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I would like your help with the following question regarding computing a revenue for a seller of an VCG (vickrey clarke groves) auction, I'm really new to this auctions\game theory so I'd really appreciate an explanation.
The auction is as follows: Three bidders,one item. each bidder has a valuation out of a uniform distribution $[0,1]$. How do I calculate the revenue of such auction?
With a single item, the VCG payment rule is just that the winner pays the second-highest bid*. So the expected revenue is the expected second-highest bid. As noted at http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)#Order_statistics , for $3$ bidders distributed uniformly and independently on $[0,1]$, this is $0.5$.
* (Edit) perhaps it is helpful to explain this. If the winner were not in the auction, the total utility of the remaining bidders would be the second-highest bid, since that bidder would get the item and gain that utility. If the winner is in the auction, the total utility of the remaining bidders would be zero because the item goes to the winner. So the difference in welfare is the second-highest bid, so this is what the winner must pay.
