7

A Matter of Calculation

The result of torture, then, is a matter of calculation ….

—CESARE BECCARIA, “OF TORTURE,” § 16, Of Crimes and Punishments, 1764

The calculation steps described in the Chapter 6, albeit different from what Beccaria had in mind, generate nine perfect Bayesian equilibria when applied to all the possible pure strategies of the Detainee. This chapter discusses general features of these equilibria, focusing on the belief thresholds that partly define them to aid the discussion of the individual equilibria in the chapters that follow. In order to better do this, I introduce a way of visualizing the equilibria which we will employ throughout the rest of the book.

I also explore properties of the thresholds by examining the mathematical and empirical properties of both the parameters constituting the thresholds and the thresholds themselves. Some of these properties will turn out to have important implications for the equilibria and the comparison with the benchmarks and thus for the argument of the book. These I will set apart and identify as observations, propositions, and implications.

Since my usage of these three terms differs a bit from the norm, let me first define each. By observation I mean a mathematical statement that follows directly and obviously from some other mathematical statement emerging from the model such as the equation for a threshold. By proposition I mean a true mathematical statement which follows not quite as directly or obviously from a mathematical statement or statements derived from the model. In addition, these propositions will sometimes rely on empirical assumptions about parameters in the model.¹ In Appendix C, I prove all the observations and propositions in this and the following chapters, proceeding in the same order as in the main text. Finally, by implication I mean a broader claim or assertion about interrogational torture which is implied or entailed by an observation or proposition.

It is important to acknowledge up front that, to the extent we make any empirical assumptions, we are stepping away from the rigor of the model and entering a world which is more arbitrary. The assumptions we will make, however, are both reasonable and entirely consistent with torture practices and programs past and present. Moreover, given our larger goal of saying something about torture as it is actually practiced in the world, it is worthwhile doing so. I will signpost very clearly any propositions resting on empirical assumptions and summarize these assumptions at the end of the chapter.

THE EQUILIBRIA

The RIT model’s equilibria are summarized by their moves in Table 7.1. Two equilibria occur under leading questioning only (ambiguous information, selective torture and false confirmation, selective torture) and one occurs under objective questioning only (valuable information, surprise torture). The remaining three equilibria occur under both objective and leading questioning (valuable information, selective torture; no information, torture; no information, no torture), so there are really six substantively equivalent equilibria.

Table 7.1 RIT GAME PURE STRATEGY EQUILIBRIA

Objective Questioning	Leading Questioning
Valuable information, Surprise torture	Ambiguous information, Selective torture
	False confirmation, Selective torture
Valuable information, Selective torture
No information, Torture
No information, No torture

As we saw in the equilibrium we found together in the Chapter 6, however, equilibria are defined not just by the actions or moves of the players but also by their beliefs. If, for example, the Detainee’s belief q that the Interrogator is Pragmatic and not Sadistic falls below the threshold , the valuable information, surprise torture equilibrium we found in Chapter 6 collapses (i.e., does not exist). Not surprisingly, it turns out that three important beliefs—whether the Interrogator is Pragmatic (q), whether the Detainee has revealed all his information (f), and whether the Detainee is Innocent after the move “no information” (p)—also define the other equilibria as thresholds. Since these beliefs form a crucial part of the definition, they must be included.

The formal structure for doing this in words is: (actions, beliefs) or, more thoroughly,

(move by Cooperative Detainee type, move by Resistant Detainee type, move by Innocent Detainee type),(move by Pragmatic Interrogator after “information,” move by Pragmatic Interrogator after “no information”)(move by Sadistic Interrogator after “information,” move by Sadistic Interrogator after “no information”), relevant beliefs q and/or f of the Detainee, relevant beliefs of the Pragmatic Interrogator (e.g., , ), including posterior beliefs and beliefs off the equilibrium path of play.

Using our equilibrium from Chapter 6 as an example and employing symbols to reduce the clutter gives ; (), (): , for , where i is “reveal information,” is “not reveal information,” t is “torture,” (spoken as “mew sub i, C”) is the Interrogator’s posterior belief, i.e. the belief updated using Bayes’ Rule after the Detainee’s move i that the Detainee is type C, and (“mew sub bar i, I”) is the Interrogator’s posterior (updated) beliefs using Bayes’ Rule after the Detainee’s move that the Detainee is Innocent.

If the statement in natural language has the advantage of familiarity but the disadvantage of wordiness, the symbolism has the advantage of being compact but the disadvantage of esoteric abstraction. Neither is terribly helpful, especially if we want to think of all nine equilibria and how they compare to each other and to the benchmarks.

A picture may not be worth a thousand words, but it will be well worth some unfamiliar math symbols and Greek letters. Recall the diagram of the information hiding threshold in Chapter 6 (Figure 6.1). There we had the Interrogator’s belief (f) about whether the Detainee had revealed all his information. This belief was arrayed on a line ranging from 0 to 1. The threshold is somewhere on that line. A value of f below the threshold means that the Interrogator is not confident enough that the Detainee has revealed all he knows and thus tortures. For values of f above the threshold, she is confident enough and thus chooses not to torture.

A high threshold means that the space occupied by “no torture” above the threshold is smaller whereas the space occupied by “torture” under the threshold is larger. In other words, a wider range of beliefs supports the Interrogator’s use of torture. In this sense, all else being equal, torture is more likely. A low threshold for choosing “not torture,” conversely, means that torture is less likely. The range of beliefs supporting the Interrogator’s use of torture has shrunk, represented by the shrinking space in that region.

If we add to the belief f the other two key beliefs, q and p, and give each of them an axis as well, we create a three-dimensional space—a cube—inhabited by the equilibria, as in Figure 7.1. The various thresholds from the derivations of the equilibria will lie along these axes, slicing up the interior space. Each equilibrium will thus take up a portion of this space, defined by whether it exists above or below the various thresholds. This provides a handy way to visually represent the equilibria, see how they change as a function of those thresholds, and, in Chapter 12, compare the RIT outcomes to the normative predictions in Chapter 3.

Figure 7.1 Equilibria Parameter Space

The vertical axis, q, represents the Cooperative and Innocent Detainees’ belief that the Interrogator is Pragmatic rather than Sadistic. The closer to the top (to one), the higher the likelihood the Interrogator is Pragmatic and not Sadistic in the estimation of the Detainee. This belief is important because it partly determines whether a Cooperative Detainee and an Innocent Detainee under leading questioning want to choose “information” or not.

Similarly, the diagonal axis p represents the Pragmatic Interrogator’s belief that the Detainee is Innocent upon observing “no information.” The closer to one on the p axis, the higher the likelihood the Detainee is Innocent and knows nothing in the estimation of the Pragmatic Interrogator. The horizontal axis f represents the Pragmatic Interrogator’s belief that, upon observing “information,” the Detainee has revealed all he knows. The closer to one on the f axis, the more likely it is that the Detainee has told the Interrogator all he knows (or that which he is willing to confirm if Innocent under leading questioning). These beliefs are also important because they determine whether or not the Pragmatic Interrogator prefers to torture (because the Interrogator wants to torture after “information” if she thinks the Detainee is hiding more information and does not want to torture after “no information” if the Detainee is Innocent since the Pragmatic Interrogator does not want to torture Innocent Detainees).

THINKING ABOUT BELIEVING

Since the equilibria depend on values of q, f, and p being above (below) the different thresholds, it will be important to consider for each equilibrium how the changes in the thresholds affect the equilibrium. The thresholds themselves are defined by the various costs and benefits in the model, captured by different variables or parameters (e.g., the ratio of v to k for the threshold ). Thus, we must approach the thresholds in two stages and on two tracks in each stage, one formal and mathematical, one contingent and empirical.

First we focus on the variables, examining both (a) what happens to the thresholds as different parameters in the model change (formal mathematical) and (b) the likely values of those parameters in the real world (empirical). Second, we explore some of the properties of the thresholds. Some of these properties will (a) be mathematically derived, formal properties, while others will (b) be contingent, empirical characteristics based on the likely values of the parameters.

Plus ça Change … Not

However widely the French proverb may apply in life, it does not apply to our thresholds; they will indeed change as their constituent parameters change. To see these effects, we can do some “comparative statics,” which requires some basic derivative calculus to identify what happens to one quantity as another quantity increases or decreases. (We won’t actually do any of that calculus here.) The idea is to see what happens to a threshold (say ) as one of the variables in the formula defining it (say c) increases (decreases).

Take the Cooperative Detainee’s information revelation threshold as an example. It is the ratio of the value of the information v to the costs of torture k (both from the perspective of the Detainee). The first thing to note is that this is a real constraint. Recall that the Cooperative Detainee prefers to give up information rather than be tortured; this is what defines a Cooperative Detainee. This preference is represented by the ordering . Multiply both sides by to get rid of the negatives and it reverses the direction of the inequality but preserves the relationship, . Because both v and k are positive and v is less than k, the threshold must lie between zero and one and thus is a real probability.

Now think about what happens to this fraction as v increases but k stays the same.

It gets bigger and would eventually approach 1 as v approaches k.

What happens as k increases? The fraction—threshold—approaches zero as k grows large relative to v.

For such a simple fraction we don’t need calculus to tell us that the threshold increases as v increases but decreases as k increases. Once we start adding in other variables, however, as is the case for the other thresholds, these effects will not be as obvious (at least to me). Calculus helps us figure out what happens to them as the other factors change. The results of performing these calculations on the other thresholds and their variables in Appendix B are summarized in Table 7.2.

Table 7.2 EFFECTS OF MODEL VARIABLES ON THRESHOLDS

Notice how the parameters r, c, and a figure in the remaining five thresholds. The probability of full information revelation u and the probabilities for each of the three Detainee types p_C, p_R, and p_I are parameters of subsets of these five. To see the effects of each variable or parameter, I organize the discussion by parameter, explaining the effect of each on the relevant threshold(s).

REPUTATION COSTS, r

As reputation costs (r) go up, so do both the information hiding (, , and ) and innocent detainee recognition thresholds ( and ).² This makes sense; it is natural that these costs would affect the Interrogator’s decision about whether or not to torture. Recall that the information hiding threshold is the point at which the Interrogator switches from “torture” to “no torture.” Moving from left to right along the bottom axis of Figure 7.1, then, a high threshold means a threshold closer to the right side, leaving most of the space occupied by torture.

Now take a look at the p axis, the diagonal line. Moving from the front toward the back from 0 to 1, the thresholds and are the points at which the Interrogator switches from “torture” to “no torture” after observing “no information.” A high threshold is one toward the back, leaving the space in front occupied by torture. The higher the perceived costs of failing to torture a knowledgeable Detainee who refuses to provide information, the more likely is the Interrogator to employ torture.

Note that this is true independent of the value of the suspected information possessed by the Detainee. Even if the information value is low, this threshold could still be high if the reputation costs are high because reputation costs are independent of whatever information a Detainee possesses.

Empirically, these costs are likely to be high for any state that tortures. If torture is to work as a threat, it must be credible. To make it credible, the state must torture if it does not receive valuable information. Failure to do so makes it less likely for future detainees to provide valuable information (or for the same detainee if we assumed multiple rounds of the game). This is why, for example, the Interrogator will torture a Resistant Detainee on whom she knows it has no effect. Interrogators who rely on torture must actually use it if they fail to get the desired information.

TORTURE COSTS, c and a

Consistent with intuition, the higher the costs to torturing, whether the general costs c or the extra cost a to “unnecessary” torture, both the information hiding ( and ) and innocent detainee recognition thresholds ( and ) go down, reducing the space supporting torture. Conversely, when these costs drop, when the state is less concerned about either cost of torture, both thresholds go up and torture is more likely.

These costs differ, of course, across states and in the same state over time. In the case of the United States, these costs were probably lowest shortly after 9/11, when Vice President Cheney spoke ominously about going to the “dark side” and CIA counterterrorism chief Cofer Black said the “gloves were off.” They may have increased over time, especially once the program started to be revealed and the United States began to suffer a different kind of (international) reputation cost. Nevertheless, these costs cannot be too high for any state that practices torture.

Note, though, that saying the costs are “low” here does not mean they are the same and close to zero. Since , it is always the case that the costs of torturing an innocent are higher than the reputation costs. So although it would appear that after 9/11, these costs may have lowered, they still exceed the reputation costs (which as we said were high).

This is appropriate, since our model is not a strawman and we are building in incentives against torture in order to let proponents make the best case that they can. It does mean, however, that we need to think of “low” differently for c than for a. It may very well be the case that c approaches zero; the costs of ‘necessary’ torture just were not very high. A low(er) cost for torturing innocents a, however, means that it approaches r so that they are not much more than the reputation costs. It will be helpful to keep these relationships in mind when we turn to the likely empirical values of the thresholds.

INFORMATION CLARITY, u

The final parameter for the Interrogator’s information hiding threshold is the clarity parameter u. As it increases, the value of the information is clearer to the Interrogator and the threshold is lowered, decreasing the space supporting torture. This also makes sense; the clearer the value of the information, the less likely the Interrogator is to misinterpret it as not valuable and use torture, thinking the Detainee is lying. Conversely, the more the Interrogator believes the information divulged by the Detainee is not valuable, the more likely the Interrogator is to employ torture. As u falls from 1 toward 0, the threshold grows, increasing the space supporting torture.

It is difficult to say in general whether u is likely to be low or high. No doubt it varies greatly depending not just on the specific information provided by the detainee, but also on the other information available to interrogators. Information from a detainee given one set of background information might make sense and be understood as valuable. The same information in the context of different background information, however, may make little sense and be perceived as lying or misdirection. It seems, then, prudent and conservative not to hazard a guess about the likely empirical value of u.

DETAINEE PROBABILITIES, p

Detainee probabilities—the likelihood that the Detainee is one of the three types—are parameters in two of the thresholds. The probability the Detainee is Cooperative (p_C) and the probability the Detainee is Innocent (p_I) are parameters in the leading questioning version of the information hiding threshold . The probability the Detainee is Resistant (p_R) is a parameter in the innocent detainee recognition threshold .

Whereas an increase in p_C increases , an increase in p_I lowers it. This too matches intuition. As the Interrogator believes that it is more likely the Detainee is Cooperative (and thus has information he would reveal or confirm), the space for torture increases. In contrast, as the Interrogator becomes convinced that the Detainee is in fact Innocent, the threshold drops, replacing that space supporting “torture” with space supporting “no torture.”

Here I think we can say something generally and with confidence, even if it is an empirical claim. It seems highly unlikely that an interrogator will think that the shivering, naked, hooded, shackled, sleep- and food-deprived detainee sitting across from her in the basement of a top-secret mini-prison is innocent. Imagine the two alternatives. Imagine you’re a professional CIA case officer dedicated to keep America safe and you’ve been specially selected and trained as an interrogator/debriefer. Your analyst(s) at the black site as well as back in Langley are providing you with information specific to that detainee, information spelling out all sorts of connections and evidence saying he’s a bad guy. Every day you read urgent cables from headquarters reiterating the importance of the information the detainee possesses and the need to elicit it pronto. How likely is it that you’ll think “no information” means “he’s innocent” or even “he’s a bad guy, but he doesn’t know that information”? Not very.

How about the probability he’s resistant and just cannot be broken? This is perhaps more likely than being innocent. After all, you know there are some tough terrorists out there, and that is far more plausible than a completely innocent man ending up in diapers, chains, and a hood in a secret CIA prison in an old equestrian academy on the outskirts of Vilnius, Lithuania. So p_R is likely higher than p_I.

Even so, interrogational torture generally and the CIA program in particular rest on the assumption, to quote the Hollywood film Zero Dark Thirty, “everybody breaks, bro” (Boal 2011, p. 6). This suggests that p_R should be low; resistance is ultimately futile and (almost) all detainees are effectively cooperative (i.e., p_C). After all, you’ve been trained in the techniques and told that they are scientifically engineered to induce “learned helplessness”—that is, to make a detainee completely dependent upon the interrogator so that he will comply with any request for information. You’ve even heard rumors whispered around the proverbial water-cooler back in Langley about officers who “broke” even the most resistant detainees (Carle 2011, p. 15). And, don’t forget, your job is to break the detainee sitting across from you. Your bosses, your colleagues, the American people are counting on you to break him.

How likely are you to think, “sorry, this is just one of those guys who can’t be broken”? Instead, even if others have been unsuccessful, you’re going to be the one to break him. Finally, note that the CIA’s own program assumed that all detainees could be broken. The agency did not have provisions for what to do with someone who never became compliant. Moreover, multiple apologists for the program have all maintained the same claim: Everyone broke eventually, just some faster than others (Bush 2011, Cheney and Cheney 2012, Rodriguez Jr and Harlow 2012, Tenet and Harlow 2007, Thiessen 2010).

In short, while it is an empirical claim and not a logical deduction, it seems very probable that interrogators will believe the Cooperative Detainee is the most likely type, followed by the far less likely Resistant type, and then, finally, the even less likely Innocent type (i.e., ), where the ≫ symbol means “much greater than.” This is a reasonable assumption. If so, if p_C is high and p_I low, then, all things being equal, the space supported by torture along this dimension is likely greater.

The final parameter is the probability that the Detainee is Resistant, p_R. As this probability increases, also increases and moves toward the back of the p axis. The space supporting torture increases. The more the Interrogator believes that the Detainee is the Resistant type and not the Innocent type, the more likely she is to torture. Note that this is true even though the Resistant types never divulge information—and the Interrogator knows this. The Interrogator must torture after no information in order to maintain the credibility of that threat even though she knows it will not work on this particular individual. As the probability the Detainee is Resistant decreases, the threshold also decreases, shrinking the space supporting torture because an Innocent Detainee is more likely.

Locating the Thresholds

With the knowledge of what happens to the thresholds as the parameters change as well as what the likely empirical values of (some of) those parameters are, we can apply a parallel strategy to the thresholds themselves. For each threshold, we first explore some of its formal mathematical properties. This will, in some cases at least, constrain the values the threshold can take. Second, we will use the empirically likely values of their constituent parameters, summarized in Table 7.3, to try and narrow the likely range of the thresholds a bit further. If we were to discover that a particular threshold is high (low), for mathematical and/or for empirical reasons, then we’ll have a better idea of the conditions necessary for that equilibrium to hold. This, in turn, gives us a sense of how likely—or rare—the equilibrium might be in the real world.

Table 7.3 LIKELY EMPIRICAL VALUES OF MODEL PARAMETERS

Parameter	Value
Reputation costs (r)	High
Torture costs (c)	Low
Additional torture costs (a)	Low
Information hiding (f)	High
Information clarity (u)	—
Probability Detainee is Innocent (pI)	Very low
Probability Detainee is Cooperative (pC)	High
Probability Detainee is Resistant (pR)	Low

INNOCENT DETAINEE RECOGNITION THRESHOLDS, ,

There are two Innocent Detainee recognition thresholds, one we encountered in the equilibrium we discovered together () in the last chapter and one after “no information,” covering both the “no information, torture” and “no information, no torture” equilibria, each under both objective and leading questioning (). Begin with the latter: .

Think about this fraction a bit, recalling that . Note that r is both in the numerator and the denominator; absent the other variables, the threshold would be 1. What makes it less than one and thus a fraction is that r is reduced by c in the numerator but is increased by a in the denominator. Actually, we can be a little more precise. Since , the denominator is at least doubled by adding a to r; even if a is just a bit greater than r, then the denominator is going to be plus some change. If the numerator is less than r and the denominator is greater than , then the threshold is less than one-half. We have, then, our first observation:

Observation 7.1 (). The Interrogator’s Innocent Detainee recognition threshold is less than one-half.

As we will see in a moment, Observation 7.1 will help us think about the other innocent detainee recognition threshold. It also, however, will help us think about two of the equilibria in the chapters to come. We now know that the threshold for will have to fall somewhere between zero and one-half. It is also important to point out that, given the assumptions of the model, this must be true. This statement does not rely on any contingent empirical assertions or claims or assumptions which may or may not turn out to be true or with which someone might not agree.

So we have narrowed down to between zero and somewhere below one-half. Can we say anything else? Can we narrow it any further? We can if we are willing to use the empirically likely values of the parameters. If we substitute in from Table 7.3 the likely empirical values of the parameters relevant for , we get something like .

Not terribly math-y,’tis true, but still illuminating. Recalling that the cost of torturing innocents a approaches r (but cannot drop lower than r), we can think of just doubling the denominator. If the cost of torturing innocents were very high, far exceeding r, then this denominator would get larger and the threshold would drop toward zero. But because those costs are actually quite low, the denominator is likely closer to . Since the permanent costs to torture c are very low, not much above zero, the numerator approaches r. Together then, these considerations suggest the following proposition:

Proposition 7.1 (). As c approaches zero and a approaches r, the Interrogator’s Innocent Detainee recognition threshold approaches one-half from below.

Now consider the other Innocent Detainee recognition threshold: . The numerator is the same as in , but the denominator is smaller (because ). Thus, absent p_R, would be greater than . Since p_R is a probability and is thus between zero and one, values of it below one reduce the numerator. The question is, how much?

Math alone will not help us here. With one very reasonable empirical assumption, however, it can. We said above that the Interrogator is likely to believe that any Detainee chained to a chair in her interrogation room is Cooperative—that is, a Detainee who both possesses information and will give it up under (the threat of [more]) torture. If that is the case, if the Interrogator believes that the Cooperative type is more likely than the other two, then we can state the following proposition:

Proposition 7.2 (). If the Pragmatic Interrogator believes it more likely that the Detainee she faces is Cooperative than both of the other two types, then the Innocent Detainee recognition threshold is less than one-half; the closer p_R is to zero, the closer is to zero.

The proof is in Appendix C, but the upshot is that a little bit of algebra shows that the greatest probability the Interrogator could assign to the Detainee being Innocent after observing “no information” is something a little under 1/2. The lower the prior probability p_R, the closer is to 0. The more the Interrogator believes everyone breaks—as is likely—the lower is . Both Innocent Detainee recognition thresholds are thus below one-half, with close to one-half and closer to zero. These constraints will become important when we compare the model results against the benchmarks in Chapter 12.

INFORMATION HIDING THRESHOLDS, , ,

There are three versions of the Interrogator’s information hiding thresholds, , , and . These are the version under objective questioning, the Detainee’s version of the Interrogator’s version under objective questioning, and the version under leading questioning, respectively. We take each in turn.

As we saw in Chapter 6, the Interrogator’s version under objective questioning is . Note that, without the u, this threshold is equivalent to and thus less than one-half. Since, however, u varies from just above zero to one, and this has a dramatic effect on , causing it to range from just above zero all the way to one (and beyond), we cannot constrain on purely mathematical grounds. Given the wide variability of u in the real world of torture (sometimes it will be high and sometimes it will be low), it is difficult to narrow u on empirical grounds as well.

If, however, we continue to assume c approaching zero and a approaching r, we can place at one-half or greater, giving us the following proposition:

Proposition 7.3 (). As c approaches zero and a approaches r, the Interrogator’s information hiding threshold is greater than or equal to one-half.

Now take a look at the Detainee’s version of the Interrogator’s information hiding threshold: . The Detainee’s version is in fact equivalent to since it does not have the u parameter. As a result, we can make the same observation we did with , giving us an observation constraining another threshold:

Observation 7.2 (). The Detainee’s version of the Interrogator’s information hiding threshold approaches one-half from below.

Thus, the Detainee thinks that the Interrogator’s information threshold is somewhere between zero and one-half, but can’t exceed one-half. Moreover, we can narrow this remaining range a bit more exactly as we did with above since the equations are identical. That is, given a very small c and an a just above r, is likely to be close to one-half.

Now compare the two versions, and . Suppose u is 1. Then the two thresholds—the Interrogator’s actual threshold and what the Detainee thinks it is —are equal. This is the only value of u for which the thresholds are the same, for which there is no misunderstanding between them. As u drops from 1, it makes the denominator in smaller, which makes the fraction larger. We can make, then, the following observation:

Observation 7.3 (). The Interrogator’s and the Detainee’s beliefs will “agree” on the information hiding threshold only in the special case when the Interrogator understands perfectly the information’s value; in all other cases the Detainee’s version is less than the Interrogator’s.³

Finally, consider the Interrogator’s version of the information hiding threshold under leading questioning: . This version is, again technically speaking, a bit goopier than the others. Even so, some algebra establishes that, if the prior probability of a Cooperative Detainee is more likely than that of an Innocent Detainee () as we argued above, is also less than one-half.

The details are in Appendix C, but the basic idea is this: Suppose is not less than one-half—that is, that it is equal to or greater than one-half. If so, then if you doubled the fraction, it would be equal to or greater than one. When you do that, you find that it contradicts the assumption that and so it cannot be true. Thus, .⁴

If we continue to assume a very low prior probability that the Detainee is Innocent (i.e., p_I approaching zero but not getting there), a torture cost c very low (also approaching zero), and a very low “unnecessary” torture cost a (a approaching but always greater than r), we can constrain a bit further. The left-hand term in the numerator will approach while the right-hand term in the numerator will approach . A very low p_I will reduce , making the numerator close to . With the denominator approaching , approaches one-half. In short:

Proposition 7.4 (). As both p_I and c approach zero and as a approaches r, the Interrogator’s information hiding threshold approaches one-half from below.

By comparing to , we can say even more. If we assume that there is some chance that a Detainee could be Innocent, even the smallest probability as long as it’s positive, then yet more algebra shows that . The assumption that the Detainee could be Innocent rather than Cooperative or Resistant motivated altering the BIT model and building the RIT model with this type included, so this should not be problematic and provides us with the following proposition:

Proposition 7.5 (). If there is a positive probability that the Detainee is Innocent, the Interrogator’s information hiding threshold under leading questioning is less than the Detainee’s version under objective questioning: for , .

It follows that . In other words, the Interrogator’s information hiding threshold is less under leading questioning than under objective questioning. This makes sense. If one is asking leading questions, there is no uncertainty about the value of the information (that’s why there is no u in the payoffs under leading questioning). And if the detainee is talking, then it is more likely that you’ll get the information you want (since you’re providing him with the answers to your questions). In contrast, under objective questioning, getting the answers you want may take longer and will be less likely. It also follows that a lower information hiding threshold under leading questioning also means that there should be less torture than under objective questioning, where the higher threshold shrinks the region supporting “no torture.”

This relationship between questioning type, information, and torture has an important implication:

Implication 7.1 (Torture–Information Trade-off 1). There is a trade-off between information and torture across questioning types. All else being equal, leading questioning results in less torture and poorer information; objective questioning results in more torture and better information.

From Proposition 7.5, . Since the former is the Interrogator’s threshold for torture under leading questioning and the latter her threshold under objective questioning, it follows that, all else being equal, the region supporting torture is smaller under leading questioning, which generates ambiguous information and false confirmation, than is the region under objective questioning, which can generate better information. While objective questioning (potentially) provides better information, it is necessarily accompanied by more torture than leading questioning, which, however, provides—at best—ambiguous information.

Before leaving f as a threshold, we should note that f is a parameter as well. If you recall the discussion in Chapter 5 and the RIT game tree diagram (Figure 5.4), you’ll remember that f is the Interrogator’s belief (the probability) that the Detainee has revealed all his information (or said as much as he is willing under leading questioning). As we worked through what became the valuable information, surprise torture equilibrium in the Chapter 6, we solved for f at one point because that helped us think about what the Interrogator needed to assess.

We could have, however, solved for u—the probability the Interrogator understands the information as valuable. Had we done so, we would have ended up with . Compare this to , and you can see that they are equivalent, just with f and u switched. While we found it difficult to posit any empirical likelihoods about u, the same is not true for f. As suggested in the section on information hiding in Chapter 5 and elsewhere, it is an interrogator’s entire job to be suspicious about whether detainees have told them all they know. They know that a cooperative detainee has an incentive to give away as little as possible. The entire logic of torture in interrogations, then, suggests that f is likely to be low.

We also know a low f to be true not just logically, but empirically as well. The history of torture offers more than enough empirical support to suggest that the interrogator’s prior belief f is likely to be low.⁵ Since f behaves mathematically just like u and we saw above that as u decreases, the threshold increases, this means that the threshold would behave the same way for low values of f. If we were to relabel the parameter space in Figure 7.1, replacing f on the bottom axis with u, then we would see the threshold closer to the right, with a wide range supporting torture to its left. Obviously, as it moves past one-half, where the Detainee threshold reaches its maximum, space opens up for the valuable information, surprise torture equilibrium.

INFORMATION REVELATION THRESHOLDS, ,

The last two thresholds and are those of the Cooperative and the Innocent Detainees, respectively, and . Since they have the same functional form and are subject to the same constraint (both are positive and fractions), they behave identically. In such a simple case there is nothing more to be said mathematically beyond the fact that these thresholds increase as v (l) increases and decrease as k increases. Unfortunately, it is not clear we can say much empirically.

One might think that a knowledgeable but potentially cooperative detainee would have a higher threshold. If they’re professionals, they’ll be suspicious of interrogators and unlikely to believe their promises not to torture. This is all the more true of modern terrorists in the Middle East, who are very well aware of the horrific tortures they would face in the prisons of Damascus and Cairo. Moreover, they actually may have trained to resist this torture. Clearly the CIA thought that modern Islamic terrorists are a new breed of hardened terrorists trained and committed to resisting interrogations.

And yet, some prisoners clearly do end up believing their interrogators and provide (some) information. Moreover, this threshold depends on the value of the information. Recalling that increases as v gets larger and decreases as k gets larger, it follows that less valuable information is more likely to be divulged (the threshold is lower for low v), whereas more valuable information is less likely to be given up (the threshold is higher). Of course, the threshold’s position also depends on the value of k, which varies across even cooperative detainees even for the same torture technique. In short, the empirical ground is too shaky to constrain the Cooperative Detainee’s threshold .

Before turning to the Innocent Detainee’s threshold, note something significant about what we just said about the value of information v and its effect on the information obtained through torture. A Cooperative Detainee is more likely to give up less valuable information than valuable information. This may seem trivially true and obvious but it is important to realize that it emerges logically from the model. We get this result—along with all the other results—from one consistent logical framework. Many authors from classical antiquity forward have spoken about the various things that can happen under torture, but they do not start with one logical framework and generate all of those outcomes. This relationship suggests the following implication:

Implication 7.2 (Torture–Information Trade-off 2). There is a trade-off between information and torture irrespective of questioning type. All things being equal, interrogators are more likely to get less valuable information than highly valuable information from torture.

Of course this is true of interrogations generally. The point is that torture does not do any better.

As far as Innocent Detainees are concerned, one might think the threshold would be very low. How and why would an innocent, non-terrorist resist and try to hold out under torture rather than tell the interrogator what she wants to hear? He might be more likely to believe the interrogator’s promises too, not having been in such a situation before. And yet, the sad and grim history of torture is replete with stories of innocents who resisted tendering false confessions despite the lure of being lowered from the rack. In short, there is too much variability here as well to constrain the location of this threshold.

Table 7.4 summarizes the likely values of those thresholds about which we can say something theoretically, empirically, or both. The squiggly lines under the inequality signs mean “approaches,” and those with the line through it mean “not approaches.” So, for example, means “greater than but approaching one-half” while means “less than and not approaching one-half.”

Table 7.4 LIKELY VALUES OF MAJOR THRESHOLDS

Given both the density of this chapter and the importance of these values in the rest of the argument, it will be useful to summarize the empirical assumptions we have made along the way which support some of these values:

1. The Interrogator’s torture costs c are very low, approaching zero.

2. The Interrogator’s additional costs to “unnecessary” torture a are also very low, approaching their minimum just above r.

3. The probability of an Innocent Detainee is positive (i.e., some amount, even tiny, above zero), but the Cooperative Detainee is the most probable type, followed (far behind) by the Resistant and then Innocent types ().

Once again, given the history of torture, these appear to be very reasonable assumptions, but it’s important to put them up front and be clear about them.

Figure 7.2 locates these thresholds in the equilibrium parameter space “cube.” All the equilibria emerging from the model will inhabit some portion of this space, their boundaries defined by different combinations of the thresholds. The placement of the thresholds and thus the size of the equilibria corresponds to their likely values, as just summarized. I have arbitrarily placed the Cooperative Detainee’s threshold a little north of halfway for the moment, but we will relax this assumption later and nothing rests on this placement.

Figure 7.2 Equilibria Parameter Space with Thresholds

The parameter space will be helpful to us in three ways. First, it aids in understanding the equilibria. Instead of a collection of inequalities—“less than this, more than that, less than this”—we have readily identifiable regions standing in relation to one another. Second, since we now know what determines the values of the thresholds, what causes them to move up or down, we can use this visualization to identify the factors supporting, and not supporting, each equilibrium, helping us interpret each of the narrative case studies. Finally and perhaps most importantly, by locating the equilibria within the entire parameter space, we can assess not just their positions relative to each other, but also the proportion occupied by each equilibrium relative to the parameter space as a whole. In other words we will have a ready way to identify the region(s) predicting both torture and elicitation of information. This will be helpful in evaluating the proponent model’s predictions and normative benchmarks about the reliability of information as well as about the frequency and severity of torture in interrogational torture programs.

The important point for the moment is that we draw on both the values in Table 7.4 and the threshold locations in Figure 7.2 as we proceed through each of the equilibria in the next four chapters. For each of the equilibria, then, we will explore how changes in the thresholds affect the nature of the equilibrium, providing us a better understanding of how they connect to the real world. We will also connect them to the real world by illustrating each equilibrium with a case study of torture corresponding to that equilibrium. We begin with surprise torture and the case of an English doctor in Pinochet’s Chile: Sheila Cassidy.

The argument thus far has shown that:

1. EITs are torture and the effectiveness of interrogational torture is an open question. (Chapter 2)

2. The Bush program approximates closely the ideal model of interrogational torture and includes limits on torture; the Bush and ideal models provide benchmarks for comparison with the game theory models to come. (Chapter 3)

3. The Bush model generates strange, quixotic outcomes. (Chapter 4)

4. The Bush interrogational torture program is more realistically modeled as objective and leading question variants of an incomplete information game, with three types of detainees, two types of interrogators, and uncertainty about the amount and value of information provided. (Chapter 5)

5. By positing a set of Detainee strategies, calculating the Interrogator’s expected utility using Bayes’ Rule to identify her best response, and checking for incentives to deviate by any of the Detainee types, it is possible to derive a perfect Bayesian equilibrium in which a Detainee is tortured after providing information. (Chapter 6)

6. The RIT model generates nine perfect Bayesian equilibria, the formal and empirical characteristics of which generate important observations, propositions, and implications, including:

(a) The Interrogator’s thresholds for believing that a Detainee is Innocent after “no information” are less than one-half, with close to one-half and closer to zero.

(b) The Interrogator’s information hiding threshold under objective questioning is greater than or equal to one-half, whereas her information hiding threshold under leading questioning as well as the Detainee’s version of are a little less than one-half.

(c) Objective questioning (potentially) provides better information, but is necessarily accompanied by more torture, than leading questioning, which, however, provides less valuable information.

(d) All things being equal, Interrogators are more likely to get less valuable information than highly valuable information. (Chapter 7)

The next step in the argument is to describe and interpret each of the RIT outcomes, starting with the valuable information, surprise torture equilibrium.