APPENDICES

These appendices contain formal descriptions and arguments associated with the RIT model in Chapters 5 through 12, including proofs of the equilibria (A), comparative statics analyses of the thresholds (B), and relevant observations and proofs of propositions (C).

APPENDIX A

The RIT Model

This appendix contains the formal description of the RIT game and proofs of the pure strategy perfect Bayesian equilibria.

A.1 THE GAME

The game, reproduced as Figure A.1, begins with two independent moves by Nature. The first move selects the Detainee’s type, , from the space , , with the common prior probability distribution , , and , where , is the probability the Detainee is type j, and . Nature’s second move selects the Interrogator’s type, , from the space , with the common prior probability distribution and , where , is the probability the Interrogator is type k, and .

Figure A.1 Realistic Interrogational Torture Game (RIT)

The Interrogator can engage in two kinds of questioning: objective or leading. Under objective questioning, the Interrogator does not tell the Detainee what she wants to hear. Under leading questioning, the Interrogator does let the Detainee know what would please her. In the leading questioning version, then, each chooses a strategy from , where i is reveal valuable information (“Information” in Figure A.1) and is not reveal valuable information (“~Information” in Figure A.1). Move is equivalent to keeping silent as well as providing information which is not valuable.

Under objective questioning, when the Interrogator does not reveal what she wants to hear, has move only. Strategies for are given as indicating that chooses , chooses , and chooses .

Following ’s move, each Interrogator type chooses to torture (t) or not torture () from , with () denoting that chooses when it observes i and chooses when it observes and likewise for with ().

Let denote ’s beliefs about the Detainee type y at her x information set, i.e., . As examples, is the Interrogator’s updated belief that the Detainee is Cooperative after observing “information” and is the Interrogator’s updated belief that the Detainee is Innocent after observing “no information.”

Both the Cooperative and Resistant Detainees pay costs for i and receive a payoff of 0 for . They also suffer costs if they are tortured by the Interrogator and receive a payoff of 0 for no torture. The preference orderings for each are: and . Since, as we shall see, is the Resistant Detainee’s dominant strategy, the threshold pertains to the Cooperative Detainee only and so it is unnecessary to index the costs k to each type. The Innocent Detainee’s payoff ordering is identical to that of the Cooperative Detainee, with l taking the place of v for the cost of i.

Both Interrogator types pay a cost , if they fail to torture after move from a knowledgeable (Cooperative or Resistant) Detainee and 0 for not torturing after move from an Innocent Detainee. bears a cost , for torturing any D_i and an additional cost , (with ), for “unnecessary” torture of an Innocent Detainee who chooses (i.e., tells the truth) or of any Detainee who chooses i. In contrast, receives a benefit to torture after any move by .

Both Interrogator types receive a payoff of V for a Cooperative Detainee’s move i under objective questioning that provides all the information they have to the Interrogator; for fractions less than full information, the Interrogators receive a payoff of . Since the value of i is uncertain, the Interrogators have only the common prior belief that i provides V with probability f and with probability , with .

In the objective questioning variant of the model, i is perceived by as i with probability u and is perceived as the nonvaluable with probability , . This uncertainty is ’s private information; the Detainee assumes that the Interrogator recognizes i as valuable () and plays accordingly. assumes that the prior belief u is common knowledge and plays accordingly. Three points of clarification are in order here. First, the Interrogator’s perception (with probability ) of the information as nonvaluable does not change her information set. Although her payoffs are the same as those of the information set ( after torture and after no torture), she knows she is receiving some type of information from a Cooperative Detainee. She must, however, decide whether or not to torture prior to fully understanding the information’s value. Second, the uncertainty captured by u occurs under objective questioning only—there is no uncertainty over the value of information under leading questioning. Third, the Interrogator’s belief about whether i is valuable (u) is independent of the Interrogator’s belief about whether the Detainee is hiding information (f).

A.2 PROOFS OF EQUILIBRIA

This section contains the proofs and formal statements of the equilibria discussed in Chapter 8 and beyond. I solve for pure strategy perfect Bayesian equilibria. I make the following knifepoint assumptions to rule out indifference between strategy choices for and : If payoff-indifferent between choosing i and , and prefer i; if payoff-indifferent between t and , prefers .

A.2.1 Objective Questioning

Under objective questioning, ’s payoffs after i are weighted by u, but any playing i believes . Since dominates i for , and only has move under objective questioning, there are only two pure strategies to consider, (i,,) and (,,).

A.2.1.1

Suppose plays the strategy ; using Bayes’ Theorem, ’s beliefs at the i information set are , , and at the information set are , , . Given these beliefs, the expected utility of t at the i information set is . The expected utility of at the i information set is . therefore prefers to torture after i for

(A.1)

Solving for f, we obtain

(A.2)

These are the information recognition and information hiding thresholds, respectively. Recalling the Detainee’s assumption that any i is recognized with certainty (), it will be useful to define the Detainee’s belief about the Interrogator’s information hiding threshold as

(A.3)

’s expected utility for t at her information set is . Her expected utility for after is . therefore plays t after for

(A.4)

This is an innocent detainee recognition threshold. By simple inspection of equations (A.2) and (A.3), it is clear that for all . Equations (A.2), (A.3), and (A.4) thus define six subcases.

A.2.1.1.1 and

For this combination of beliefs, plays (). always prefers torture to not torture. It remains to check whether () is ’s best response to these choices. The strategy dominates i for ; and under objective questioning, is ’s only strategy so they will not deviate. Because , would anticipate ’s response of t after i, providing with an incentive to switch to . Consequently, this set of strategies and beliefs cannot constitute a PBE.

A.2.1.1.2 and

For this combination of beliefs, plays (). always prefers torture to not torture. It remains to check whether () is ’s best response to these choices. The strategy dominates i for and under objective questioning is ’s only strategy so they will not deviate. Because believes that , he believes that plays rather than t after i. For , the expected utility of i is , or and the expected utility of is or . Thus, prefers i to for , or

(A.5)

This is the Cooperative Detainee’s information revelation threshold. With no incentive to deviate to , the strategy profile ; (), (): , for constitutes a PBE. This is the valuable information, surprise torture equilibrium.

A.2.1.1.3 and

For this combination of beliefs, chooses () and chooses (). It remains to check whether () is ’s best response to these choices. From equation (A.5), prefers i to for . The strategy dominates i for ; and under objective questioning, is ’s only strategy so they will not deviate. Thus, the strategy profile ; (), (): for constitutes a PBE. This is a valuable information, selective torture equilibrium.

A.2.1.1.4 and

For this combination of beliefs, plays (). always prefers torture to not torture. It remains to check whether () is ’s best response to these choices. The strategy dominates i for ; and under objective questioning, is ’s only strategy so they will not deviate. Because , would anticipate ’s response of t after i. Since plays after , has an incentive to deviate to and so this strategy profile and belief combination cannot be part of a PBE.

A.2.1.1.5 and

For this combination of beliefs, plays (). always prefers torture to not torture. It remains to check whether () is ’s best response to these choices. The strategy dominates i for ; and under objective questioning, is ’s only strategy, so they will not deviate. Because believes that , he believes plays rather than t after . nevertheless has an incentive to deviate because plays after , making preferable to i for any q and preventing this strategy profile and combination of beliefs from constituting a PBE.

A.2.1.1.6 and

For this combination of beliefs, plays (). always prefers torture to not torture. Since plays after , has an incentive to deviate to and so this strategy profile and belief combination cannot be part of a PBE.

A.2.1.2

Suppose plays the strategy (); using Bayes’ Theorem, ’s beliefs at the information set are , , and . Given these beliefs, ’s expected utility from t after is . Her expected utility from after is . Thus plays t after for

(A.6)

This is the other innocent detainee recognition threshold, providing two cases.

A.2.1.2.1

For this set of beliefs, plays t; chooses the dominant strategy t. It remains to check whether () is ’s best response to these choices. The strategy dominates i for and under objective questioning, is ’s only strategy, so they will not deviate. Under objective questioning, only can play i, so, applying the Intuitive Criterion, (Cho and Kreps 1987). This is identical to Case A.2.1.1 above, so the expected utility of t and are given by and , respectively. From equation (A.2), therefore prefers to torture after i if its off-path beliefs satisfy

(A.2)

Further, for this off-path move to prevent ’s deviation, must believe that will play t after i—that is, . Thus, the strategy profile ); (), (): ( or and ); ( for and is a PBE. This is the no information, torture equilibrium.

A.2.1.2.2

For this set of beliefs, plays after ; chooses the dominant strategy (). It remains to check whether () is ’s best response to these choices. No can do better, and so the strategy profile for and is a PBE. This is the no information, no torture equilibrium.

A.2.2 Leading Questioning

In this case the Interrogator’s approach is leading questioning, causing u to drop out of ’s payoffs and making strategy i now available to . Because continues to dominate i for , there are four pure strategies to consider: , , , and .

A.2.2.1

Suppose plays the strategy (); using Bayes’ Theorem, ’s beliefs at the i information set are , , and at the information set are , . Given these beliefs, ’s expected utility for t after i is . The expected utility for is . therefore plays t after i for

(A.7)

This is the information hiding threshold under leading questioning. ’s expected utilities after are for t and for , so plays t after . There are thus two cases based on .

A.2.2.1.1

For this set of beliefs, plays (). always prefers torture to not torture. It remains to check whether () is ’s best response to these choices. The strategy dominates i for . Both and , however, can do better by switching to for any q, and this combination of beliefs and strategies cannot be part of a PBE.

A.2.2.1.2

For this set of beliefs, plays . always prefers torture to not torture. It remains to check whether () is ’s best response to these choices. The strategy dominates i for . From equation (A.5) earlier, we know that prefers i to for . For , the expected utility of i is and the expected utility of is . Thus, prefers i to for

(A.8)

This is the innocent detainee’s information revelation threshold. Thus, the strategy profile and ; for , , and is a PBE. This is the ambiguous information, selective torture equilibrium.

A.2.2.2

Suppose plays the strategy (); using Bayes’ Theorem, ’s beliefs at the i information set are and at the information set are . Given these beliefs, ’s expected utility for t after i is and his expected utility for is V.

’s expected utility for t after is and his expected utility for is , so chooses (). chooses (). It remains to check whether is ’s best response to these choices. From equation (A.5), prefers to i when i is not pivotal to avoid torture, which happens when and when and . The strategy dominates i for . From case A.2.2.1.2, prefers i to for .

Thus, the strategy profile , and , and for , , and constitutes a PBE. This is a false confirmation, selective torture equilibrium.

A.2.2.3

This set of strategies on the part of is identical to case A.2.1.1, where had move only. Therefore, ’s beliefs at the i information set are and at the information set are .

Recalling that u drops from ’s payoffs under leading questioning, the expected utility of t at the i information set is . The expected utility of at the i information set is . Identical to equation (A.3) above, therefore prefers to torture after i if

(A.3)

It likewise follows from case A.2.1.1 that ’s expected utility for t at her information set is and her expected utility for after is and so, from equation (A.4), I_P plays t after for

(A.4)

This defines four subcases.

A.2.2.3.1 and

For this combination of beliefs, chooses () and chooses (). It remains to check whether is ’s best response to these choices. Since plays t after i, has an incentive to deviate to , and this strategy profile and belief combination cannot be part of a PBE.

A.2.2.3.2 and

For this combination of beliefs, chooses () and chooses (). It remains to check whether is ’s best response to these choices. From equation (A.5), prefers i to for . Strategy dominates i for . From equation (A.8), prefers to i for .

Thus, the strategy profile ; , for and constitutes a PBE. This is a valuable information, selective torture equilibrium.

A.2.2.3.3 and

For this combination of beliefs, chooses () and chooses (). But since plays after , has an incentive to switch to , and this strategy profile and set of beliefs cannot be part of a PBE.

A.2.2.3.4 and

chooses () and chooses (). Again, since plays after , has an incentive to switch to , and this strategy profile and set of beliefs cannot be part of a PBE.

A.2.2.4

Once again, this strategy profile is identical to its counterpart under objective questioning in A.2.1.2, but, given that now has move i in addition to move , it is necessary to check whether would deviate in each of the two subcases of A.2.1.2 defined by equation (A.6), .

A.2.2.4.1

For this set of beliefs, plays t; chooses the dominant strategy t. It remains to check whether is the best response for both and under leading questioning. By equation (A.5), prefers to i for and thus will not deviate; the same is true for for .

For , expects to play t after i and so will not deviate to i even for . For and , however, expects to play after i and thus has an incentive to deviate to i. also has an incentive to deviate for .

To prevent deviation to i by and , would have to play t after i. Since under leading questioning, both and can choose i but never does so, let be ’s off-path belief that the Detainee is , and be ’s off-path belief that the Detainee is , upon observing i.

The expected utility of t is . The expected utility of is . therefore prefers to torture after i for off-path beliefs satisfying

(A.9)

This off-path belief is a real constraint (i.e., ) for

(A.6)

Thus, the strategy profile and ) or ( and and ) or ( and and ) or ( and and ) with for and is a PBE. This is the no information, torture equilibrium.

A.2.2.4.2

For this set of beliefs, plays ; chooses the dominant strategy (). It remains to check whether is ’s best response to these choices. No can do better and so the strategy profile for is a PBE. This is the no information, no torture equilibrium.