With second price auction which distributions should I use to model the winning bids and 2nd bids (separately)? I'm thinking of using Gaussian. However for the winning bids r.v, it has to satisfy:
$$ p(b_1|b_1>b_2, \cdots, b_1>b_N) $$
And the 2nd bids r.v, it has to satisfy:
$$ p(b_2|b_1>b_2, b_2>b_3, \cdots, b_2>b_N) $$
I guess this could give something more than general Gaussian? I'm not a mathematician so forgive me if it's too obvious.
The usual way to model auctions is by treating them as a Bayesian game. It turns out that a second price auction with independent values leads to truthful bidding ones own valuation as an equilibrium in weakly dominant strategies. So the winning bid is simply the largest valuation in that case, and its distribution will depend on the distribution of the valuations.
In general, the valuations should be nonnegative, since nobody would bid for something worse than nothing in usual models. So the distribution of winning bids shouldn't allow for negative values either. In particular, it should not be Gaussian.