User:Dschafer/Vickrey-Clarke-Groves auction

A Vickrey-Clarke-Groves auction is an type of sealed-bid auction where multiple items are up for bid, and each bidder submits a different value for each item. The auction system assigns the items in a socially optimal manner, while ensuring each bidder receives at most one item. This system charges each individual the "harm they cause to the rest of the world." , and ensures that the optimal strategy for a bidder is to bid the true valuations of the objects. It is a generalization of a Vickrey auction for multiple items.

Assigning Items
Assume that there are two bidders $$b_1$$ and $$b_2$$, and two items, $$t_1$$ and $$t_2$$. We let $$v_{i,j}$$ be bidder $$b_i$$'s valuation for item $$t_j$$. Assume $$v_{1,1} = 10$$, $$v_{1,2} = 5$$, $$v_{2,1} = 5$$, and $$v_{2,2} = 2$$. We see that both $$b_1$$ and $$b_2$$ would prefer to receive item $$t_1$$; however, the socially optimal assignment gives item $$t_1$$ to bidder $$b_1$$ (so his achieved value is $$10$$) and item $$t_2$$ to bidder $$b_2$$ (so his achieved value is $$3$$). Hence, the total achieved value is $$13$$, which is optimal.

Computing Costs
If person $$b_2$$ were not in the auction, person $$b_1$$ would still be assigned to $$t_1$$, and hence no harm was done for that bidder. Hence, $$b_2$$ is charged nothing.

If person $$b_1$$ were not in the auction, person $$b_2$$ would be assigned to $$t_2$$, and would have valuation $$5$$. $$b_1$$ thus caused $$2$$ harm to $$b_2$$, and hence $$b_1$$ is charged $$2$$.