Our Gemara on Amud Beis discusses various psychological techniques when considering negotiating with those who kidnap for ransom. 

אֵין פּוֹדִין אֶת הַשְּׁבוּיִין יוֹתֵר עַל כְּדֵי דְמֵיהֶם, מִפְּנֵי תִּקּוּן הָעוֹלָם. הָא בִּכְדֵי דְמֵיהֶן פּוֹדִין, אַף עַל גַּב דְּפִרְקוֹנָהּ יוֹתֵר עַל כְּתוּבָּתָהּ.

One does not redeem captives at more than their value. This policy is for the betterment of the world, because if captives are ransomed at exorbitant prices, this will encourage their captors to kidnap more people. 

Halakha Le’maase, the poskim make distinctions between public policy and private action.  That is, the congregation should not feel burdened with excessive ransoms but family members and even the person himself might be able to pay whatever amount demanded upon them.  (See Shulkan Arukh YD:252:4 Shach and Bach.)

In a modern sense, Israel and America have often publicly declared that they “Do not negotiate with terrorists.”  While the logic of this is probably similar to our Gemara, namely that we do not want to encourage escalation of these tactics, what does the research actually show?

According to researcher Brian Michael Jennings, the findings are not as clear-cut ( Jenkins, Brian Michael, Does the U.S. No-Concessions Policy Deter Kidnappings of Americans?. Santa Monica, CA: RAND Corporation, 2018. https://www.rand.org/pubs/perspectives/PE277.html )

Here are the findings:

Deterrence is offered as the principal reason for the U.S. adherence to a no-concessions policy. Logically, a no-concessions policy should be a deterrent to kidnapping. No concessions means denying a reward to the kidnappers, thereby removing the incentive to kidnap Americans.

• The deterrent effect of the no-concessions policy, however, may be eroded by the fact that kidnappers are not aware of U.S. policy, do not believe it, or may not care because other objectives will still be served by holding Americans hostage.

• The RAND Corporation’s research in the early 1970s was unable to find persuasive evidence supporting the assertion that a no-concessions policy provided an effective deterrent. There was little correlation between the different negotiating policies adopted by various governments and the absence or occurrence of further kidnappings.

• Different precedents and practices—ransom payments in domestic kidnappings, payments by private parties abroad, not resisting the demands of airline hijackers (until 2001),a highly publicized breach of policy by the U.S. government itself, prisoner exchanges following or during armed conflicts, and concessions by other governments to obtain the release of U.S. hostages—have blurred perceptions of U.S. policy.

• The U.S. no-concessions policy may often be irrelevant to the kidnappers’ aims. Terrorists take hostages to extract ransom or other concessions from other governments, to attract attention and make themselves important players in the region, to use the hostages as shields against govern- ment military operations, to discourage foreign investment, to create political crises that will embarrass the hostage-takers’ foes, or to make demands they know will not be met, thereby giving them an excuse to murder their hostages while blaming the government for its callous obstinacy.

• The most important factor in reducing further kidnappings appears to be the fate of the kidnappers. Where they are apprehended and face stiff penalties and their gangs and groups are destroyed, kidnappings decline.

• Despite the U.S. no-concessions policy, U.S. citizens continue to top the list of nationalities kidnapped by terrorists. This may be explained by the prominent role and perceived influence of the United States and the ubiquity of U.S. citizens around the world. Nationals of the United Kingdom, which also has a no-concessions policy, are second on the list.

• Research by RAND and other institutions shows little evidence of nationality-specific targeting. In conflict zones, terrorists (or criminal gangs who sell hostages to terrorists) kidnap opportunistically.

• While a no-concessions policy may not deter kidnappings, it may affect the treatment of hostages in captivity and determine their ultimate fate. According to a 2015 study published by West Point, Americans held hostage by jihadist groups are nearly four times as likely to be murdered as other Western hostages (Loertscher and Milton, 2015). The no-concessions policy may be only part of the reason. Another factor would be the jihadists’ intense hostility toward the United States.

• While the U.S. no-concessions policy has not deterred kidnappings, there is some evidence that political concessions and ransom payments appear to encourage further kidnappings and escalating demands.

• And although it did not produce any demonstrable decline in kidnappings of U.S. citizens, a 2016 study published in the European Journal of Political Economy argues that, without the no-concessions policy, there would have been even more kidnappings of U.S. nationals (Brandt, George, and Sandler, 2016).

• Finally, the no-concessions policy serves goals other than deterrence. Terrorists use ransoms to finance further operations, releasing prisoners would undermine the judicial system, and making other political concessions raises issues of governance.

Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation

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