Environment
Suppose $n$ bidders participate in a second price sealed-bid auction, in which one object is being sold. Each bidder $i$ values the object at $v_i$, and each $v_i$ is independently and uniformly distributed on $[0,1]$. The $v_i$'s are private information.
The rules of the auction is that the highest bidder, say $i$, wins the object and pays the a price equal to $\max_{j\ne i}v_j$. If one doesn't win the object, then payment is zero.
We know that it's optimal for each bidder to submit a bid equal to their true value, i.e. $b_i=v_i$ for all $i=1,\dots,n$.
Question
For a bidder $i$ with value $v_i$, what is his expected payment?
What I've done…
I know that the seller's expected revenue is just the the second highest order statistic from the $n$ independent draws from the uniform distribution, and I have no problem in calculating that to be $(n-1)/(n+1)$.
From a bidder's perspective, expected payment should be
$$
\mathbb E\left[\max_{j\ne i}v_j\middle\vert v_j\le v_i,\;j\ne i\right]\Pr(v_j\le v_i,\;j\ne i)
$$
It is computing the conditional expectation that I'm having trouble with. In particular, I'm having difficulty deriving the conditional distribution for the second highest order statistic given that $v_i$ is the highest one.
Any help would be appreciated.
Best Answer
Hint: The probability of bidder $i$ winning is the probability that all other valuations are lower than his. What is that?. Given that he wins, the other valuations are now uniform on $[0,v_i)$. What is the expected maximum of $n-1$ of these?