right condition, thousands of these organisms coalesce into a unified collective called a plasmodium; in this form, the slime mold is bright yellow and big enough to be visible to the naked eye. In the wild it lives on rotting plants. In the laboratory it really likes oats.

  You wouldn’t think there’d be much to say about the psychology of the plasmodial slime mold, which has no brain or anything that could be called a nervous system, let alone feelings or thoughts. But a slime mold, like every living creature, makes decisions. And the interesting thing about the slime mold is that it makes pretty good decisions. In the slime mold’s limited world, these decisions more or less come down to “move toward things I like” (oats) and “move away from things I don’t like” (bright light). Somehow, the slime mold’s decentralized thought process is able to get this job done very effectively. As in, you can train a slime mold to run through a maze. (This takes a long time and a lot of oats.) Biologists hope that by understanding how the slime mold navigates its world, they can open a window into the evolutionary dawn of cognition.

  And even here, in the most primitive kind of decision-making imaginable, we encounter some puzzling phenomena. Tanya Latty and Madeleine Beekman of the University of Sydney were studying the way slime molds handled tough choices. A tough choice for a slime mold looks something like this: On one side of the petri dish is three grams of oats. On the other side is five grams of oats, but with an ultraviolet light trained on it. You put a slime mold in the center of the dish. What does it do?

  Under those conditions, they found, the slime mold chooses each option about half the time; the extra food just about balances out the unpleasantness of the UV light. If you were a classical economist of the kind Daniel Ellsberg worked with at RAND, you’d say that the smaller pile of oats in the dark and the bigger pile under the light have the same amount of utility for the slime mold, which is therefore ambivalent between them.

  Replace the five grams with ten grams, though, and the balance is broken; the slime mold goes for the new double-size pile every time, light or no light. Experiments like this teach us about the slime mold’s priorities and how it makes decisions when those priorities conflict. And they make the slime mold look like a pretty reasonable character.

  But then something strange happened. The experimenters tried putting the slime mold in a petri dish with three options: the three grams of oats in the dark (3-dark), the five grams of oats in the light (5-light), and a single gram of oats in the dark (1-dark). You might predict that the slime mold would almost never go for 1-dark; the 3-dark pile has more oats in it and is just as dark, so it’s clearly superior. And indeed, the slime mold just about never picks 1-dark.

  You might also guess that, since the slime mold found 3-dark and 5-light equally attractive before, it would continue to do so in the new context. In the economist’s terms, the presence of the new option shouldn’t change the fact that 3-dark and 5-light have equal utility. But no: when 1-dark is available, the slime mold actually changes its preferences, choosing 3-dark more than three times as often as it does 5-light!

  What’s going on?

  Here’s a hint: the small, dark pile of oats is playing the role of H. Ross Perot in this scenario.

  The mathematical buzzword in play here is “independence of irrelevant alternatives.” That’s a rule that says, whether you’re a slime mold, a human being, or a democratic nation, if you have a choice between two options A and B, the presence of a third option C shouldn’t affect which of A and B you like better. If you’re deciding whether you’d rather have a Prius or a Hummer, it doesn’t matter whether you also have the option of a Ford Pinto. You know you’re not going to choose the Pinto. So what relevance could it have?

  Or, to keep it closer to politics: in place of an auto dealership, put the state of Florida. In place of the Prius, put Al Gore. In place of the Hummer, put George W. Bush. And in place of the Ford Pinto, put Ralph Nader. In the 2000 presidential election, George Bush got 48.85% of Florida’s votes and Al Gore got 48.84%. The Pinto got 1.6%.

  So here’s the thing about Florida in 2000. Ralph Nader was not going to win Florida’s electoral votes. You know that, I know that, and every voter in the state of Florida knew that. What the voters of the state of Florida were being asked was not actually

  “Should Gore, Bush, or Nader get Florida’s electoral votes?”

  but

  “Should Gore or Bush get Florida’s electoral votes?”

  It’s safe to say that virtually every Nader voter thought Al Gore would be a better president than George Bush.* Which is to say that a solid 51% majority of Florida voters preferred Gore over Bush. And yet the presence of Ralph Nader, the irrelevant alternative, means that Bush takes the election.

  I’m not saying the election should have been decided differently. But what’s true is that votes produce paradoxical outcomes, in which majorities don’t always get their way and irrelevant alternatives control the outcome. Bill Clinton was the beneficiary in 1992, George W. Bush in 2000, but the mathematical principle is the same: it’s hard to make sense of “what the voters really want.”

  But the way we settle elections in America isn’t the only way. That might seem weird at first—what choice, other than the candidate who got the most votes, could possibly be fair?

  Here’s how a mathematician would think about this problem. In fact, here’s the way one mathematician—Jean-Charles de Borda, an eighteenth-century Frenchman distinguished for his work in ballistics—did think about the problem. An election is a machine. I like to think of it as a big cast-iron meat grinder. What goes into the machine is the preferences of the individual voters. The sausagey goop that comes out, once you turn the crank, is what we call the popular will.

  What bothers us about Al Gore’s loss in Florida? It’s that more people p
referred Gore to Bush than the reverse. Why doesn’t our voting system know that? Because the people who voted for Nader had no way to express their preference for Gore over Bush. We’re leaving some relevant data out of our computation.

  A mathematician would say, “You shouldn’t leave out information that might be relevant to the problem you’re trying to solve!”

  A sausage maker would put it, “If you’re grinding meat, use the whole cow!”

  And both would agree that you ought to find a way to take into account people’s full set of preferences—not just which candidate they like the most. Suppose the Florida ballot had allowed voters to list all three candidates in their preferred order. The results might have looked something like this:

  Bush, Gore, Nader

  49%

  Gore, Nader, Bush

  25%

  Gore, Bush, Nader

  24%

  Nader, Gore, Bush*

  2%

  The first group represents Republicans and the second group liberal Democrats. The third group is conservative Democrats for whom Nader is a little too much. The fourth group is, you know, people who voted for Nader.

  How to make use of this extra information? Borda suggested a simple and elegant rule. You can give each candidate points according to their placement: if there are three candidates, give 2 for a first-place vote, 1 for second, 0 for third. In this scenario, Bush gets 2 points from 49% of the voters and 1 point from 24% more, for a score of

  2 × 0.49 + 1 × 0.24 = 1.22.

  Gore gets 2 points from 49% of the voters and 1 point from another 51%, or a score of 1.49. And Nader gets 2 points from the 2% who like him best, and another point from the liberal 25%, coming in last at 0.29.

  So Gore comes in first, Bush second, Nader third. And that jibes with the fact that 51% of the voters prefer Gore to Bush, 98% prefer Gore to Nader, and 73% prefer Bush to Nader. All three majorities get their way!

  But what if the numbers were slightly shifted? Say you move 2% of the voters from “Gore, Nader, Bush” to “Bush, Gore, Nader.” Then the tally looks like this:

  Bush, Gore, Nader

  51%

  Gore, Nader, Bush

  23%

  Gore, Bush, Nader

  24%

  Nader, Gore, Bush

  2%

  Now a majority of Floridians like Bush better than Gore. In fact, an absolute majority of Floridians have Bush as their first choice. But Gore still wins the Borda count by a long way, 1.47 to 1.26. What puts Gore over the top? It’s the presence of Ralph “Irrelevant Alternative” Nader, the same guy who spoiled Gore’s bid in the actual 2000 election. Nader’s presence on the ballot pushes Bush down to third place on many ballots, costing him points; while Gore enjoys the privilege of never being picked last, because the people who hate him hate Nader even more.

  Which brings us back to the slime mold. Remember, the slime mold doesn’t have a brain to coordinate its decision making, just thousands of nuclei enclosed in the plasmodium, each pushing the collective in one direction or another. Somehow the slime mold has to aggregate the information available to it into a decision.

  If the slime mold were deciding purely on food quantity, it would rank 5-light first, 3-dark second, and 1-dark third. If it used only darkness, it would rank 3-dark and 1-dark tied for first, with 5-light third.

  Those rankings are incompatible. So how does the slime mold decide to prefer 3-dark? What Latty and Beekman speculate is that the slime mold uses some form of democracy to choose between these two options, via something like the Borda count. Let’s say 50% of the slime mold nuclei care about food and 50% care about light. Then the Borda count looks like this:

  5-light, 3-dark, 1-dark

  50%

  1-dark and 3-dark tied, 5-light

  50%

  5-light gets 2 points from the half of the slime mold that cares about food, and 0 from the half of the slime mold that cares about light, for a point total of

  2 × (0.5) + 0 × (0.5) = 1.

  In a tie for first, we give both contestants 1.5 points; so 3-dark gets 1.5 points from half the slime mold and 1 from the other half, ending up with 1.25. And the inferior option 1-dark gets nothing from the food-loving half of the slime mold, which ranks it last, and 1.5 from the light-hating half, which has it tied for first, for a total of 0.75. So 3-dark comes in first, 5-light second, and 1-dark last, in exact conformity with the experimental result.

  What if the 1-dark option weren’t there? Then half the slime mold would rate 5-light above 3-dark, and the other half would rate 3-dark above 5-light; you get a tie, which is exactly what happened in the first experiment, where the slime mold chose between the dark three-gram pile of oats and the bright five-gram pile.

  In other words: the slime mold likes the small, unlit pile of oats about as much as it likes the big, brightly lit one. But if you introduce a really small unlit pile of oats, the small dark pile looks better by comparison; so much so that the slime mold decides to choose it over the big bright pile almost all the time.

  This phenomenon is called the “asymmetric domination effect,” and slime molds are not the only creatures subject to it. Biologists have found jays, honeybees, and hummingbirds acting in the same seemingly irrational way.

  Not to mention humans! Here we need to replace oats with romantic partners. Psychologists Constantine Sedikides, Dan Ariely, and Nils Olsen offered undergraduate research subjects the following task:

  You will be presented with several hypothetical persons. Think of these persons as prospective dating partners. You will be asked to choose the one person you would ask out for a date. Please assume that all prospective dating partners are: (1) University of North Carolina (or Duke University) students, (2) of the same ethnicity or race as you are, and (3) of approximately the same age as you are. The prospective dating partners will be described in terms of several attributes. A percentage point will accompany each attribute. This percentage point reflects the relative position of the prospective dating partner on that trait or characteristic, compared to UNC (DU) students who are of the same gender, race, and age as the prospective partner is.

  Adam is in the 81st percentile of attractiveness, the 51st percentile of dependability, and the 65th percentile of intelligence, while Bill is the 61st percentile of attractiveness, 51st of dependability, and 87th of intelligence. The college students, like the slime mold before them, faced a tough choice. And just like the slime mold, they went 50-50, half the group preferring each potential date.

  But things changed when Chris came into the picture. He was in the 81st percentile of attractiveness and 51st percentile of dependability, just like Adam, but in only the 54th percentile of intelligence. Chris was the irrelevant alternative; an option that was plainly worse than one of the choices already on offer. You can guess what happened. The presence of a slightly dumber version of Adam made the real Adam look better; given the choice between dating Adam, Bill, and Chris, almost two-thirds of the women chose Adam.

  So if you’re a single guy looking for love, and you’re deciding which friend to bring out on the town with you, choose the one who’s pretty much exactly like you—only slightly less desirable.

  Where does irrationality come from? We’ve seen already that the apparent irrationality of popular opinion can arise from the collective behavior of perfectly rational individual people. But individual people, as we know from experience, are not perfectly rational. The story of the
slime mold suggests that the paradoxes and incoherencies of our everyday behavior might themselves be explainable in a more systematic way. Maybe individual people seem irrational because they aren’t really individuals! Each one of us is a little nation-state, doing our best to settle disputes and broker compromises between the squabbling voices that drive us. The results don’t always make sense. But they somehow allow us, like the slime molds, to shamble along without making too many terrible mistakes. Democracy is a mess—but it kind of works.

  USING THE WHOLE COW, IN AUSTRALIA AND VERMONT

  Let me tell you how they do it in Australia.

  The ballot down under looks a lot like Borda’s. You don’t just mark your ballot with the candidate you like best; you rank all the candidates, from your favorite to the one you hate the most.

  The easiest way to explain what happens next is to see what Florida 2000 would have looked like under the Australian system.

  Start by counting the first-place votes, and eliminate the candidate who got the fewest. In this case, that’s Nader. Toss him! Now we’re down to Bush vs. Gore.

  But just because we threw Nader out doesn’t mean we have to throw out the ballots of the people who voted for him. (Use the whole cow!) The next step—the “instant runoff”—is the really ingenious one. Cross Nader’s name off every ballot and count the votes again, as if Nader had never existed. Now Gore has 51% of the first-place votes: the 49% he had from the first round, plus the votes that used to go to Nader. Bush still has the 49% he started with. He has fewer first-place votes, so he’s eliminated. And Gore is the victor.

  What about our slightly modified version of Florida 2000, where we moved 2% from “Gore, Nader, Bush” to “Bush, Gore, Nader”? In that situation, Gore still won the Borda count. By Aussie rules, it’s a different story. Nader still gets knocked off in the first round; but now, since 51% of the ballots place Bush higher than Gore, Bush takes the prize.