War and politics were never far from Amos and Danny’s minds or their conversations. They watched their fellow Israelis closely in the aftermath of the Yom Kippur war. Most regretted that Israel had been caught by surprise. Some regretted that Israel had not attacked first. Few regretted what both Danny and Amos thought they should most regret: the Israeli government’s reluctance to give back the territorial gains from the 1967 war. Had Israel given back the Sinai to Egypt, Sadat would quite likely never have felt the need to attack in the first place. Why didn’t people regret Israel’s inaction? Amos and Danny had a thought: People regretted what they had done, and what they wished they hadn’t done, far more than what they had not done and perhaps should have. “The pain that is experienced when the loss is caused by an act that modified the status quo is significantly greater than the pain that is experienced when the decision led to the retention of the status quo,” Danny wrote in a memo to Amos. “When one fails to take action that could have avoided a disaster, one does not accept responsibility for the occurrence of the disaster.”
They set out to build a theory of regret. They were uncovering, or thought they were uncovering, what amounted to the rules of regret. One rule was that the emotion was closely linked to the feeling of “coming close” and failing. The nearer you came to achieving a thing, the greater the regret you experienced if you failed to achieve it.? A second rule: Regret was closely linked to feelings of responsibility. The more control you felt you had over the outcome of a gamble, the greater the regret you experienced if the gamble turned out badly. People anticipated regret in Allais’s problem not from the failure to win a gamble but from the decision to forgo a certain pile of money.
That was another rule of regret. It skewed any decision in which a person faced a choice between a sure thing and a gamble. This tendency was not merely of academic interest. Danny and Amos agreed that there was a real-world equivalent of a “sure thing”: the status quo. The status quo was what people assumed they would get if they failed to take action. “Many instances of prolonged hesitation, and of continued reluctance to take positive action, should probably be explained in this fashion,” wrote Danny to Amos. They played around with the idea that the anticipation of regret might play an even greater role in human affairs than it did if people could somehow know what would have happened if they had chosen differently. “The absence of definite information concerning the outcomes of actions one has not taken is probably the single most important factor that keeps regret in life within tolerable bounds,” Danny wrote. “We can never be absolutely sure that we would have been happier had we chosen another profession or another spouse. . . . Thus, we are often protected from painful knowledge concerning the quality of our decisions.”
They spent more than a year working and reworking the same basic idea: In order to explain the paradoxes that expected utility could not explain, and create a better theory to predict behavior, you had to inject psychology into the theory. By testing how people choose between various sure gains and gains that were merely probable, they traced the contours of regret.
Which of the following two gifts do you prefer?
Gift A: A lottery ticket that offers a 50 percent chance of winning $1,000
Gift B: A certain $400
or
Which of the following gifts do you prefer?
Gift A: A lottery ticket that offers a 50 percent chance of winning $1 million Gift B: A certain $400,000
They collected great heaps of data: choices people had actually made. “Always keep one hand firmly on data,” Amos liked to say. Data was what set psychology apart from philosophy, and physics from metaphysics. In the data, they saw that people’s subjective feelings about money had a lot in common with their perceptual experiences. People in total darkness were extremely sensitive to the first glimmer of light, just as people in total silence were alive to the faintest sound, and people in tall buildings were quick to detect even the slightest swaying. As you turned up the lights or the sound or the movement, people became less sensitive to incremental change. So, too, with money. People felt greater pleasure going from 0 to $1 million than they felt going from $1 million to $2 million. Of course, expected utility theory also predicted that people would take a sure gain over a bet that offered an expected value of an even bigger gain. They were “risk averse.” But what was this thing that everyone had been calling “risk aversion?” It amounted to a fee that people paid, willingly, to avoid regret: a regret premium.
Expected utility theory wasn’t exactly wrong. It simply did not understand itself, to the point where it could not defend itself against seeming contradictions. The theory’s failure to explain people’s decisions, Danny and Amos wrote, “merely demonstrates what should perhaps be obvious, that non-monetary consequences of decisions cannot be neglected, as they all too often are, in applications of utility theory.” Still, it wasn’t obvious how to weave what amounted to a collection of insights about an emotion into a theory of how people make risky decisions. They were groping. Amos liked to use an expression he’d read someplace: “carving nature at its joint.” They were trying to carve human nature at its joint, but the joints of an emotion were elusive. That was one reason Amos didn’t particularly like to think or talk about emotion; he didn’t like things that were hard to measure. “This is indeed a complex theory,” Danny confessed one day in a memo. “In fact it consists of several mini-theories, which are rather loosely connected.”
In reading about expected utility theory, Danny had found the paradox that purported to contradict it not terribly puzzling. What puzzled Danny was what the theory had left out. “The smartest people in the world are measuring utility,” he recalled. “As I’m reading about it, something strikes me as really, really peculiar.” The theorists seemed to take it to mean “the utility of having money.” In their minds, it was linked to levels of wealth. More, because it was more, was always better. Less, because it was less, was always worse. This struck Danny as false. He created many scenarios to show just how false it was: Today Jack and Jill each have a wealth of 5 million.
Yesterday, Jack had 1 million and Jill had 9 million.
Are they equally happy? (Do they have the same utility?) Of course they weren’t equally happy. Jill was distraught and Jack was elated. Even if you took a million away from Jack and left him with less than Jill, he’d still be happier than she was. In people’s perceptions of money, as surely as in their perception of light and sound and the weather and everything else under the sun, what mattered was not the absolute levels but changes. People making choices, especially choices between gambles for small sums of money, made them in terms of gains and losses; they weren’t thinking about absolute levels. “I came back to Amos with that question, expecting that he would explain it to me,” Danny recalled. “Instead Amos says, ‘You’re right.’”
* * *
* I apologize for this, but it must be done. Those whose minds freeze when confronted with algebra can skip what follows. A simpler proof of the paradox, devised by Danny and Amos, will come later. But here, more or less reproduced from Mathematical Psychology: An Elementary Introduction, is the proof of Allais’s point that Amos asked Danny to ponder.
Let u stand for utility.
In situation 1:
u(gamble 1) > u(gamble 2)
and hence
1u(5) > .10u(25) + .89u(5) + .01u(0)
so
.11u(5) > .10u(25) + .01u(0)
Now turn to situation 2, where most people chose 4 over 3. This implies
u(gamble 4) > u(gamble 3)
and hence
.10u(25) + .90u(0) > .11u(5) + .89u(0)
so .10u(25) + .01u(0) > .11u(5) Or the exact reverse of the choice made in the first gamble.
? Two decades later, in 1995, the American psychologist Thomas Gilovich, who collaborated in turn with Danny and Amos, coauthored a study that examined the relative happiness of silver and bronze medal winners at the 1992 Summer Olympics. From video footage, subjects judged the bronze medal winners to be happier than the silver medal winners. The silver medalists, the authors suggested, dealt with the regret of not having won gold, while the bronze medalists were just happy to be on a podium.
10