$$

\text{Pr}(A|r)=\frac{\pi_A \text{Pr}(r|A)}{\pi_A \text{Pr}(r|A)+ (1-\pi_A) \text{Pr}(r|A^{'})}

$$ (eq:one)

$$ (eq:util-rand-one)

## Zoo of Concepts

\text{or}&\\

\text{Pr}(A^{'}|r)&>1-\pi_A

\end{aligned}

$$ (eq:two)

$$ (eq:util-rand-two)

From Bayes's rule:

$$

\frac{\text{Pr}(A|r)}{\text{Pr}(A^{'}|r)}\times \frac{(1-\pi_A)}{\pi_A} = \frac{\text{Pr}(r|A)}{\text{Pr}(r|A^{'})}

$$ (eq:three)

$$ (eq:util-rand-three)

If this expression is greater (less) than unity, it follows that r is jeopardizing with respect to $A$($A^{'}$). Then, the natural measure of jeopardy will be:

&\text{and}\\

g(r|A^{'})&=\frac{\text{Pr}(r|A^{'})}{\text{Pr}(r|A)}

\end{aligned}

$$ (eq:four)

$$ (eq:util-rand-four)

Suppose, without loss of generality, that $\text{Pr}(\text{yes}|A)>\text{Pr}(\text{yes}|A^{'})$, then a yes (no) answer is jeopardizing with respect $A$($A^{'}$), that is,

$$

\max \left\{ \text{Pr}(A|\text{yes}) , \text{Pr}(A|\text{no}) \right\}

$$ (eq:five)

$$ (eq:util-rand-five-a)

Holding this measure constant, he explained under what conditions the smallest variance of the estimate was achieved with the unrelated question model or Warner's (1965) original model.

$$

\frac{1-\max \left\{ \text{Pr}(A|\text{yes}) , \text{Pr}(A|\text{no}) \right\}}{1-\pi_A}

$$ (eq:six)

$$ (eq:util-rand-six)

### 2.4 Greenberg, Kuebler, Abernathy, and Horvitz (1977)

$$

\text{Pr}(\text{yes}|A)\times \text{Pr}(A|\text{yes})+\text{Pr}(\text{no}|A)\times \text{Pr}(A|\text{no})

$$ (eq:seven-a)

$$ (eq:util-rand-seven-a)

Similarly, the hazard for an individual who does not belong to $A$ would be

$$

\text{Pr}(\text{yes}|A^{'})\times \text{Pr}(A|\text{yes})+\text{Pr}(\text{no}|A^{'}) \times \text{Pr}(A|\text{no})

$$ (eq:seven-b)

$$ (eq:util-rand-seven-b)

Greenberg et al. (1977) also considered an alternative related measure of hazard that "is likely to be closer to the actual concern felt by a respondent."

The "limited hazard" for an individual in $A$ and $A^{'}$ is

$$

\text{Pr}(\text{yes}|A)\times \text{Pr}(A|\text{yes})

$$ (eq:eight-a)

$$ (eq:util-rand-eight-a)

and

$$

\text{Pr}(\text{yes}|A^{'})\times \text{Pr}(A|\text{yes})

$$ (eq:eight-b)

$$ (eq:util-rand-eight-b)

This measure is just the first term in $(7)$, i.e., the probability that an individual answers "yes" and is perceived to belong to A.

$$

\frac{\partial U_i\left(\text{Pr}(A|r_i),\phi_i\right) }{\partial \text{Pr}(A|r_i)} <0, \text{ for } \phi_i \in \left\{\text{truth},\text{lie}\right\}

$$ (eq:nine-a)

$$ (eq:util-rand-nine-a)

and

$$

U_i\left(\text{Pr}(A|r_i),\text{truth}\right)>U_i\left(\text{Pr}(A|r_i),\text{lie}\right) , \text{ for } \text{Pr}(A|r_i) \in [0,1]

$$ (eq:nine-b)

$$ (eq:util-rand-nine-b)

Suppose now that correct answer for individual $i$ is "yes".

Individual $i$ would choose to answer truthfully if

$$

U_i\left(\text{Pr}(A|\text{yes}),\text{truth}\right)\geq U_i\left(\text{Pr}(A|\text{no}),\text{lie}\right)

$$ (eq:ten-a)

$$ (eq:util-rand-ten-a)

If the correct answer is "no," individual $i$ would volunteer the correct answer only if

$$

U_i\left(\text{Pr}(A|\text{no}),\text{truth}\right)\geq U_i\left(\text{Pr}(A|\text{yes}),\text{lie}\right)

$$ (eq:ten-b)

$$ (eq:util-rand-ten-b)

Assume that

$$

U_i\left(\text{Pr}(A|\text{yes}),\text{truth}\right)= U_i\left(\text{Pr}(A|\text{no}),\text{lie}\right)

$$ (eq:eleven)

$$ (eq:util-rand-eleven)

Equation $(11)$ defines a "truth border".

Differentiating $(11)$ with respect to the conditional probabilities shows that the truth border has a positive slope in the space of conditional probabilities:

$$

\frac{\partial \text{Pr}(A|\text{no})}{\partial \text{Pr}(A|\text{yes})}=\frac{\frac{\partial U_i\left(\text{Pr}(A|\text{yes}),\text{truth}\right) }{\partial \text{Pr}(A|\text{yes})}}{\frac{\partial U_i\left(\text{Pr}(A|\text{no}),\text{lie}\right) }{\partial \text{Pr}(A|\text{no})}}>0

$$ (eq:twelve)

$$ (eq:util-rand-twelve)

The source of the positive relationship is:

V(\text{Pr}(A|\text{yes}) , \text{Pr}(A|\text{no}))

= &\frac{{\pi_A}^2 (1-\pi_A)^2}{n}\times \frac{1}{\text{Pr}(A|\text{yes})-\pi_A}\times \frac{1}{\pi_A-\text{Pr}(A|\text{no})}

\end{aligned}

$$ (eq:thirteen)

$$ (eq:util-rand-thirteen)

where the random sample with replacement consists of $n$ individuals.

$$

\frac{d \text{ Pr}(A|\text{no})}{d\text{ Pr}(A|\text{yes})}\bigg|_{\text{constant variance}}=\frac{\pi_A-\text{Pr}(A|\text{no})}{\text{Pr}(A|\text{yes})-\pi_A}>0

$$ (eq:fourteen-a)

$$ (eq:util-rand-fourteen-a)

$$

\frac{d^2 \text{ Pr}(A|\text{no})}{d\text{ Pr}(A|\text{yes})^2}\bigg|_{\text{constant variance}}=- \frac{2 \left[\pi_A-\text{Pr}(A|\text{no})\right]}{\left[\text{Pr}(A|\text{yes})-\pi_A \right]^2}<0

$$ (eq:fourteen-b)

$$ (eq:util-rand-fourteen-b)

From expression $(13)$ and $(14)$ we can see that:

$$

\max \left\{ \text{Pr}(A|\text{yes}) , \text{Pr}(A|\text{no}) \right\}

$$ (eq:five)

$$ (eq:util-rand-five-b)

Following Lanke's suggestion, the statistician should find the highest possible $\text{ Pr}(A|\text{yes})$ consistent with truth telling while $\text{ Pr}(A|\text{no})$ is fixed at 0. The variance is then minimized at point $X$ in Figure 3.

$$

\text{Pr}(\text{yes}|A)\times \text{Pr}(A|\text{yes})+\text{Pr}(\text{no}|A)\times \text{Pr}(A|\text{no})

$$ (eq:seven-a)

$$ (eq:util-rand-seven-aa)

The hazard for an individual who does not belong to $A$ is

$$

\text{Pr}(\text{yes}|A^{'})\times \text{Pr}(A|\text{yes})+\text{Pr}(\text{no}|A^{'}) \times \text{Pr}(A|\text{no})

$$ (eq:seven-a)

$$ (eq:util-rand-seven-bb)

They also considered an alternative related measure of hazard that they said "is likely to be closer to the actual concern felt by a respondent."

Their "limited hazard" for an individual in $A$ and $A^{'}$ is

$$

\text{Pr}(\text{yes}|A)\times \text{Pr}(A|\text{yes})

$$ (eq:eight-a)

$$ (eq:util-rand-eight-aa)

and

$$

\text{Pr}(\text{yes}|A^{'})\times \text{Pr}(A|\text{yes})

$$ (eq:eight-b)

$$ (eq:util-rand-eight-bb)

According to Greenberg et al. (1977), a respondent commits himself or herself to answer truthfully on the basis of a probability in $(7)$ or $(8)$ **before** randomly selecting the question to be answered.