Amos’s introduction to mathematical psychology sketched the controversy and argument that had ensued after Allais posed his paradox. On the American end, the argument was spearheaded by a brilliant American statistician and mathematician named L. J. (Jimmie) Savage, who had made important contributions to utility theory and who admitted that he, too, had been suckered by Allais into contradicting himself. Savage found an even more complicated way to restate Allais’s gambles so that at least a few devotees of expected utility theory, himself included, looked at the second situation and picked option number 3 instead of option number 4. That is, he demonstrated—or thought he had demonstrated—that the Allais “paradox” was not a paradox at all, and that people behaved just as expected utility predicted they would behave. Amos, along with pretty much everyone else who took an interest in such things, remained dubious.
As Danny read up on decision theory, Amos helped him to understand what was important about it and what was not. “He just had impeccable taste,” said Danny. “He knew what the problems were. He knew how to situate himself in the broad field. I didn’t have that.” What was important, Amos said, were the unresolved puzzles. “Amos said, ‘This is the story, this is the game. The game is to solve the Allais paradox.’”
Danny wasn’t inclined to see the paradox as a problem of logic. It looked to him more like a quirk in human behavior. “I wanted to understand the psychology of what was going on,” he said. He sensed that Allais himself hadn’t given much thought to why people might choose in a way that violated the major theory of decision making. But to Danny the reason seemed obvious: regret. In the first situation people sensed that they would look back on their decision, if it turned out badly, and feel they had screwed up; in the second situation, not so much. Anyone who turned down a certain gift of $5 million would experience far more regret, if he wound up with nothing, than a person who turned down a gamble in which he stood a slight chance of winning $5 million. If people mostly chose option 1, it was because they sensed the special pain they would experience if they chose option 2 and won nothing. Avoiding that pain became a line item on the inner calculation of their expected utility. Regret was the ham in the back of the deli that caused people to switch from turkey to roast beef.
Decision theory had approached the seeming contradiction at the heart of the Allais paradox as a technical problem. Danny found that silly: There was no contradiction. There was just psychology. The understanding of any decision had to account not just for the financial consequences but for the emotional ones, too. “Obviously it is not regret itself that determines decisions—no more than the actual emotional response to consequences ever determines the prior choice of a course of action,” Danny wrote to Amos, in one of a series of memos on the subject. “It is the anticipation of regret that affects decisions, along with the anticipation of other consequences.” Danny thought that people anticipated regret, and adjusted for it, in a way they did not anticipate or adjust for other emotions. “What might have been is an essential component of misery,’” he wrote to Amos. “There is an asymmetry here, because considerations of how much worse things could have been is not a salient factor in human joy and happiness.”
Happy people did not dwell on some imagined unhappiness the way unhappy people imagined what they might have done differently so that they might be happy. People did not seek to avoid other emotions with the same energy they sought to avoid regret.
When they made decisions, people did not seek to maximize utility. They sought to minimize regret. As the starting point for a new theory, it sounded promising. When people asked Amos how he made the big decisions in his life, he often told them that his strategy was to imagine what he would come to regret, after he had chosen some option, and to choose the option that would make him feel the least regret. Danny, for his part, personified regret. Danny would resist a change to his airline reservations, even when the change made his life a lot easier, because he imagined the regret he would feel if the change led to some disaster. It’s not a stretch to say that Danny anticipated anticipating regret. He was perfectly capable of anticipating the regret provoked by events that might never occur and decisions that he might never need to make. Once, at a dinner with Amos and their wives, Danny went on at length and with great certainty about his premonition that his son, then still a boy, would one day join the Israeli military; that war would break out; and that his son would be killed. “What were the odds of all that happening?” said Barbara Tversky. “Minuscule. But I couldn’t talk him out of it. It was so unpleasant talking with him about these small probabilities that I just gave up.” It was as if Danny thought that by anticipating his feelings he might dull the pain they would inevitably bring.
By the end of 1973, Amos and Danny were spending six hours a day with each other, either holed up in a conference room or on long walks across Jerusalem. Amos hated smoke; he hated being around people who smoked. Danny was still smoking two packs of cigarettes a day, and yet Amos never said a word. All that mattered was the conversation. When they weren’t with each other, they were writing memos to each other, to clarify and extend what had been said. If they happened to find themselves at the same social function, they inevitably wound up in the corner of a room, talking to each other. “We just found each other more interesting than anyone else,” said Danny. “Even if we had just spent the entire day working together.” They’d become a single mind, creating ideas about why people did what they did, and cooking up odd experiments to test them. For instance, they put this scenario to subjects:
You have participated in a lottery at a fair, and have bought a single expensive ticket in the hope of winning the single large prize that is offered. The ticket was drawn blindly from a large urn, and its number is 107358. The results of the lottery are now announced, and it turns out that the winning number is 107359.
They asked their subjects to rate their unhappiness on a scale from 1 to 20. Then they went to two other groups of subjects and gave them the same scenario, but with one change: the winning number. One group of subjects was told that the winning number was 207358; the second group was told that the winning number was 618379. The first group professed greater unhappiness than the second. Weirdly—but as Danny and Amos had suspected—the further the winning number was from the number on a person’s lottery ticket, the less regret they felt. “In defiance of logic, there is a definite sense that one comes closer to winning the lottery when one’s ticket number is similar to the number that won,” Danny wrote in a memo to Amos, summarizing their data. In another memo, he added that “the general point is that the same state of affairs (objectively) can be experienced with very different degrees of misery,” depending on how easy it is to imagine that things might have turned out differently.
Regret was sufficiently imaginable that people conjured it out of situations they had no control over. But it was of course at its most potent when people might have done something to avoid it. What people regretted, and the intensity with which they regretted it, was not obvious.