Consider an auction of a single unit of indivisible good. There are $n$ buyers whose values of the object is drawn independently from the uniform distribution on $[0,1]$. The buyers have interim linear utility:

$$ u_i(k(\hat{\theta}), t(\hat{\theta}));\theta_i) = \theta_i k_i(\hat{\theta}) + t_i(\hat{\theta})), $$

where $\hat{\theta}$ is the profile of bids, $k \in \mathbb{R}^n$ is the allocation ($ k_i(\hat{\theta})$ is $1$ is player $i$ gets object and $0$ otherwise), and $t \in \mathbb{R}^n$ is the transfer/payment profile.

* Assumption/Requirement* We only consider auction mechanisms where in the resulting Bayesian Nash equilibrium, the object goes to the bidder with highest value, i.e.

$$ k_i(\theta) = 1 \Leftrightarrow \theta_i = \max_j \theta_j. $$

Quoting the revelation principle, I have replaced $\hat{\theta}$ by the true type profile $\theta$ above.

Buyer $i$'s interim expected utility as a function of his type $\theta_i$ is

$$ U_i(\theta_i) = \theta_i \bar{k}_i (\theta_i) + \bar{t}_i (\theta_i), $$

where $\bar{k}_i(\theta_i)$ is the probability he gets the good and $\bar{t}_i(\theta_i)$ is expected transfer.

From the seller's perspective, the expected payment from a buyer of type $\theta_i$ is

$$ - \bar{t}(\theta_i) = - [ \int_0 ^{\theta_i} \bar{k}_i (\theta_i ') d \theta_i ' + U_i(0) - \bar{k}_i(\theta_i) \theta_i]. $$

The requirement above means $\bar{k}_i(\theta_i) = \theta_i^n$. So when the seller is risk-neutral, the maximum expected revenue for the seller is achieved by any mechanism that makes $U_i(0) = 0$, i.e. a buyer of type $0$ has interim expected utility $0$.

Question: What if the seller is risk-averse, for example with utility function $w(t) = \sqrt{t}$? What characterizes the seller's optimal auction in this case?

According to my calculations, the first price auction gives the seller expected utility $$ \sqrt{n(n-1)} \frac{1}{n + \frac{1}{2}}, $$

while the second price auctions gives

$$ n(n-1) \frac{1}{n + \frac{1}{2}} \frac{1}{n - \frac{1}{2}}. $$

So revenue equivalence no longer holds.