In auction theory, a Vickrey–Clarke–Groves (VCG) auction is a type of sealed-bid auction of multiple items. Bidders submit bids that report their valuations for the items, without knowing the bids of the other people in the auction. The auction system assigns the items in a socially optimal manner: it charges each individual the harm they cause to other bidders. It also gives bidders an incentive to bid their true valuations, by ensuring that the optimal strategy for each bidder is to bid their true valuations of the items. It is a generalization of a Vickrey auction for multiple items.
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For any set of auctioned items and any set of bidders , let be the social value[clarification needed] of the VCG auction for a given bid-combination. For a bidder and item , let the bidder's bid for the item be .
A bidder whose bid for an item , namely , is an "overbid" wins the item, but pays , which is the social cost of his winning that is incurred by the rest of the agents.
Indeed, the set of bidders other than is . When item is available, they could attain welfare The winning of the item by reduces the set of available items to , however, so that the attainable welfare is now . The difference between the two levels of welfare is therefore the loss in attainable welfare suffered by the rest bidders, as predicted, given the winner got the item . This quantity depends on the offers of the rest agents and is unknown to agent .
- Winner's utility
The winning bidder whose bid is his true value for the item , derives maximum utility
See the example with apples in the Generalization section of Vickrey Auction.
Assume that there are two bidders, and , two items, and , and each bidder is allowed to obtain one item. We let be bidder 's valuation for item . Assume , , , and . We see that both and would prefer to receive item ; however, the socially optimal assignment gives item to bidder (so his achieved value is ) and item to bidder (so his achieved value is ). Hence, the total achieved value is , which is optimal.
If person were not in the auction, person would still be assigned to , and hence person can gain nothing. The current outcome is hence is charged .
If person were not in the auction, would be assigned to , and would have valuation . The current outcome is 3 hence is charged .
Here we will look at a multiple item auction. Consider the situation when there are bidders, houses, and values , representing the value player has for house . Possible outcomes in this auction are characterized by bipartite matchings, pairing houses with people. If we know the values, then maximizing social welfare reduces to computing a maximum weight bipartite matching.
If we do not know the values, then we instead solicit bids , asking each player how much he would wish to bid for house . Define if bidder receives house in the matching . Now compute , a maximum weight bipartite matching with respect to the bids, and compute
The first term is another max weight bipartite matching, and the second term can be computed easily from .
Optimality of Truthful Bidding
The following is a proof that bidding one's true valuations for the auctioned items is optimal.
For each bidder , let be his true valuation of an item , and suppose (without loss of generality) that wins upon submitting his true valuations. Then the net utility attained by is given by . As is independent of , the maximization of net utility is pursued by the mechanism along with the maximization of corporate gross utility for the declared bid .
Let us form the difference between net utility of under truthful bidding gotten item , and net utility of bidder under non-truthful bidding for item gotten item on utility .
is the maximum corporate gross utility obtained with the non-truthful bidding. But the allocation assigning to is different from the allocation of to which gets maximum true gross corporate utility. Hence and q.e.d.
More general setting
We can consider a more general setting of the VCG mechanism. Consider a set of possible outcomes and bidders which have different valuations for each outcome. This can be expressed as, function
for each bidder which expresses the value it has for each alternative. In this auction, each bidder will submit his valuation and the VCG mechanism will choose the alternative that maximizes and charge prices given by:
where , that is, is a function that only depends on the valuation of the other players. This alone gives a truthful mechanism, that is, a mechanism where bidding the true valuation is a dominant strategy.
We could take, for example, , but we would have all prices negative, which might not be desirable. We would rather prefer that players give money to the mechanism than the other way around. The function:
is called Clark pivot rule.
On the other hand, if we do not know the values , we can solicit bids . The mechanism then chooses maximizing the revenue , treating the bids like the values. We then set
Intuitively, the mechanism charges player his externality, or the decrease in optimal social welfare when he is included in the auction.
The Clark pivot rule has some very good properties as:
- it is individually rational, i.e., . It means that all the players are getting positive utility by participating in the auction. No one is forced to bid.
- it has no positive transfers, i.e., . The mechanism does not need to pay anything to the bidders.
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