Priceless: The Myth of Fair Value (and How to Take Advantage of It) (9 page)

This was a direct thrust at the American’s jugular. As Allais knew, one of Savage’s axioms of reasonable decision making says (in essence) that when deciding between a burger with diet soda or pizza with diet soda, you can ignore the diet soda because you’re getting it in either case. The only thing that matters is whether you like burgers or pizza better. In general, according to Savage, deciders should ignore the common elements of choices and choose based on the differences.

This sounds reasonable to just about everyone. Allais spotted a subtle flaw. By Savage’s logic, the choice in Riddle 3 shouldn’t depend on what’s in the box. Whether you choose (a) or (b), you get the same 89 percent chance at winning the same box.

This doesn’t mean that the box’s contents are unimportant. The box could contain a billion dollars, or a deadly tarantula, or the phone number of that cool person you met on the subway. But according to Savage, the box shouldn’t bear on the choice between (a) and (b). That choice should be based solely on whether it’s better to have an 11 percent chance of $1 million or a 10 percent chance of $2.5 million.

In other words, the answer to Riddle 3 should be the same as to Riddle 2. That’s not all. Suppose we open the box and discover a million dollars in there. Then the choice in Riddle 3 ends up being identical to that in Riddle 1. In short, the answer to all three riddles should be the same, either (a) or (b) with no flip-flopping. Allais had tricked Savage into betraying his own rule.

A few months later, Allais gave a similar pop quiz to Milton Friedman. Friedman did
not
fall into the trap Savage did and gave consistent answers. I suppose you have to wonder whether Savage clued him in.

•   •   •

In a 1953
Econometrica
article, published in French, Allais took issue with the axioms of
l’école Américaine
(meaning Detroit-born Jimmie Savage and his Brooklyn-born friend Milton Friedman). The Americans were saying that everyone’s got a price (utility) for everything. These subjective prices determine all decisions. Humans are more complex than that, Allais argued. Choices depend on context, and no single number can express how one feels about uncertain outcomes.

This demonstration has since become known as Allais’ paradox. Don’t worry if you’re still unclear on what Allais was driving at and why it’s important. Let me give a remix of the paradox, conceived by Richard Zeckhauser of Harvard. You are a contestant on a popular new game show,
Your Money or Your Life
. Like most game shows, it simply recycles an old parlor game. Unfortunately for you, that old parlor game is Russian roulette.

At the beginning of every show, Tiffany, the “Bullet Lady,” spins Fortune’s Wheel. The wheel is divided into six equal slices. The spin tells Tiffany how many bullets, from one to six, to load into a six-barrel gun and hand to the show’s host, Brian. After a brief commercial break, Brian spins the gun’s barrel and points it directly at your left temple. Just before he pulls the trigger, he proposes a financial arrangement that you will doubtless find interesting.

You can buy a bullet.
Should you and Brian agree on a price, he will extract one bullet at random from the gun’s barrel and hand it to you in exchange for the money you give him. He will then spin the barrel again, point the gun at your temple again, and pull the trigger.

Here’s the odd thing. You’d probably be willing to pay a higher price for a bullet when it’s the only one in the barrel. Buy that one bullet, and you’re 100 percent certain to survive (versus having a 1 in 6 chance of not making it to the commercial break). You’d pay a lot for that, right?

Just for the sake of comparison, suppose there are four bullets in the barrel. Now how much would you pay to buy one bullet—to have just three bullets rather than four? Somehow this weakens the case for raising every last penny for that bullet. You might even feel you’d be willing to take your chances with the four bullets.

Isn’t the human mind a funny thing? A bullet is a bullet, dead is
dead. The reduction in probability of your demise is precisely the same in both cases. Why isn’t your price the same?

Or imagine there are six bullets in the gun. You’re a corpse unless you buy a bullet. This may cause you to flip-flop again and conclude that the bullet is priceless, worth paying everything you’ve got.

Both this game and Allais’ original puzzle reveal a
certainty effect
. There is often a huge subjective difference between an
absolute, 100 percent sure thing
and something that is only 99 percent likely. This difference is expressed in prices as well as choices. Meanwhile, the difference between a 10 and an 11 percent chance is shrugged off.

To a select following of economists, psychologists, and philosophers, the Allais paradox became a sword in the stone. Great minds tested themselves against it, few managing to get much of a grip. In later years, Allais himself thought and wrote extensively about his puzzle. In true economist fashion, he tried to lay out axioms of human decision making and show that they were subtly incompatible, leading to contradiction.

“His paradox was great,” one scholar said of Allais. “But if you read his own papers on what he thought the right theory was, they’re very hard to understand . . . He’s also cantankerous. There were a few conferences of a group called FUR, Foundations of Uncertainty and Risk, and I went to a couple of them. Allais would give this talk, and someone would say, ‘Your axioms are wrong, you claim you’ve proved something that’s not proved.’ Allais would bluster, and [UC San Diego economist] Mark Machina would literally stand up and try to defend Allais. Then Allais would turn on Machina.”

Allais continued his attack in a prickly 1995 paper subtitled “Unceasingly Repeated Errors or Contradictions of Mark Machina.” (“As a matter of fact,” Allais wrote, “I haven’t been able until now to answer Machina’s paper. My time has been entirely used up, on one hand by the task of editing the first printed version of my 1943 work, in view of which I have [been] awarded the 1988 Nobel Prize in Economic Science, with a new and long introduction, and on another hand by the task of publishing an important book on Europe . . . The reader will understand that I cannot accept to spend too much of a scarce time to correct Machina’s mistakes,
line after line
. . .”)

The long-suffering Machina has posted Allais’s paper on his website, under the heading “News, Gossip & Games.” I’ll confine myself to saying a little about why Allais’ paradox was so intractable. The stumbling block isn’t the certainty effect per se. It’s the way that smart people are influenced by mere words, by the way the choices are framed. As Amos Tversky later wrote, “We choose between descriptions of options, rather than between the options themselves.” For the most part, economists were not ready to accept that fact of life.

For a psychologist, Ward Edwards could display startling insensitivity to the feelings of others. Sarah Lichtenstein found him exasperating. Fresh out of Swarthmore, she arrived at Ann Arbor for graduate work, with Edwards as her advisor. Edwards proposed that she collaborate on an article with another grad student, Paul Slovic. “When we had written it up and were talking about the order of names to publish it under, Ward very graciously agreed to be the third name,” Lichtenstein said. “He suggested—it was stronger than that—that Paul be the first author because he, being a man, would have to earn a living.” The article appeared in a 1965 issue of
The American Journal of Psychology
, credited to Paul Slovic, Sarah Lichtenstein, and Ward Edwards. Slovic was three years younger than Lichtenstein.

The patriarchal times dictated Lichtenstein’s moves after grad school. “I sort of followed hubby around for several years.” Husband Ed was a clinical psychologist who took a job in Los Angeles. When he got an offer from the University of Oregon in 1966, one selling point was that Sarah might be able to land a job at the Oregon Research Institute. “It was a terrific inducement,” she explained. ORI “was a marvelous place to work at that time.”

Paul Slovic was already there. He had accepted a job after graduation in 1964 and lobbied for ORI to hire Lichtenstein. The two resumed an agreeable collaboration that, among other things, studied how people assign prices to gambles.

For example: A wager offers a 1 in 8 chance of winning $77. How much would you be willing to pay for the privilege of playing this bet?

The obvious approach is to compute how much you can expect to win, on the average, each time you play. This comes to ⅛ times $77, or $9.63. Of course, the numbers here make it hard to do the math in your head. The psychologists were interested in intuitive judgments, and they observed that the prices subjects assigned to simple bets were usually too high. People apparently paid more attention to the prize amount than to the chance of winning it.

This could explain why lotteries are so popular. A lottery offers, let’s say, a one-in-a-zillion chance of winning $58 million. Players are essentially buying the right to fantasize about the jackpot. The “one in a zillion” is in the fine print, literally
and
in the minds of players. When lottery boards want to drum up business, they raise the jackpots, not the chances of winning.

A similar phenomenon pertains to losing bets. How much would you pay to get out of a situation in which you have a 1 in 12 chance of losing $63? People were typically willing to pay
more
than the average loss. The dollar amount of the penalty loomed more important than the probability in their decision making.

This suggests an explanation for why people buy insurance. They are willing to pay “too much” for coverage because they worry more about the dollar value of catastrophes than the remoteness of the odds.

Lichtenstein and Slovic asked some of their subjects to rate the “attractiveness” of bets on a scale of 1 to 5. They found that the ratings correlated most strongly with the probability of winning. People liked bets that were easy to win.

Okay, fine. But the
prices
assigned to bets correlated with the amount to win. It was as if people had two ways of valuing bets, and they were subtly in conflict.

“I remember we were in Paul’s office, I can’t tell you what year it was,” Lichtenstein said. “We were getting an idea of what subjects were paying attention to. I don’t recall who said it first, or whether we said it at the same time. But it struck us that we could design bets that would encourage subjects to do one thing under one response mode and another under another response mode. When we saw it and said it aloud, we were sure it was going to work—and it did.”

Their brainstorm was that prices might not reflect what people want.
They could invent a pair of bets—call them A and B—such that most people would say they preferred A, but, when asked to assign prices to them, they would give a higher value to B.

The strangeness of this might be easier to appreciate if you pretend that A and B are fancy gift boxes wrapped in paper and bows. I don’t know for sure what’s in either box. I have had a chance to shake them and form some opinion about what’s inside. Okay, I’ve decided that I’m willing to pay $40 for Box A and $70 for Box B. I’ve also decided that I’d rather have Box A.

This is crazy! My prices don’t jibe with my desires or actions. Lichtenstein and Slovic found something crazier yet. For certain types of gambles,
most
people have valuations just like this.

They called this a “preference reversal,” and here’s an example. In the figure below, the two circles represent dartboards. Pick one; then a “dealer” is going to toss a dart at the center of your chosen target, so that the dart is equally likely to land anywhere within the circle. That determines how much (if anything) you win. Which target would you rather use?

The target on the left offers an 80 percent chance of winning $5 (otherwise nothing). The one on the right has a 10 percent chance of winning $40, and otherwise nothing.

The expected value happens to be the same for both bets ($4), so that
provides no grounds for choosing. Yet a majority prefer the target on the left. Lichtenstein and Slovic termed gambles like the one on the left
P
(for
probability
) bets. A P bet offers a high chance of winning. The bet on the right is a
$
(
money
) bet offering a bigger prize and a lesser chance of winning it. When asked to choose, most people prefer P bets to $ bets.

There is nothing peculiar about that. Choosing the P bet increases the odds of walking away a winner. What
is
odd is that the same subjects regularly assign higher prices to $ bets, like the one on the right above. The prices contradict the preferences.