I'm not familiar enough with the auction theory to know where to look, but this seems close to what seems to be known as the "standard auction model".


Say an asset is up for auction.The true value of this asset follows a normal distribution $tv \sim N(0,\eta^2)$. All bidders share that prior.
On top of that, all bidders separately observe the true value with a certain precision, ie bidder $i$ observes $V\_i \sim N(tv,\sigma^2)$.


We now hold a first price auction - the highest bidder pays its own bid, receives the asset, and therefore earns $tv - bid$.
At the Nash equilibrium, assuming bidders maximize their expected profit, bidder $i$ will bid $f(V\_i)$, with $f$ an increasing function of the observed value, depending on $\eta$, $\sigma$ and n. Are there closed form formulas for $f$? What if bidders only pay the second best price?



standard auction model
This problem is known as the "mineral rights" model, which is a special case of pure common value auctions. A simple and classical result by Milgrom and Weber (see Proposition 6.1 in p.89 of Krishna's "Auction Theory" book) states that the SECOND-price equilibrium is for each bidder to bid $E[V| X\_i=x, max\_{X\_{-i}}=x]$, i.e., the expected value of V conditioned on the event that the second highest signal matches exactly the signal of the i'th player. If fact, this result applied much more generally when the true value is a *symmetric* function of the signals X\_i. Not sure what happens for 1st-price though. 


