## 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

Source:
Does Torture Work?
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 $qP+qS=1$.

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: $DC=0>−v>−k>−v−k$ and $DR=0>−k>−v>−v−k$. 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 $V−h$. Since the value of i is uncertain, the Interrogators have only the common prior belief that i provides V with probability f and $V−h$ with probability $1−f$, 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 $1−u$, $u∈(0,1]$. 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 $1−u$) 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 $uV−uh−ufa+ufh−c$. The expected utility of $tˉ$ at the i information set is $uV−uh+ufh+ufr−r$. $IP$ therefore prefers to torture after i for

(A.1)
$Display mathematics$

Solving for f, we obtain

(A.2)
$Display mathematics$

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)
$Display mathematics$

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

(A.4)
$Display mathematics$

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

#### A.2.1.1.1$f and $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, $DC$ 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∗ and $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∗, he believes that $IP$ plays $tˉ$ rather than t after i. For $DC$, the expected utility of i is $q(−v)+(1−q)(−v−k)⇔−qv−v−k+qv+qk$, or $qk−v−k$ and the expected utility of $iˉ$ is $−kq+(1−q)−k$ or $−k$. Thus, $DC$ prefers i to $iˉ$ for $qk−v−k≥k⇔qk≥v$, or

(A.5)
$Display mathematics$

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

#### A.2.1.1.3$f∗≤fˆ and $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 $q≥qˆ$. 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 ${i,iˉ,iˉ}$; ($tˉ,t$), ($t,t$): $q≥qˆ,f∗≤fˆ for $μi,C=1,μiˉ,I=p constitutes a PBE. This is a valuable information, selective torture equilibrium.

#### A.2.1.1.4$f and $p≥p∗$

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, $DC$ 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∗ and $p≥p∗$

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∗, 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$f∗≤fˆ and $p≥p∗$

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 $−c−pIa$. Her expected utility from $tˉ$ after $iˉ$ is $−r(pC+pR)$. Thus $IP$ plays t after $iˉ$ for

(A.6)
$Display mathematics$

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

#### A.2.1.2.1$p

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 $uV−uh−ufa+ufh−c$ and $uV−uh+ufh+ufr−r$, respectively. From equation (A.2), $IP$ therefore prefers to torture after i if its off-path beliefs satisfy

(A.2)
$Display mathematics$

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

#### A.2.1.2.2$p≥pˆ$

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 $μiˉ,I=p≥pˆ$ is a PBE. This is the no information, no torture equilibrium.

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 $V+−pCc−pCh−pCfa+pCfh−pIc−pIapC+pI$. The expected utility for $tˉ$ is $V+−pCh−pCr+pCfh+pCfrpC+pI$. $IP$ therefore plays t after i for

(A.7)
$Display mathematics$

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

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 $q≥qˆ$. For $DI$, the expected utility of i is $qk−l−k$ and the expected utility of $iˉ$ is $−k$. Thus, $DI$ prefers i to $iˉ$ for

(A.8)
$Display mathematics$

This is the innocent detainee’s information revelation threshold. Thus, the strategy profile ${(i,iˉ,i),(tˉ,t),(t,t):q≥qˆ$ and $q≥q∗;f>f˜$; $(μi,μiˉ)}$ 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 $V−c−a$ 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 and when $q≥qˆ$ and $f. The strategy $iˉ$ dominates i for $DR$. From case A.2.2.1.2, $DI$ prefers i to $iˉ$ for $q≥q∗$.

Thus, the strategy profile ${(iˉ,iˉ,i);(tˉ,t),(tˉ,tˉ):q, $q≥qˆ$ and $f, and $q≥q∗;(μi,μiˉ)}$ 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 $V−c−h−fa+fh$. The expected utility of $tˉ$ at the i information set is $V−h−r+fh+fr$. Identical to equation (A.3) above, $IP$ therefore prefers to torture after i if

(A.3)
$Display mathematics$

It likewise follows from case A.2.1.1 that $IP$’s expected utility for t at her $iˉ$ information set is $−c−pIpR+pI(a)$ 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)
$Display mathematics$

This defines four subcases.

#### (p.263) A.2.2.3.1$f and $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

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 $q≥qˆ$. Strategy $iˉ$ dominates i for $DR$. From equation (A.8), $DI$ prefers $iˉ$ to i for $q≤q∗$.

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

#### A.2.2.3.3$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), $pˆ≡r−cr+a$.

#### A.2.2.4.1$p

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 and thus will not deviate; the same is true for $DI$ for $q.

For $f, $DC$ expects $IP$ to play t after i and so will not deviate to i even for $q≥qˆ$. For $q≥qˆ$ 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 $q≥q∗$.

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 $1−μi,C$ be $IP$’s off-path belief that the Detainee is $DI$, upon observing i.

The expected utility of t is $V−c−a−μfa−μh−μfh+μa$. The expected utility of $tˉ$ is $μV−μh−μr+μfh+μfr+V$. $IP$ therefore prefers to torture after i for off-path beliefs satisfying (p.264)

(A.9)
$Display mathematics$

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

(A.6)
$Display mathematics$

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

#### A.2.2.4.2$p≥pˆ$

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 $μiˉ,I=p≥pˆ$ is a PBE. This is the no information, no torture equilibrium. □