I'm facing difficulty in understanding how they in the book, jumped for (12.13) to (12.14). what is given is that $b_1$ and $V_2$ are uniformly distributed between $[0,1]$. I could not post a picture but here are the equations:
(12.12) $=P(b_1\gt V_2/2)(v_1-b_1)$
(12.13) $=P(2b_1\gt V_2)(v_1-b_1)$
(12.14) $=\min\{{2b_1,V_2\}}(v_1-b_1)$
(This is a first price auction symmetric equilibrium problem)
If all you know is that $b_1$ and $V_2$ are uniform on $[0,1], P(2b_1 \gt V_2)=\frac 34)$ To see this, $2b_1$ is uniform on $[0,2]$. With chance $\frac 12, 2b_1 \gt 1$ and is certainly greater than $V_2$. Otherwise, it is uniform on $[0,1]$ the the chance of each being greater is $\frac 12$ by symmetry.