As for sweetening the pot, in general the greater the monetary incentive involved in the deal, the greater the disgust from respondents. Israelis and Palestinians alike often reacted as though we had asked them to sell their children. This strongly implies that using the standard approaches of “business-like negotiations” favored by Western diplomats will only backfire.

Many Westerners seem to ignore these clearly expressed “irrational” preferences, because in a sensible world they ought not to exist. Diplomats hope that peace and concrete progress on material and quality-of-life matters (electricity, water, agriculture, the economy and so on) will eventually make people forget the more heartfelt issues. But this is only a recipe for another Hundred Years’ War  progress on everyday material matters will simply heighten attention on value-laden issues of “who we are and want to be.”

Fortunately, our work also offers hints of another, more optimistic course.

Absolutists who violently rejected offers of money or peace for sacred land were considerably more inclined to accept deals that involved their enemies making symbolic but difficult gestures. For example, Palestinian hard-liners were more willing to consider recognizing the right of Israel to exist if the Israelis simply offered an official apology for Palestinian suffering in the 1948 war. Similarly, Israeli respondents said they could live with a partition of Jerusalem and borders very close to those that existed before the 1967 war if Hamas and the other major Palestinian groups explicitly recognized Israel’s right to exist.

Remarkably, our survey results were mirrored by our discussions with political leaders from both sides. For example, Mousa Abu Marzook (the deputy chairman of Hamas) said no when we proposed a trade-off for peace without granting a right of return. He became angry when we added in the idea of substantial American aid for rebuilding: “No, we do not sell ourselves for any amount.”

But when we mentioned a potential Israeli apology for 1948, he brightened: “Yes, an apology is important, as a beginning. It’s not enough because our houses and land were taken away from us and something has to be done about that.” His response suggested that progress on sacred values might open the way for negotiations on material issues, rather than the reverse.

We got a similar reaction from Benjamin Netanyahu, the hard-line former Israeli prime minister. We asked him whether he would seriously consider accepting a two-state solution following the 1967 borders if all major Palestinian factions, including Hamas, were to recognize the right of the Jewish people to an independent state in the region. He answered, “O.K., but the Palestinians would have to show that they sincerely mean it, change their textbooks and anti-Semitic characterizations.”

Making these sorts of wholly intangible “symbolic” concessions, like an apology or recognition of a right to exist, simply doesn’t compute on any utilitarian calculus. And yet the science says they may be the best way to start cutting the knot.