Eight months later, Ms. Foley would next see her son in a video showing him kneeling in the sand, an executioner’s knife at his neck.

The United States and Britain are among the only countries that abide by a zero-concession policy, refusing to accede to terrorists’ demands, arguing that doing so encourages more kidnapping. By contrast, European countries have repeatedly paid to free their citizens, despite signing numerous declarations vowing not to, prompting condemnation from former American officials and analysts.

“What is hard to prove is how many Americans have not been kidnapped as a result of the fact that the enemy knows they will not get a penny from us,” said Gen. John R. Allen, who recently retired as the top commander in Afghanistan. “In the aftermath of this horrific event it makes it hard to explain this policy. But the fact that there are Americans in the region who were never taken because they knew there was no advantage to doing so needs to be factored in.”

The willingness to pay ransoms for kidnapped victims is a source of debate and mounting tension between the United States and Britain on one side and their European allies on the other. From the families’ perspective, there is another dividing line between the two approaches: Many European nations take charge of the situation from the moment their citizens are captured and aggressively begin a negotiation. By contrast, relatives in the United States said they were left to puzzle through the crisis largely on their own.

While the F.B.I. declined to comment on its handling of the hostage crisis, a senior law enforcement official said that the bureau used every tool at its disposal. He conceded, however, that the bureau is bound by American law and cannot engage in a discussion over a potential ransom. “Those are the lanes in the road we are left to work with,” the official said.

From the perspective of the families, however, the policy of not bargaining with terrorists is itself controversial. They argue that the death of even one person seems a high price to pay for the broader goal of not encouraging further kidnappings.