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Does Torture Work?$

John W. Schiemann

Print publication date: 2015

Print ISBN-13: 9780190262365

Published to Oxford Scholarship Online: November 2015

DOI: 10.1093/acprof:oso/9780190262365.001.0001

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(p.255) Appendix A The RIT Model

(p.255) Appendix A The RIT Model

Source:
Does Torture Work?
Author(s):

John W. Schiemann

Publisher:
Oxford University Press

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, Dj, from the space {Cooperative,Resistant,Innocent}, {DC,DR,DI}, with the common prior probability distribution pC, pR, and pI, where pj, is the probability the Detainee is type j, and pC+pR+pI=1. Nature’s second move selects the Interrogator’s type, Ik, from the space {Pragmatic,Sadistic}, {IP,IS} with the common prior probability distribution qP and qS, where qk, is the probability the Interrogator is type k, and qP+qS=1.

Appendix A The RIT Model

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 Dj chooses a strategy from {i,iˉ}, where i is reveal valuable information (“Information” in Figure A.1) and iˉ is not reveal valuable information (“~Information” in Figure A.1). Move iˉ 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, DI has move iˉ only. Strategies for Dj are given as (α1,α2,α3) indicating that DC chooses (α1), DR chooses (α2), and DI chooses (α3).

Following Dj’s move, each Interrogator type Ik chooses to torture (t) or not torture (tˉ) from {t,tˉ}, with (β1,β2) denoting that IP chooses β1 when it observes i and chooses β2 when it observes iˉ and likewise for IS with (γ1,γ2).

(p.256) Let μx,y denote Ik’s beliefs about the Detainee type y at her x information set, i.e., (x,y){i,iˉ}×{C,R,I}. As examples, μi,C is the Interrogator’s updated belief that the Detainee is Cooperative after observing “information” and μiˉ,I is the Interrogator’s updated belief that the Detainee is Innocent after observing “no information.”

Both the Cooperative and Resistant Detainees pay costs v,v>0 for i and receive a payoff of 0 for iˉ. They also suffer costs k,k>0 if they are tortured by the Interrogator and receive a payoff of 0 for no torture. The preference orderings for each are: DC=0>v>k>vk and DR=0>k>v>vk. Since, as we shall see, iˉ is the Resistant Detainee’s dominant strategy, the vk 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.

(p.257) Both Interrogator types pay a cost r, r>0 if they fail to torture after move iˉ from a knowledgeable (Cooperative or Resistant) Detainee and 0 for not torturing after move iˉ from an Innocent Detainee. IP bears a cost c, c>0 for torturing any Di and an additional cost a, a>0 (with c>r>a), for “unnecessary” torture of an Innocent Detainee who chooses iˉ (i.e., tells the truth) or of any Detainee who chooses i. In contrast, IS receives a benefit s,s>0 to torture after any move by Di.

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 Vh. Since the value of i is uncertain, the Interrogators have only the common prior belief that i provides V with probability f and Vh with probability 1f, with f(0,1).

In the objective questioning variant of the model, i is perceived by IP as i with probability u and is perceived as the nonvaluable iˉ with probability 1u, u(0,1]. This uncertainty is IP’s private information; the Detainee assumes that the Interrogator recognizes i as valuable (u=1) and plays accordingly. IP 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 1u) of the information as nonvaluable does not change her information set. Although her payoffs are the same as those of the iˉ information set (c after torture and r 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 Dj and IP: If payoff-indifferent between choosing i and iˉ, DC and DI prefer i; if payoff-indifferent between t and tˉ, IP prefers tˉ.

A.2.1 Objective Questioning

Under objective questioning, IP’s payoffs after i are weighted by u, u(0,1] but any Di playing i believes u=1. Since iˉ dominates i for DR, and DI only (p.258) has move iˉ under objective questioning, there are only two pure strategies to consider, (i,iˉ,iˉ) and (iˉ,iˉ,iˉ).

A.2.1.1 {i,iˉ,iˉ}

Suppose Di plays the strategy (i,iˉ,iˉ); using Bayes’ Theorem, IP’s beliefs at the i information set are μi,C=1, μi,R=0, μi,I=0 and at the iˉ information set are μiˉ,C=0, μiˉ,R=pRpR+pI, μiˉ,I=pIpR+pI. Given these beliefs, the expected utility of t at the i information set is uVuhufa+ufhc. The expected utility of tˉ at the i information set is uVuh+ufh+ufrr. IP therefore prefers to torture after i for

(A.1)
u<rcf(r+a)uˆ.

Solving for f, we obtain

(A.2)
f<rcu(r+a)fˆ.

These are the information recognition and information hiding thresholds, respectively. Recalling the Detainee’s assumption that any i is recognized with certainty (u=1), it will be useful to define the Detainee’s belief about the Interrogator’s information hiding threshold as

(A.3)
f<rcr+af.

IP’s expected utility for t at her iˉ information set is cpIpR+pI(a). Her expected utility for tˉ after iˉ is pRpR+pI(r). IP therefore plays t after iˉ for

(A.4)
pI<rcc+apRp.

This is an innocent detainee recognition threshold. By simple inspection of equations (A.2) and (A.3), it is clear that ffˆ for all u,u(0,1]. Equations (A.2), (A.3), and (A.4) thus define six subcases.

A.2.1.1.1 f<ffˆ and p<p

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

(p.259) A.2.1.1.2 f<f<fˆ and p<p

For this combination of beliefs, IP plays (t,t). IS always prefers torture to not torture. It remains to check whether (i,iˉ,iˉ) is Di’s best response to these choices. The strategy iˉ dominates i for DR and under objective questioning iˉ is DI’s only strategy so they will not deviate. Because DC believes that f<f, he believes that IP plays tˉ rather than t after i. For DC, the expected utility of i is q(v)+(1q)(vk)qvvk+qv+qk, or qkvk and the expected utility of iˉ is kq+(1q)k or k. Thus, DC prefers i to iˉ for qkvkkqkv, or

(A.5)
qvkqˆ.

This is the Cooperative Detainee’s information revelation threshold. With no incentive to deviate to iˉ, the strategy profile {i,iˉ,iˉ}; (t,t), (t,t): qqˆ, f<f<fˆ;(μi,μiˉ)} for μi,C=1,μiˉ,I=p<p constitutes a PBE. This is the valuable information, surprise torture equilibrium.

A.2.1.1.3 ffˆ<f and p<p

For this combination of beliefs, IP chooses (tˉ,t) and IS chooses (t,t). It remains to check whether (i,iˉ,iˉ) is Di’s best response to these choices. From equation (A.5), DC prefers i to iˉ for qqˆ. The strategy iˉ dominates i for DR; and under objective questioning, iˉ is DI’s only strategy so they will not deviate. Thus, the strategy profile {i,iˉ,iˉ}; (tˉ,t), (t,t): qqˆ,ffˆ<f;(μi,μiˉ)} for μi,C=1,μiˉ,I=p<p constitutes a PBE. This is a valuable information, selective torture equilibrium.

A.2.1.1.4 f<f<fˆ and pp

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

A.2.1.1.5 f<f<fˆ and pp

For this combination of beliefs, IP plays (t,tˉ). IS always prefers torture to not torture. It remains to check whether (i,iˉ,iˉ) is Di’s best response to these choices. The strategy iˉ dominates i for DR; and under objective questioning, iˉ is DI’s only strategy, so they will not deviate. Because DC believes that f<f, he believes IP plays tˉ rather than t after iˉ. DC nevertheless has an incentive to deviate because IP (p.260) plays tˉ after iˉ, making iˉ preferable to i for any q and preventing this strategy profile and combination of beliefs from constituting a PBE.

A.2.1.1.6 ffˆ<f and pp

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

A.2.1.2 {iˉ,iˉ,iˉ}

Suppose Di plays the strategy (iˉ,iˉ,iˉ); using Bayes’ Theorem, IP’s beliefs at the iˉ information set are pC, pR, and pI. Given these beliefs, IP’s expected utility from t after iˉ is cpIa. Her expected utility from tˉ after iˉ is r(pC+pR). Thus IP plays t after iˉ for

(A.6)
pI<rcr+apˆ.

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

A.2.1.2.1 p<pˆ

For this set of IP beliefs, IP plays t; IS chooses the dominant strategy t. It remains to check whether (iˉ,iˉ,iˉ) is Di’s best response to these choices. The strategy iˉ dominates i for DR; and under objective questioning, iˉ is DI’s only strategy, so they will not deviate. Under objective questioning, only DC can play i, so, applying the Intuitive Criterion, μi,C=1 (Cho and Kreps 1987). This is identical to Case A.2.1.1 above, so the expected utility of t and tˉ are given by uVuhufa+ufhc and uVuh+ufh+ufrr, respectively. From equation (A.2), IP therefore prefers to torture after i if its off-path beliefs satisfy

(A.2)
f<rcu(r+a)f^

Further, for this off-path move to prevent DC’s deviation, DC must believe that IP will play t after i—that is, f<ffˆ. Thus, the strategy profile {(iˉ,iˉ,iˉ); (t,t), (t,t): (q<qˆ or qqˆ and f<ffˆ); (μi,μiˉ)} for μi,C=1 and μiˉ,I=p<pˆ is a PBE. This is the no information, torture equilibrium.

A.2.1.2.2 ppˆ

For this set of IP beliefs, IP plays tˉ after iˉ; IS chooses the dominant strategy (t,t). It remains to check whether (iˉ,iˉ,iˉ) is Di’s best response to these choices. No Di can do better, and so the strategy profile (iˉ,iˉ,iˉ);(β1,tˉ),(t,t):(q(0,1));μi,μiˉ for μi=0 and μiˉ,I=ppˆ is a PBE. This is the no information, no torture equilibrium.

(p.261) A.2.2 Leading Questioning

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

(p.262) A.2.2.1 {i,iˉ,i}

Suppose Di plays the strategy (i,iˉ,i); using Bayes’ Theorem, IP’s beliefs at the i information set are μi,C=pCpC+pI, μi,R=0, μi,I=pIpC+pI and at the iˉ information set are μiˉ,C=0, μiˉ,R=1,μiˉ,I=0. Given these beliefs, IP’s expected utility for t after i is V+pCcpChpCfa+pCfhpIcpIapC+pI. The expected utility for tˉ is V+pChpCr+pCfh+pCfrpC+pI. IP therefore plays t after i for

(A.7)
f<pC(rc)+pI(ac)pC(r+a)f˜.

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

A.2.2.1.1 f<f˜

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

A.2.2.1.2 f>f˜

For this set of beliefs, IP plays (tˉ,t). IS always prefers torture to not torture. It remains to check whether (i,iˉ,i) is Di’s best response to these choices. The strategy iˉ dominates i for DR. From equation (A.5) earlier, we know that DC prefers i to iˉ for qqˆ. For DI, the expected utility of i is qklk and the expected utility of iˉ is k. Thus, DI prefers i to iˉ for

(A.8)
q>lkq.

This is the innocent detainee’s information revelation threshold. Thus, the strategy profile {(i,iˉ,i),(tˉ,t),(t,t):qqˆ and qq;f>f˜; (μi,μiˉ)} for μi,C=pCpC+pI, μi,I=pIpC+pI, and μiˉ,R=1 is a PBE. This is the ambiguous information, selective torture equilibrium.

A.2.2.2 {iˉ,iˉ,i}

Suppose Di plays the strategy (iˉ,iˉ,i); using Bayes’ Theorem, IP’s beliefs at the i information set are μi,C=0,μi,R=0,μi,I=1 and at the iˉ information set are μiˉ,C=pCpC+pR,μiˉ,R=pRpC+pR,μiˉ,I=0. Given these beliefs, IP’s expected utility for t after i is Vca and his expected utility for tˉ is V.

IP’s expected utility for t after iˉ is c and his expected utility for tˉ is r, so IP chooses (tˉ,t). IS chooses (t,t). It remains to check whether (iˉ,iˉ,i) is Di’s best response to these choices. From equation (A.5), DC prefers iˉ to i when i is not pivotal to avoid torture, which happens when q<qˆ and when qqˆ and f<f. The strategy iˉ dominates i for DR. From case A.2.2.1.2, DI prefers i to iˉ for qq.

Thus, the strategy profile {(iˉ,iˉ,i);(tˉ,t),(tˉ,tˉ):q<qˆ, qqˆ and f<f, and qq;(μi,μiˉ)} for μr,I=1, μiˉ,C=pCpC+pR, and μiˉ,R=pRpC+pR constitutes a PBE. This is a false confirmation, selective torture equilibrium.

A.2.2.3 {i,iˉ,iˉ}

This set of strategies on the part of Di is identical to case A.2.1.1, where DI had move iˉ only. Therefore, IP’s beliefs at the i information set are μi,C=1,μi,R=0,μi,I=0 and at the iˉ information set are μiˉ,C=0,μiˉ,R=pRpR+pI,μiˉ,I=pIpR+pI.

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

(A.3)
f<rcr+af*

It likewise follows from case A.2.1.1 that IP’s expected utility for t at her iˉ information set is cpIpR+pI(a) and her expected utility for tˉ after iˉ is pRpR+pI(r) and so, from equation (A.4), IP plays t after iˉ for

(A.4)
pI<rcc+apRp*

This defines four subcases.

(p.263) A.2.2.3.1 f<f and p<p

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

A.2.2.3.2 f>f and p<p

For this combination of beliefs, IP chooses (tˉ,t) and IS chooses (t,t). It remains to check whether (i,iˉ,iˉ) is Di’s best response to these choices. From equation (A.5), DC prefers i to iˉ for qqˆ. Strategy iˉ dominates i for DR. From equation (A.8), DI prefers iˉ to i for qq.

Thus, the strategy profile {(i,i¯,i¯); (tˉ,t),(t,t):qqˆ,q<q, f>f;(μi,μiˉ)} for μi,C=1 and μiˉ,I=p<p constitutes a PBE. This is a valuable information, selective torture equilibrium.

A.2.2.3.3 f<f and p>p

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

A.2.2.3.4 f>f and p>p

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

A.2.2.4 {iˉ,iˉ,iˉ}

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

A.2.2.4.1 p<pˆ

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

For f<f, DC expects IP to play t after i and so will not deviate to i even for qqˆ. For qqˆ and f>f, however, DC expects IP to play tˉ after i and thus has an incentive to deviate to i. DI also has an incentive to deviate for qq.

To prevent deviation to i by DC and DI, IP would have to play t after i. Since under leading questioning, both DC and DI can choose i but DR never does so, let μi,C be IP’s off-path belief that the Detainee is DC, and 1μi,C be IP’s off-path belief that the Detainee is DI, upon observing i.

The expected utility of t is Vcaμfaμhμfh+μa. The expected utility of tˉ is μVμhμr+μfh+μfr+V. IP therefore prefers to torture after i for off-path beliefs satisfying (p.264)

(A.9)
μi,C>c+a(1f)(a+r)μi.

This off-path belief is a real constraint (i.e., μi,C(0,1) ) for

(A.6)
f<rcr+apˆ.

Thus, the strategy profile {(iˉ,iˉ,iˉ);(t,t),(t,t):(q<qˆ and q<q) or (qqˆ and qq and f<f) or (qqˆ and q<q and f<f) or (q<qˆ and qq and f<f) with (μi,μiˉ)} for μi,C>μi and μiˉ,I=p<pˆ is a PBE. This is the no information, torture equilibrium.

A.2.2.4.2 ppˆ

For this set of IP beliefs, IP plays (β1,tˉ); IS chooses the dominant strategy (t,t). It remains to check whether (iˉ,iˉ,iˉ) is Di’s best response to these choices. No Di can do better and so the strategy profile {(iˉ,iˉ,iˉ);(β1,tˉ),(t,t)}:q(0,1);(μi,μiˉ) for μiˉ,I=ppˆ is a PBE. This is the no information, no torture equilibrium. □