A Vickrey auction or sealed-bid second-price auction (SBSPA) is a type of sealed-bid
auction
An auction is usually a process of buying and selling goods or services by offering them up for bids, taking bids, and then selling the item to the highest bidder or buying the item from the lowest bidder. Some exceptions to this definition e ...
. 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. This type of auction is strategically similar to an
English auction
An English auction is an open-outcry ascending dynamic auction. It proceeds as follows.
* The auctioneer opens the auction by announcing a suggested opening bid, a starting price or reserve for the item on sale.
* Then the auctioneer accepts incre ...
and gives bidders an
incentive to bid their true value. The auction was first described academically by
Columbia University
Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...
professor
Professor (commonly abbreviated as Prof.) is an academic rank at universities and other post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "person who professes". Professo ...
William Vickrey
William Spencer Vickrey (21 June 1914 – 11 October 1996) was a Canadian-American professor of economics and Nobel Laureate. Vickrey was awarded the 1996 Nobel Memorial Prize in Economic Sciences with James Mirrlees for their research into the e ...
in 1961 though it had been used by
stamp collectors
Stamp collecting is the collecting of postage stamps and related objects. It is an area of philately, which is the study (or combined study and collection) of stamps. It has been one of the world's most popular hobbies since the late nineteen ...
since 1893. In 1797
Johann Wolfgang von Goethe
Johann Wolfgang von Goethe (28 August 1749 – 22 March 1832) was a German poet, playwright, novelist, scientist, statesman, theatre director, and critic. His works include plays, poetry, literature, and aesthetic criticism, as well as t ...
sold a manuscript using a sealed-bid, second-price auction.
Vickrey's original paper mainly considered auctions where only a single, indivisible good is being sold. The terms ''Vickrey auction'' and ''second-price sealed-bid auction'' are, in this case only, equivalent and used interchangeably. In the case of multiple identical goods, the bidders submit inverse demand curves and pay the opportunity cost.
Vickrey auctions are much studied in economic literature but uncommon in practice. Generalized variants of the Vickrey auction for
multiunit auction
A multiunit auction is an auction in which several homogeneous items are sold. The units can be sold each at the same price (a uniform price auction) or at different prices (a discriminatory price auction).
Uniform price auction
A uniform pric ...
s exist, such as the
generalized second-price auction
The generalized second-price auction (GSP) is a non-truthful auction mechanism for multiple items. Each bidder places a bid. The highest bidder gets the first slot, the second-highest, the second slot and so on, but the highest bidder pays the pri ...
used in Google's and Yahoo!'s online advertisement programmes (not
incentive compatible
A mechanism is called incentive-compatible (IC) if every participant can achieve the best outcome to themselves just by acting according to their true preferences.
There are several different degrees of incentive-compatibility:
* The stronger d ...
) and the
Vickrey–Clarke–Groves auction
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 bidders. The auction system assigns the items in a ...
(incentive compatible).
Properties
Self-revelation and incentive compatibility
In a Vickrey auction with private values each bidder maximizes their
expected utility The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on the ...
by bidding (revealing) their valuation of the item for sale. These type of auctions are sometimes used for specified pool trading in the agency mortgage-backed securities (MBS) market.
Ex-post efficiency
A Vickrey auction is decision 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. It is only ex-post efficient (sum of transfers equal to zero) if the seller is included as "player zero," whose transfer equals the negative of the sum of the other players' transfers (i.e. the bids).
Weaknesses
* It does not allow for
price discovery
In economics and finance, the price discovery process (also called price discovery mechanism) is the process of determining the price of an asset in the marketplace through the interactions of buyers and sellers.
Overview
Price discovery is diff ...
, that is, discovery of the market price if the buyers are unsure of their own valuations, without sequential auctions.
* Sellers may use
shill
A shill, also called a plant or a stooge, is a person who publicly helps or gives credibility to a person or organization without disclosing that they have a close relationship with said person or organization. Shills can carry out their operatio ...
bids to increase profit.
The
Vickrey–Clarke–Groves (VCG) mechanism has the additional shortcomings:
* It is vulnerable to bidder
collusion
Collusion is a deceitful agreement or secret cooperation between two or more parties to limit open competition by deceiving, misleading or defrauding others of their legal right. Collusion is not always considered illegal. It can be used to att ...
. If all bidders in Vickrey auction reveal their valuations to each other, they can lower some or all of their valuations, while preserving who wins the auction.
* It is vulnerable to a version of shill bidding in which a buyer uses multiple identities in the auction in order to maximize its profit.
* It does not necessarily maximize seller revenues; seller revenues may even be zero in VCG auctions. If the purpose of holding the auction is to maximize profit for the seller rather than just allocate resources among buyers, then VCG may be a poor choice.
* The seller's revenues are non-
monotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
with regard to the sets of bidders and offers.
The non-monotonicity of seller's revenues with respect to bids (without introducing the VCG opportunity-cost mechanism described at the bottom of this article) can be shown by the following example. Consider three bidders A, B, and C, and two homogeneous items bid upon, Y and Z.
* A wants both items and bids $2 for the package of Y and Z.
* B and C both bid $2 each for a single item (bid $2 for Y or Z), as they really want one item but don't care if they have the second.
Now, Y and Z are allocated to B and C, but the price is $0, as can be found by removing either B or C respectively. If C bid $0 instead of $2, then the seller would make $2 instead of $0. Because the seller's revenue can go up when bids are either increased or decreased, the seller's revenues are non-monotonic with respect to bids.
Proof of dominance of truthful bidding
The dominant strategy in a Vickrey auction with a single, indivisible item is for each bidder to bid their true value of the item.
Let
be bidder i's value for the item. Let
be bidder i's bid for the item.
The payoff for bidder i is
The strategy of overbidding is dominated by bidding truthfully. Assume that bidder i bids
.
If
then the bidder would win the item with a truthful bid as well as an overbid. The bid's amount does not change the payoff so the two strategies have equal payoffs in this case.
If
then the bidder would lose the item either way so the strategies have equal payoffs in this case.
If
then only the strategy of overbidding would win the auction. The payoff would be negative for the strategy of overbidding because they paid more than their value of the item, while the payoff for a truthful bid would be zero. Thus the strategy of bidding higher than one's true valuation is dominated by the strategy of truthfully bidding.
The strategy of underbidding is dominated by bidding truthfully. Assume that bidder i bids
.
If
then the bidder would lose the item with a truthful bid as well as an underbid, so the strategies have equal payoffs for this case.
If
then the bidder would win the item either way so the strategies have equal payoffs in this case.
If
then only the strategy of truthfully bidding would win the auction. The payoff for the truthful strategy would be positive as they paid less than their value of the item, while the payoff for an underbid bid would be zero. Thus the strategy of underbidding is dominated by the strategy of truthfully bidding.
Truthful bidding dominates the other possible strategies (underbidding and overbidding) so it is an optimal strategy.
Revenue equivalence of the Vickrey auction and sealed first price auction
The two most common auctions are the sealed first price (or high-bid) auction and the open ascending price (or English) auction. In the former each buyer submits a sealed bid. The high bidder is awarded the item and pays his or her bid. In the latter, the auctioneer announces successively higher asking prices and continues until no one is willing to accept a higher price. Suppose that a buyer's valuation is
and the current asking price is
. If