What goes wrong in this joint probability calculation? There are two independent RV's $v_1, v_2 \sim U[0,1]$, and two parameters $p_1, p_2$, where $p_1$ may be larger, equal, or smaller than $p_2$ (I think that my calculation does not depend on this).
I want to find $Pr(v_2 - p_2 > v_1 - p_1 \land v_2 > p_2 \land v_1 > p_1)$, but unfortunately, my two approaches yield different results. 
Approach 1:
$$Pr(v_2 - p_2 > v_1 - p_1 \land v_2 > p_2 \land v_1 > p_1) = Pr(v_2 > v_1 - p_1 + p_2 \land v_2 > p_2 \land v_1 > p_1) = Pr(v_2 > \max(v_1 - p_1 + p_2, p_2) \land v_1 > p_1) $$
As $v_1 > p_1$ implies $\max(v_1 - p_1 + p_2, p_2) = v_1 - p_1 + p_2$, I can calculate this expression with:
$$ \ldots = Pr(v_2 > v_1 - p_1 + p_2 \land v_1 > p_1) = \int_{p_1}^1 \int_{v_1-p_1 + p_2}^1 dv_2 dv_1 =  \int_{p_1}^1 1-v_1+p_1-p_2 dv_1 = 1/2-p_2+p_1 p_2 -\frac{p_1^2}{2} $$
Approach 2
$$Pr(v_2 - p_2 > v_1 - p_1 \land v_2 > p_2 \land v_1 > p_1) = Pr(v_1 < v_2 - p_2 + p_1 \land v_2 > p_2 \land v_1 > p_1) = \int_{p_2}^1 \int_{p_1}^{v_2-p_2+p_1} dv_1 dv_2 = \int_{p_2}^1 v_2-p_2 dv_2 = 1/2 - p_2 + \frac{p_2^2}{2}$$
So even though the two approaches are equivalent from my point of view, the result obviously is not. What goes wrong here?
 A: Comment:
Here is an attempt to simulate your problem in R for the special case
in which $v_1 = 1/4$ and $v_2 = 3/4$. I hope I have transcribed
your problem and approaches to an answer correctly. You can
expect about two place accuracy in the simulation.
 # Simulated results
 m = 10^5;  v1 = runif(m);  v2 = runif(m)
 p1 = 1/4;  p2 = 3/4
 mean((v2 > v1-p1+p2)&(v1 > p1)&(v2 > p2))
 ## 0.03116
  mean((v2 > v1-p1+p2)&(v1 > p1)&(v2 > p2))
 ## 0.03116

 # Computations of analytic results
 .5 - p2 + p1*p2 - p1^2/2  # approach 1
 ## -0.09375               #  gives impossible probability
 .5 - p2 + p2^2/2          # approach 2
 ## 0.03125                #  seems consistent with simulation

It seems that your first approach gives a formula that can have
nonsensical negative values. Your second approach gives a value
that is consistent with the simulation in the run shown (and in
a subsequent additional run).
Of course, a simulation with a particular pair of values for $v_1$ and $v_2$ does not solve your problem or definitively check your
answers. But it may help you to interpret some of the other
Comments and give you an idea where to look for errors. (Because
it is not known which of $p_1$ and $p_2$ is smaller, you may need
to consider a couple of different cases. As in the first Comment by @A.S., it is important to visualize the region of integration
at the start.)
Below is a plot of the 100,000 simulated $(V_1,V_2)$-pairs shown in 
light grey. The pairs that satisfy the condition of your Problem
are shown in dark blue. Grid lines help to show that the dark blue
points are about 1/32 of all points, for my choices of the
constants $p_1$ and $p_2.$
 cond = (v2 > v1 - p1 + p2)&(v1 > p1)&(v2 > p2)
 plot(v1, v2, pch=".", col="grey")
 points(v1[cond], v2[cond], pch=".", col="darkblue")
 abline(v = (0:4)/4, col="darkgreen");  bline(h = (0:4)/4, col="darkgreen")


A: As @A.S. pointed out, the problem lies in the boundaries: In Approach 1, the lower bound of the second integral might be below 0; and in Approach 2 the upper bound of the second integral might exceed 1.
Once I account for boundary conditions, both approaches yield the same results.
