Auction theory
Encyclopedia
Auction theory is an applied branch of economics
which deals with how people act in auction
markets and researches the properties of auction markets. There are many possible designs (or sets of rules) for an auction and typical issues studied by auction theorists include the efficiency
of a given auction design, optimal and equilibrium bidding strategies, and revenue comparison. Auction theory is also used as a tool to inform the design of real-world auctions; most notably auctions for the privatisation of public-sector companies or the sale of licenses for use of the electromagnetic spectrum.
(i) They may be used to sell any item and so are universal, also
(ii) The outcome of the auction does not depend on the identity of the bidders; i.e., auctions are anonymous.
Most auctions have the feature that participants submit bids, amounts of money they are willing to pay. Standard auctions require that the winner of the auction is the participant with the highest bid. A nonstandard auction does not require this (e.g. a lottery).
Most auction theory revolves around these four "standard" auction types. However, other auction types have also received some academic study, such as:
) available to each player, and a payoff vector corresponding to each combination
of strategies. Generally, the players are the buyer(s) and the seller(s). The action set of each player is a set of bid functions
or reservation price
s. Each bid function maps the player's value
(in the case of a buyer) or cost
(in the case of a seller) to a bid price
. The payoff of each player under a combination of strategies is the expected utility
(or expected
profit) of that player under that combination of strategies.
Game-theoretic models of auctions and strategic bidding generally fall into either of the following two categories. In a private value model, each participant (bidder) assumes that each of the competing
bidders obtains a random private value from a probability distribution
. In a common value
model, each participant assumes that any other participant obtains a random signal from a probability distribution common to all bidders. Usually, but not always, a private values model assumes that the values are independent
across bidders, whereas a common value model usually assumes that the values are independent up to the common parameter
s of the probability distribution.
When it is necessary to make explicit assumptions about bidders' value distributions
, most of the published research assumes symmetric
bidders. This means that the probability distribution from which the bidders obtain their values (or signals) is identical across bidders. In a private values model which assumes independence, symmetry implies that the bidders' values are independently and identically distributed (i.i.d.).
An important example (which does not assume independence) is Milgrom
and Weber
's "general symmetric model" (1982). One of the earlier published theoretical research addressing properties of auctions among asymmetric bidders is Keith Waehrer's 1999 article. Later published research include Susan Athey
's 2001 Econometrica article, as well as Reny and Zamir (2004).
In a simple first-price auction model with two buyers bidding for an object, each buyer might assume that the rival buyer's private value is drawn from the uniform distribution over the interval [0,1], with the cumulative distribution function 
. (Since F is symmetric between the two buyers, this is an auction model with symmetric bidders.) Assuming that: (i) the value of the object for the seller is 0, and (ii) the seller's reservation price is also 0, each buyer's expected utility U as a function of his/her bid price p is equal to the consumer surplus that the buyer will receive conditional on winning 
, multiplied by the likelihood that he or she is going to be the buyer with the highest bid price. That likelihood is given by the probability that this buyer's bid price p exceeds the other buyer's bid price B (expressed as a function of the other buyer's value 
). Express this probability as 
. Then, 
. Assume that each buyer's equilibrium bid price is monotonically increasing in that buyer's value; this implies that the bid function B has an inverse function
. Let Y be the inverse of B:
. Then 
. Since 
is distributed 
, 
, which implies 
. A bid price p maximizes U if U'(p) = 0. Differentiating U with respect to p and setting to zero, 
. Since the buyers are symmetric, in equilibrium it must be the case that 
or (equivalently) 
. Therefore 
. A solution 
of this differential equation is an inverse Nash equilibrium strategy of this game.
At this point, one may conjecture that the (unique) solution is the linear function
and 
for some real number a. Substituting into 
, 
or 
. Solving for a yields 
. Therefore 
satisfies 
. 
implies 
, or 
. Thus, the (unique) Nash equilibrium strategy bidding function of this game is established as 
, at least within the set of invertible bidding functions. Lebrun (1996) provides a more general proof that an equilibrium exists in a first-price auction when the random valuations of the bidders are independent of each other.
The winning bid should be epsilon above the second highest valuation. Relaxing these assumptions
can provide valuable insights for auction design. Decision biases can also lead to predictable
non-equivalencies. Additionally, if some bidders are known to have a higher valuation for
the lot, techniques such as price discriminating against such bidders will yield higher
returns. (In other words, if a bidder is known to value the lot at $X more than the next
highest bidder, the seller can increase their profits by charging that bidder $X-delta more
than any other bidder (or equivalently a special bidding fee of $X - delta). This bidder will
still win the lot, but will pay more than she would otherwise.
is a phenomenon which can occur in common value settings—when the actual values to the different bidders are unknown but correlated, and the bidders make bidding decisions based on estimated values. In such cases, the winner will tend to be the bidder with the highest estimate, and that winner will frequently have bid too much for the auctioned item.
In an equilibrium of such a game, the winner's curse does not occur because the bidders account for the bias in their bidding strategies. Behaviorally and empirically, however, winner's curse is a common phenomenon. (cf. Richard Thaler
).
C7 is the classification for Game Theory and D44 is the classification for Auctions.
Draft edition available online
A very good modern textbook on auction theory.
. A survey.
A recent textbook; see Chapter 11, which presents auction theory from a computational perspective. Downloadable free online.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
which deals with how people act in auction
Auction
An auction is a process of buying and selling goods or services by offering them up for bid, taking bids, and then selling the item to the highest bidder...
markets and researches the properties of auction markets. There are many possible designs (or sets of rules) for an auction and typical issues studied by auction theorists include the efficiency
Efficiency (economics)
In economics, the term economic efficiency refers to the use of resources so as to maximize the production of goods and services. An economic system is said to be more efficient than another if it can provide more goods and services for society without using more resources...
of a given auction design, optimal and equilibrium bidding strategies, and revenue comparison. Auction theory is also used as a tool to inform the design of real-world auctions; most notably auctions for the privatisation of public-sector companies or the sale of licenses for use of the electromagnetic spectrum.
General idea
Auctions take many forms but always satisfy two conditions:(i) They may be used to sell any item and so are universal, also
(ii) The outcome of the auction does not depend on the identity of the bidders; i.e., auctions are anonymous.
Most auctions have the feature that participants submit bids, amounts of money they are willing to pay. Standard auctions require that the winner of the auction is the participant with the highest bid. A nonstandard auction does not require this (e.g. a lottery).
Types of auction
There are traditionally four types of auction that are used for the allocation of a single item:- First-price sealed-bid auctions in which bidders place their bid in a sealed envelope and simultaneously hand them to the auctioneer. The envelopes are opened and the individual with the highest bid wins, paying a price equal to the exact amount that he or she bid.
- Second-price sealed-bid auctions (Vickrey auctions) in which bidders place their bid in a sealed envelope and simultaneously hand them to the auctioneer. The envelopes are opened and the individual with the highest bid wins, paying a price equal to the exact amount of the second highest bid.
- Open Ascending-bid auctions (English auctions)English auctionAn English auction is a type of auction, whose most typical form is the "open outcry" auction. The auctioneer opens the auction by announcing a Suggested Opening Bid, a starting price or reserve for the item on sale and then accepts increasingly higher bids from the floor consisting of buyers with...
in which the price is steadily raised by the auctioneer with bidders dropping out once the price becomes too high. This continues until there remains only one bidder who wins the auction at the current price. - Open Descending-bid auctions (Dutch auctions)Dutch auctionA Dutch auction is a type of auction where the auctioneer begins with a high asking price which is lowered until some participant is willing to accept the auctioneer's price, or a predetermined reserve price is reached. The winning participant pays the last announced price...
in which the price starts at a level sufficiently high to deter all bidders and is progressively lowered until a bidder indicates that he is prepared to buy at the current price. He or she wins the auction and pays the price at which they bid.
Most auction theory revolves around these four "standard" auction types. However, other auction types have also received some academic study, such as:
- All-pay auctionAll-pay auctionIn economics and game theory an all-pay auction, is an auction in which all bidders must pay regardless of whether they win the prize, which is awarded to the highest bidder as in a conventional auction...
s in which bidders place their bid in a sealed envelope and simultaneously hand them to the auctioneer. The envelopes are opened and the individual with the highest bid wins, paying a price equal to the exact amount that he or she bid. All losing bidders are also required to make a payment to the auctioneer equal to their own bid in an all-pay auction. This auction format is non-standard, but can be used to understand things such as election campaigns (in which bids can be interpreted as campaign spending) or queuing for a scarce commodity (in which your bid is the amount of time for which you are prepared to queue). - Amsterdam auctions, a type of premium auction which begins as an English auction. Once only two bidders remain, each submits a sealed bid. The higher bidder wins, paying either the first or second price. Both finalists receive a premium: a proportion of the excess of the second price over the third price (at which English auction ended).
- Unique bid auctionUnique bid auctionA unique bid auction is a type of strategy game related to traditional auctions where the winner is usually the individual with the lowest unique bid, although less commonly the auction rules may specify that the highest unique bid is the winner...
s - Many homogenous item auctions, e.g., spectrum auctionSpectrum auctionA spectrum auction is a process whereby a government uses an auction system to sell the rights to transmit signals over specific bands of the electromagnetic spectrum and to assign scarce spectrum resources. Depending on the specific auction format used, a spectrum auction can last from a single...
s - Simultaneous multiple-round auctions
- Position auctions
- Generalized second-price auctionGeneralized second-price auctionThe generalized second-price auction is a non-truthful auction mechanism for multiple items. First thought of as a natural extension of the Vickrey auction, it actually doesn't conserve some good properties of the Vickrey auction . It is used mainly in the context of keyword auctions, where...
- Menu auction
- Ascending package auction
Game-theoretic models
A game-theoretic auction model is a mathematical game represented by a set of players, a set of actions (strategiesStrategy
Strategy, a word of military origin, refers to a plan of action designed to achieve a particular goal. In military usage strategy is distinct from tactics, which are concerned with the conduct of an engagement, while strategy is concerned with how different engagements are linked...
) available to each player, and a payoff vector corresponding to each combination
Combination
In mathematics a combination is a way of selecting several things out of a larger group, where order does not matter. In smaller cases it is possible to count the number of combinations...
of strategies. Generally, the players are the buyer(s) and the seller(s). The action set of each player is a set of bid functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
or reservation price
Reservation price
In microeconomics, the reservation price is the highest price a buyer is willing to pay for goods or a service; or; the smallest price at which a seller is willing to sell a good or service...
s. Each bid function maps the player's value
Value (economics)
An economic value is the worth of a good or service as determined by the market.The economic value of a good or service has puzzled economists since the beginning of the discipline. First, economists tried to estimate the value of a good to an individual alone, and extend that definition to goods...
(in the case of a buyer) or cost
Cost
In production, research, retail, and accounting, a cost is the value of money that has been used up to produce something, and hence is not available for use anymore. In business, the cost may be one of acquisition, in which case the amount of money expended to acquire it is counted as cost. In this...
(in the case of a seller) to a bid price
Price
-Definition:In ordinary usage, price is the quantity of payment or compensation given by one party to another in return for goods or services.In modern economies, prices are generally expressed in units of some form of currency...
. The payoff of each player under a combination of strategies is the expected utility
Expected utility hypothesis
In economics, game theory, and decision theory the expected utility hypothesis is a theory of utility in which "betting preferences" of people with regard to uncertain outcomes are represented by a function of the payouts , the probabilities of occurrence, risk aversion, and the different utility...
(or expected
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
profit) of that player under that combination of strategies.
Game-theoretic models of auctions and strategic bidding generally fall into either of the following two categories. In a private value model, each participant (bidder) assumes that each of the competing
Competition
Competition is a contest between individuals, groups, animals, etc. for territory, a niche, or a location of resources. It arises whenever two and only two strive for a goal which cannot be shared. Competition occurs naturally between living organisms which co-exist in the same environment. For...
bidders obtains a random private value from a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
. In a common value
Common value auction
A common value auction is a term in economics used to describe an environment in which information about the value of the object for sale is dispersed among bidders. The term is used in different ways by different people. By one definition it describes an auction in which the good being auctioned...
model, each participant assumes that any other participant obtains a random signal from a probability distribution common to all bidders. Usually, but not always, a private values model assumes that the values are independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
across bidders, whereas a common value model usually assumes that the values are independent up to the common parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
s of the probability distribution.
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
, most of the published research assumes symmetric
Symmetric game
In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If one can change the identities of the players without changing the payoff to the strategies, then a game is symmetric. ...
bidders. This means that the probability distribution from which the bidders obtain their values (or signals) is identical across bidders. In a private values model which assumes independence, symmetry implies that the bidders' values are independently and identically distributed (i.i.d.).
An important example (which does not assume independence) is Milgrom
Paul Milgrom
Paul Robert Milgrom is an American economist. He is the Shirley and Leonard Ely Professor of Humanities and Sciences at Stanford University, a position he has held since 1987. Dr. Milgrom is an expert in game theory, specifically auction theory and pricing strategies...
and Weber
Robert J. Weber
Robert J. Weber is the Frederic E. Nemmers Distinguished Professor of Decision Sciences at the J.L. Kellogg Graduate School of Management, Northwestern University.-Education:...
's "general symmetric model" (1982). One of the earlier published theoretical research addressing properties of auctions among asymmetric bidders is Keith Waehrer's 1999 article. Later published research include Susan Athey
Susan Athey
Susan Carleton Athey is an American economist. She is currently Professor of Economics at Harvard University and the first female winner of the John Bates Clark Medal.- Early life :...
's 2001 Econometrica article, as well as Reny and Zamir (2004).
. (Since F is symmetric between the two buyers, this is an auction model with symmetric bidders.) Assuming that: (i) the value of the object for the seller is 0, and (ii) the seller's reservation price is also 0, each buyer's expected utility U as a function of his/her bid price p is equal to the consumer surplus that the buyer will receive conditional on winning , multiplied by the likelihood that he or she is going to be the buyer with the highest bid price. That likelihood is given by the probability that this buyer's bid price p exceeds the other buyer's bid price B (expressed as a function of the other buyer's value ). Express this probability as . Then, . Assume that each buyer's equilibrium bid price is monotonically increasing in that buyer's value; this implies that the bid function B has an inverse functionInverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
. Let Y be the inverse of B:
. Then . Since is distributed , , which implies . A bid price p maximizes U if U'(p) = 0. Differentiating U with respect to p and setting to zero, . Since the buyers are symmetric, in equilibrium it must be the case that or (equivalently) . Therefore . A solution of this differential equation is an inverse Nash equilibrium strategy of this game.At this point, one may conjecture that the (unique) solution is the linear function
and for some real number a. Substituting into , or . Solving for a yields . Therefore satisfies . implies , or . Thus, the (unique) Nash equilibrium strategy bidding function of this game is established as , at least within the set of invertible bidding functions. Lebrun (1996) provides a more general proof that an equilibrium exists in a first-price auction when the random valuations of the bidders are independent of each other.Revenue equivalence
One of the major findings of Auction Theory is the celebrated Revenue Equivalence Theorem (first proved by Vickrey in 1961 and generalized 20 years later by Myerson, and independently by Riley and Samuelson.), which states that any allocation mechanism/auction in which (i) the bidder with the highest type/signal/valuation always wins, (ii) the bidder with the lowest possible type/valuation/signal expects zero surplus, (iii) all bidders are risk neutral and (iv) all bidders are drawn from a strictly increasing and atomless distribution will lead to the same expected revenue for the seller (and player i of type v can expect the same surplus across auction types).The winning bid should be epsilon above the second highest valuation. Relaxing these assumptions
can provide valuable insights for auction design. Decision biases can also lead to predictable
non-equivalencies. Additionally, if some bidders are known to have a higher valuation for
the lot, techniques such as price discriminating against such bidders will yield higher
returns. (In other words, if a bidder is known to value the lot at $X more than the next
highest bidder, the seller can increase their profits by charging that bidder $X-delta more
than any other bidder (or equivalently a special bidding fee of $X - delta). This bidder will
still win the lot, but will pay more than she would otherwise.
Winner's curse
The winner's curseWinner's curse
The winner's curse is a phenomenon akin to a Pyrrhic victory that occurs in common value auctions with incomplete information. In short, the winner's curse says that in such an auction, the winner will tend to overpay...
is a phenomenon which can occur in common value settings—when the actual values to the different bidders are unknown but correlated, and the bidders make bidding decisions based on estimated values. In such cases, the winner will tend to be the bidder with the highest estimate, and that winner will frequently have bid too much for the auctioned item.
In an equilibrium of such a game, the winner's curse does not occur because the bidders account for the bias in their bidding strategies. Behaviorally and empirically, however, winner's curse is a common phenomenon. (cf. Richard Thaler
Richard Thaler
Richard H. Thaler is an American economist and the Ralph and Dorothy Keller Distinguished Service Professor of Behavioral Science and Economics at the University of Chicago Booth School of Business...
).
JEL classification
In the Journal of Economic Literature Classification SystemJEL classification codes
Articles in economics journals are usually classified according to the JEL classification codes, a system originated by the Journal of Economic Literature. The JEL is published quarterly by the American Economic Association and contains survey articles and information on recently published books...
C7 is the classification for Game Theory and D44 is the classification for Auctions.
Further reading
- Cassady, R. (1967). Auctions and auctioneering. University of California PressUniversity of California PressUniversity of California Press, also known as UC Press, is a publishing house associated with the University of California that engages in academic publishing. It was founded in 1893 to publish books and papers for the faculty of the University of California, established 25 years earlier in 1868...
. An influential early survey.
- Klemperer, P. (Ed.). (1999b). The economic theory of auctions. Edward Elgar. A collection of seminal papers in auction theory.
- Klemperer, P. (1999a). Auction theory: A guide to the literature. Journal of Economic Surveys, 13(3), 227–286. A good modern survey; the first chapter of the preceding book.
Draft edition available online
A very good modern textbook on auction theory.
. A survey.
- Myerson, R. (1981). Optimal auction design. Mathematics of Operations ResearchMathematics of Operations ResearchMathematics of Operations Research is a scholarly journal published since 1976. The founding editor was Arthur F. Veinott, Jr. of Stanford University, who served as editor-in-chief 1976-1980. MOR is published quarterly by INFORMS and indexed by Journal Citation Reports...
, 6(1), 58–73. A seminal paper, introduced revenue equivalence and optimal auctions.
- Riley, J., and Samuelson, W. (1981). Optimal auctions. The American Economic Review, 71(3), 381–392. A seminal paper; published concurrently with Myerson's paper cited above.
A recent textbook; see Chapter 11, which presents auction theory from a computational perspective. Downloadable free online.
- Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. The Journal of Finance, 16(1), 8–37. A pathbreaking paper that introduced second price auctions and performed new analysis of first price.
- Wilson, R. (1987a). Auction theory. In J. Eatwell, M. Milgate, P. Newman (Eds.), The New Palgrave Dictionary of EconomicsThe New Palgrave Dictionary of EconomicsThe New Palgrave Dictionary of Economics , 2nd Edition, is an eight-volume reference work, edited by Steven N. Durlauf and Lawrence E. Blume. It contains 5.8 million words and spans 7,680 pages with 1,872 articles. Included are 1057 new articles and, from earlier, 80 essays that are designated as...
, vol. I. London: Macmillan.