Most common auction types

Perhaps it's worth to mention that there are different kind of auctions and the most famous are:

1. the ascending-bid auction (also called English aution, open or oral) is perhaps the most famous, in which the bidders offers higher and higher prices until nobody offers more
2. the descending-bid auction (also called Dutch auction, since its most famous application is the sale of flowers in Nederlands) in which the price sinks until a bidder offers to pay the current price
3. the first-price sealed auction in which the bids are conveied in private form and the auction is won by the bidder who offers and pays the highest price
4. the second-price sealed auction (or Vickrey auction) in which the bids are conveied in private form and the auction is won by the bidder who offers the highest price, but pays just the price of the second highest bid plus a small increment.

There is an important theorem, the revenue equivalence theorem, that proofs that under some circumstances these 4 auctions have the same results. I am trying to understand it, but I am still not done (I need to refresh some mathematics :o !!!) I will let you know...