Vickrey auction

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A Vickrey auction is a type of sealed-bid auction, where bidders submit written bids without knowing the bid of the other people in the auction. The highest bidder wins, but the price paid is the second highest bid. The auction was created by William Vickrey. This type of auction is strategically similar to an English auction, and gives bidders an incentive to bid their true value.

Vickrey's original paper considered only auctions where a single, indivisible good is being sold. In this case, the terms Vickrey auction and second-price sealed-bid auction are equivalent, and are used interchangeably. When multiple identical units (or a divisible good) are being sold in a single auction, the most obvious generalization is to have all bidders pay the amount of the highest non-winning bid. This is known as a uniform-price auction.

The uniform-price auction does not, however, result in bidders bidding their true valuations as they do in a second-price auction unless each bidder only has demand for a single unit. For that reason, the name "Vickrey auction" in the multi-good auction is usually reserved by economists for a more complicated pricing scheme based on opportunity cost, which does give bidders the incentive to bid truthfully. This scheme is known as the Vickrey-Clarke-Groves (VCG) mechanism. In a VCG auction, each bidder divulges its demand curve by offering a separate bid for each additional unit. The winner of each bid only pays the opportunity cost for its allocation. This opportunity cost for each winner is the sum of the N highest rejected bids, where N is the number of units allocated to the winner.

Vickrey auctions are much studied in economic literature, but are not particularly common in practice. One market in which they have been used is stamp collecting. eBay's system of proxy bidding is similar, but not identical, to a Vickrey auction. A slight variant of a Vickrey auction is known to be used in Google's online advertisement programme, AdWords, its transparency allowing real-time unmonitored auctions to take place.

Contents

1 Properties

1.1
1.2 Ex-post efficiency
1.3 Weaknesses

2 Use in Network Routing
3 External links
4 References

Properties

In a Vickrey auction each bidder maximizes his or her expected utility by bidding (revealing) his or her true valuation.

Ex-post efficiency

A Vickrey auction is ex-post efficient (the winner is the bidder with the highest valuation) under the most general circumstances; it thus provides a baseline model against which the efficiency properties of other types of auctions can be posited.

Weaknesses

Dispite the Vickrey auction's strengths, it has shortcomings:

The auction is not budget balanced. It does not maximize the seller revenues; the seller revenues may even be zero in VCG auctions. If the purpose of holding the auction is to maximize profit for the seller, as is often the case, the Vickrey auction is a poor choice.

It does not allow for Price Discovery, that is, discovery of the market price if the buyers are unsure of their own valuations, without sequential auctions.

Sellers may use shill bids to increase profit.

In iterated Vickrey auctions, the strategy of revealing true valuations is no longer dominant.

The Vickrey-Clark-Groves mechanism has the additional shortcomings:

The seller's revenues are non-monotonic with regard to the sets of bidders and offers.

It is vulnerable to collusion by losing bidders.

It is vulnerable to shill bidding with respect to the buyers.

Use in Network Routing

In network routing, VCG mechanisms are a family of payment schemes based on the added value concept. The basic idea of a VCG mechanism in network routing is to pay the owner of each link or node (depending on the network model) its declared cost plus its added value. In many routing problems, this mechanism is not only strategyproof, but also the minimum among all strategyproof mechanisms.

In the simplest, unicast case, a least cost path in graph G is calculated based on the declared costs [d_] of each of the links, and payment is calculated as follows:

Each link [e_] on the LCP is paid

[p_ = d_ + LCP(G - e_) - LCP(G)]

and each link not on the LCP is paid nothing. This routing problem is one of the cases for which VCG is strategyproof and minimum.

In 2004, it was shown that the expected VCG overpayment of an Erdös-Renyi random graph with [n] nodes and edge probability [p] [G \in G(n, p)] approaches

[ \frac ]

as [n], approaches [ \infty ]. Prior to this result, it was known that VCG overpayment in [ G(n, p)] is

[\Omega(\frac)]

and

[O(1)]

with high probability given

[np=\omega(\log n)].

External links

[Paper on the history of Vickrey auctions in stamp collecting]

References

Vijay Krishna, Auction Theory
Peter Cramton, Yoav Shoham, Richard Steinberg (Eds), Combinatorial Auctions (2006), Chapter 1. ISBN 0262033429.

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