Sandeep and I are very close to finishing a first draft of our paper on torture.  As I was working on it today, I came up with a simple three-paragraph summary of the model and some results.  Here it is.

A number of strategic considerations play a central role in shaping the equilibrium. First, the rate at which the agent can be induced to reveal information is limited by the severity of the threat.  If the principal demands too much information in a given period then the agent will prefer to resist and succumb to torture. Second, as soon as the victim reveals that he is informed by yielding to the principal’s demand, he will subsequently be forced to reveal the maximum given the amount of time remaining.  This makes it costly for the victim to concede and makes the alternative of resisting torture more attractive. Thus, in order for the victim to be willing to concede the principal must also torture a resistant suspect, in particular an uninformed suspect, until the very end.  Finally, in order to maintain principal’s incentive to continue torturing a resistant victim  the informed victim must, with positive probability, wait any number of periods before making his first concession.

These features combine to give a sharp characterization of the value of torture and the way in which it unfolds.  Because concessions are gradual and torture cannot stop once it begins, the principal waits until very close to the terminal date before even beginning to torture. Starting much earlier would require torturing an uninformed victim for many periods in return for only a small increase in the amount of information extracted from the informed.  In fact we show that the principal  starts to torture only after the game has reached the ticking time-bomb phase: the point in time after which the deadline becomes a binding constraint on the amount of information the victim can be induced to reveal. This limit on the duration of torture also limits the value of torture for the principal.

Because the principal must be willing to torture in every period, the informed victim concession probability in any given period is bounded, and this also bounds the principal’s payoff.  In fact we obtain a strict upper bound on the principal’s equilibrium payoff by considering an alternative problem in which the victim’s concession probability is maximal subject to this incentive constraint. This bound turns out to be useful for a number of results.   For example the bound enables us to derive an upper bound on the number of periods of torture that is independent of the total amount of information available.  We use this result to show that the value of torture shrinks to zero when the period length, i.e. the time interval between torture decisions, shortens.  In addition it implies that laws preventing indefinite detention of terrorist suspects entail no compromise in terms of the value of information that could be extracted in the intervening time.