We find expected revenue in first bid auction by following method.
let us say $V_1$ and $V_2$ denotes maximum amount that player 1 and player 2 willing to pay.
$V_1,V_2 \in [0,1]$
In case when we have only 2 players.
Nash equilibrium is $(\frac{V_1}{2},\frac{V_2}{2})$.
Expected revenue = $P(V_1>V_2)\frac{V_1}{2} + P(V_2>V_1) \frac{V_2}{2}$
we replace $V_1$ by $\frac{1+V_2}{2}$ (mean in range $V_2$ to 1).
Now Expected revenue = $\frac{V_2^{2}}{2} +(1-V_2)\frac{1+V_2}{4}$ = $\frac{1+V_2^{2}}{4}$
Now by integrating from 0 to 1 we get $\frac{1}{3}$ answer.
I did not Understand why we take mean.
Can anyone please explain.