APPENDIX C

Observations and Propositions

This appendix makes several observations and proves the relevant propositions from Chapter 7, following the order of presentation in the text after making two observations not in the text but which will prove useful in proving a proposition below.

C.1 OBSERVATION C.1

Observation C.1 .

Let . Since . Claiming implies that , which becomes . This is true since and .

C.2 OBSERVATION C.2

Observation C.2 If or if , then .

Proof

By assumption, p_C, p_R, and p_I are positive and . Suppose that or and suppose that . Since , this implies . But this means , which contradicts .

□

C.3 OBSERVATION 7.1:

The Interrogator’s innocent detainee recognition threshold is less than one-half.

Given

C.4 PROPOSITION 7.1:

As c approaches zero and a approaches r, the Interrogator’s innocent detainee recognition threshold approaches one-half from below.

C.5 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: If or if , then .

Proof

First, by Observation 7.1, . Note that if and only if , which is equivalent to . Since, by Observation C.1, , and, by Observation C.2, , we have . Thus and so by Observation 7.1, .

□

C.6 PROPOSITION 7.3:

As c approaches zero and as a approaches r, the Interrogator’s information hiding threshold is greater than one-half.

Proof

Recall that . Note that if and only if . Rearranging some terms gives us . Solving for u, we have . Given , as assumed, for and , . This is equivalent to which is true for , as assumed.

□

C.7 OBSERVATION 7.2:

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

Since and by Observation C.4 , it follows immediately that .

C.8 OBSERVATION 7.3:

The Interrogator’s and the Detainee’s beliefs will “agree” on the information hiding threshold f only in the special case when the Interrogator has understood perfectly the information’s value; all other cases open up the possibility for surprise torture.

By simple inspection of and , it is clear that for all , as assumed, and that if and only if .

C.9 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.

Proof

Recall that . Suppose . This implies . Simplifying yields . Rearrange some terms and we have . Given as assumed, this is a contradiction, and so . Given as assumed, when , and , .

□

C.10 PROPOSITION 7.5: FOR ,

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 , .

Proof

Note first that by observation 7.3, . Recall the definitions and . Note that if and only if . Simplifying the right side of the inequality yields . Simplifying the left side gives us . Rearrange some terms and we have , which is true for and .