Let $Y$ and $W$ be two random variables with support $(y\_1,y\_2)$ and $(w\_1,w\_2)$ and distributions $F\_Y$ and $F\_W$, both twice continuously differentiable (densities $f\_Y$ and $f\_W$). Assume that both have (finite) mean $\bar{y}$ and $\bar{w}$. Assume also that $f\_Y(y)>0$ for all $y\in (y\_1,y\_2)$ and $f\_W(w)>0$ for all $w\in (w\_1,w\_2)$.


Define $g(x)=\mathrm{E}[Y\,|\,Y>x]$. Let $A:\mathbb{R}\to \mathbb{R}$ be a (strictly) decreasing function and $\tilde{w}$ denote the unique solution $w$ to $A(w)=g^{-1}(A(\bar{w}))$. Assume that $\bar{y}< A(\bar{w})$.


Consider:


$P\_n = \int\_{w\_1}^{w\_2}\overline{F}\_Y(g^{-1}(A(w)))f\_W(w)\mathrm{d} w$


$P\_m = \frac{1}{2}\overline{F}\_Y(g^{-1}(A(\bar{w}))) + \int\_{\tilde{w}}^{w\_2}\bigl[\overline{F}\_Y(A(w))-\overline{F}\_Y(g^{-1}(A(\bar{w}))) \bigr]f\_W(w)\mathrm{d} w$


Are there sufficient conditions under which $P\_n\geq P\_m$ or $P\_m\geq P\_n$?


**MOTIVATION**


A machine is labeled "reliable" if the probability that failure does not occur (random variable $Y$) is above a threshold $A(W)$ (the threshold is stochastic).


The machine designer will eventually know the realization of $Y$. In fact, he will know it before he knows the realization of $W$. He can commit to a disclosure policy regarding $Y$. An authority who only observes the realization of $W$ and the designer's disclosure policy will decide on granting the "reliable" certification. Since $\bar{y}< A(\bar{w})$, committing to not disclosing the realization of $Y$ irrespective of its value precludes the machine from being granted the "reliable" certificate.


The machine designer can choose between two disclosure policies:


* **Option 1**


Wait until he observes the realization of $W$ to disclose. This option prescribes the following. Let $y^n$ denote the unique solution to $\mathrm{E}[Y|Y>y^n]=A(W)$.


If $y< y^n$, discloses $y$; otherwise, discloses nothing.


Thus, the ex-ante probability that the machine being granted the "reliable" certification is given by $P\_n$.


* **Option 2**


Disclosure of realization of $Y$ is made as soon as possible (if at all). This option prescribes the following. Let $y^m=A(\bar{w})$.


If $y< y^m$, discloses $y$; otherwise, discloses nothing.


Thus, the ex-ante probability that the machine being granted the "reliable" certification is given by $P\_m$.


So, under what conditions would the designer prefer option 1 over option 2? Under what conditions would he prefer option 2 over option 1?



Sufficient conditions for inequality with integral of reliability functions
Rewriting $P\_m$


\begin{align}
P\_m &= \int\_{\bar{w}}^{w\_2}\overline{F}\_Y(g^{-1}(A(\bar{w})))f\_W(w)\mathrm{d} w 
+ \int\_{\tilde{w}}^{w\_2}\bigl[\overline{F}\_Y(A(w))-\overline{F}\_Y(g^{-1}(A(\bar{w}))) \bigr]f\_W(w)\mathrm{d} w & \\
&= \int\_{\bar{w}}^{\tilde{w}}\overline{F}\_Y(g^{-1}(A(\bar{w})))f\_W(w)\mathrm{d} w 
+ \int\_{\tilde{w}}^{w\_2}\overline{F}\_Y(A(w))f\_W(w)\mathrm{d} w &
\end{align}


Thus
\begin{align}
P\_n - P\_m &= \int\_{w\_1}^{\bar{w}}\overline{F}\_Y(g^{-1}(A(w)))f\_W(w)\mathrm{d} w
+\int\_{\bar{w}}^{\tilde{w}}\bigl[\overline{F}\_Y(g^{-1}(A(w)))-\overline{F}\_Y(g^{-1}(A(\bar{w}))) \bigr]f\_W(w)\mathrm{d} w 
+ \int\_{\tilde{w}}^{w\_2}\bigl[\overline{F}\_Y(g^{-1}(A(w))) - \overline{F}\_Y(A(w)) \bigr]f\_W(w)\mathrm{d} w &
\end{align}


Since $w\geq \bar{w} \Rightarrow g^{-1}(A(w))\leq g^{-1}(A(\bar{w}))\Rightarrow \overline{F}\_Y((g^{-1}(A(w)))\geq \overline{F}\_Y(g^{-1}(A(\bar{w})))$, the second integral is non-negative.


Also, $f\_Y(y)>0 \text{ for all } y \Rightarrow g^{-1}(A(w))<A(w)\Rightarrow \overline{F}\_Y(g^{-1}(A(w)))>\overline{F}\_Y(A(w))$. Hence the third integral is positive.


It follows that $P\_n>P\_m$.


