Auction theory

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Auction theory is an applied branch of economics which deals with how bidders act in auction markets and researches how the features of auction markets incentivise predictable outcomes. Auction theory is a tool used to inform the design of real-world auctions. Sellers use auction theory to raise higher revenues while allowing buyers to procure at a lower cost. The conference of the price between the buyer and seller is an economic equilibrium. Auction theorists design rules for auctions to address issues which can lead to market failure. The design of these rulesets encourages optimal bidding strategies among a variety of informational settings.^[1] The 2020 Nobel Prize for Economics was awarded to Paul R. Milgrom and Robert B. Wilson “for improvements to auction theory and inventions of new auction formats.”^[2]

Introduction[]

Auctions facilitate transactions by enforcing a specific set of rules regarding the resource allocations of a group of bidders. Theorists consider auctions to be economic games that differ in two respects: format and information.^[3] The format defines the rules for the announcement of prices, the placement of bids, the updating of prices, the auction close, and the way a winner is picked.^[4] The way auctions differ with respect to information regards the asymmetries of information that exist between bidders.^[5] In most auctions, bidders have some private information that they choose to withhold from their competitors. For example, bidders usually know their personal valuation of the item, which is unknown to the other bidders and the seller; however, the behaviour of bidders can influence the personal valuation of other bidders.

1994 Nobel Laureate for Economic Sciences, John Nash,^[6] designed a generalized theory of auctions as a non-cooperative game which moves beyond simple zero-sum games. This theory was vital for the theorisation of auctions since the goal of auctions is to assign an object to the buyer who will make the most use of it for the highest price, thus maximising value for both buyer and seller. Nash developed a way for auctions to facilitate absolute gains for society. Vickrey (1996 Nobel Laureate) and Harsanyi (1994 Laureate) extended on Nash's equilibrium specifying ways in which equilibrium can be reached under informational settings. By the 1990s, auction theorists had defined equilibrium bidding conditions for single-object auctions under most realistic auction formats and information settings.^[7] The state-of-the-art considers how multiple-object auctions can be performed efficiently; Robert B. Wilson and Paul Milgrom won The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2020 for their work in defining these auctions.^[8] Some other state-of-the-art auction designs are Product-Mix Auctions which allows 'package bidding' which was implemented by Paul Klemperer in response to the 2007 Northern Rock Bank Run to sell troubled debt, and Position Auctions which implement a generalized Second-price auction which Google has used to sell ads on Internet search keywords efficiently.^[9]

Auction types[]

There are traditionally four types of auction that are used for the allocation of a single item:

• First-price sealed-bid auction 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 the amount bid. This form of auction requires complex game theorisation since bidders must not only consider their valuation but other bidders' valuations and what other bidders believe other bidders' valuations are.^[10]
• 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 second-highest bid. The logic of this auction type is that the dominant strategy for all bidders is to bid their true valuation ^[11] William Vickrey was the first scholar to study second-price valuation auctions, but their use goes back in history with some evidence suggesting that Goethe sold his manuscripts to a publisher using the second-price auction format.^[12] Online auctions often use an equivalent version of Vickrey's second-price auction wherein bidders provide proxy bids for items. A proxy bid is an amount an individual values some item at. The online auction house will bid up the price of the item until the proxy bid for the winner is at the top. However, the individual only has to pay one increment higher than the second-highest price, despite their own proxy valuation.^[13]
• Open ascending-bid auctions (English auctions) in which participants make increasingly higher bids, each stopping bidding when they are not prepared to pay more than the current highest bid. This continues until no participant is prepared to make a higher bid; the highest bidder wins the auction at the final amount bid. Sometimes the lot is sold only if the bidding reaches a reserve price set by the seller.
• Open descending-bid auctions (Dutch auctions) in which the price is set by the auctioneer at a level sufficiently high to deter all bidders, and is progressively lowered until a bidder is prepared to buy at the current price, winning the auction.

Most auction theory revolves around these four "basic" auction types. However, others have also received some academic study (see Auction § Types).

Benchmark model[]

The benchmark model for auctions, as defined by McAfee and McMillan (1987), offers a generalization of auction formats, and is based on four assumptions:

1. All of the bidders are risk-neutral.
2. Each bidder has a private valuation for the item independently drawn from some probability distribution.
3. The bidders possess symmetric information.
4. The payment is represented as a function of only the bids.

The benchmark model is often used in tandem with the Revelation Principle, which states that each of the basic auction types is structured such that each bidder has the incentive to report their valuation honestly. The two are primarily used by sellers to determine the auction type that maximizes the expected price. This optimal auction format is defined such that the item will be offered to the bidder with the highest valuation at a price equal to their valuation, but the seller will refuse to sell the item if they expect that all of the bidders' valuations of the item are less than their own.^[14]

Relaxing each of the four main assumptions of the benchmark model yields auction formats with unique characteristics:

• Risk-averse bidders incur some kind of cost from participating in risky behaviours, which affects their valuation of a product. In sealed-bid first-price auctions, risk-averse bidders are more willing to bid more to increase their probability of winning, which, in turn, increases their expected utility. This allows sealed-bid first-price auctions to produce higher expected revenue than English and sealed-bid second-price auctions.
• In formats with correlated values—where the bidders’ values for the item are not independent—one of the bidders perceiving their value of the item to be high makes it more likely that the other bidders will perceive their own values to be high. A notable example of this instance is the Winner’s curse, where the results of the auction convey to the winner that everyone else estimated the value of the item to be less than they did. Additionally, the linkage principle allows revenue comparisons amongst a fairly general class of auctions with interdependence between bidders' values.
• The asymmetric model assumes that bidders are separated into two classes that draw valuations from different distributions (e.g., dealers and collectors in an antique auction).
• In formats with royalties or incentive payments, the seller incorporates additional factors, especially those that affect the true value of the item (e.g., supply, production costs, and royalty payments), into the price function.^[14]

Game-theoretic models[]

A game-theoretic auction model is a mathematical game represented by a set of players, a set of actions (strategies) 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 prices (reserves). 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 , each participant (bidder) assumes that each of the competing bidders obtains a random private value from a probability distribution. In a common value model, the participants have equal valuations of the item, but they do not have perfectly accurate information about this valuation. In lieu of knowing the exact valuation of the item, each participant can assume that any other participant obtains a random signal, which can be used to estimate the true valuation, from a probability distribution common to all bidders.^[15] 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 parameters of the probability distribution.

A more general category for strategic bidding is the affiliated values model, in which the bidder's total utility depends on both their individual private signal and some unknown common value. Both the private value and common value models can be perceived as extensions of the general affiliated values model.^[16]

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 "i.i.d." – independently and identically distributed.

An important example (which does not assume independence) is Milgrom and Weber's "general symmetric model" (1982).^[17][18] One of the earlier published theoretical research addressing properties of auctions among asymmetric bidders is Keith Waehrer's 1999 article.^[19] Later published research include Susan Athey's 2001 Econometrica article,^[20] as well as Reny and Zamir (2004).^[21]

The first formal analysis of auctions was by William Vickrey (1961). Vickrey considers two buyers bidding for a single item. Each buyer's value, v, is an independent draw from a uniform distribution with support [0,1]. Vickrey showed that in the sealed first-price auction it is an equilibrium bidding strategy for each bidder to bid half his valuation. With more bidders, all drawing a value from the same uniform distribution it is easy to show that the symmetric equilibrium bidding strategy is

${\displaystyle B(v)=\left({\frac {n-1}{n}}\right)v}$.

To check that this is an equilibrium bidding strategy we must show that if it is the strategy adopted by the other n-1 buyers, then it is a best response for buyer 1 to adopt it also. Note that buyer 1 wins with probability 1 with a bid of (n-1)/n so we need only consider bids on the interval [0,(n-1)/n]. Suppose buyer 1 has value v and bids b. If buyer 2's value is x he bids B(x). Therefore, buyer 1 beats buyer 2 if

${\displaystyle B(x)=\left({\frac {n-1}{n}}\right)x<b}$ that is ${\displaystyle x<\left({\frac {n}{n-1}}\right)b}$

Since x is uniformly distributed, buyer 1 bids higher than buyer 2 with probability nb/(n-1). To be the winning bidder, buyer 1 must bid higher than all the other bidders (which are bidding independently). Then his win probability is

${\displaystyle w(b)=\Pr {{\{{{b}_{2}}<b\}}^{n-1}}={{\left({\frac {n}{n-1}}\right)}^{n-1}}{{b}^{n-1}}}$

Buyer 1's expected payoff is his win probability times his gain if he wins. That is,

${\displaystyle U(b)=w(b)(v-b)={{\left({\frac {n}{n-1}}\right)}^{n-1}}{{b}^{n-1}}(v-b)={{\left({\frac {n}{n-1}}\right)}^{n-1}}({{b}^{n-1}}v-{{b}^{n}})}$

It is readily confirmed by differentiation that U(b) takes on its maximum at

${\displaystyle B(v)=\left({\frac {n-1}{n}}\right)v}$

It is not difficult to show that B(v) is the unique symmetric equilibrium. Lebrun (1996)^[22] provides a general proof that there are no asymmetric equilibria.

Revenue equivalence[]

One of the major findings of auction theory is the revenue equivalence theorem. Early equivalence results focused on a comparison of revenue in the most common auctions. The first such proof, for the case of two buyers and uniformly distributed values was by Vickrey (1961) harvtxt error: no target: CITEREFVickrey1961 (help). In 1979 Riley & Samuelson (1981) harvtxt error: no target: CITEREFRileySamuelson1981 (help) proved a much more general result. (Quite independently and soon after, this was also derived by Myerson (1981)).The revenue equivalence theorem states that any allocation mechanism or auction that satisfies the four main assumptions of the benchmark model will lead to the same expected revenue for the seller (and player i of type v can expect the same surplus across auction types).^[14]

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 – Δ (a sum just slightly inferior to the sum is willing to pay) more than any other bidder (or equivalently a special bidding fee of $X – Δ). This bidder will still win the lot, but will pay more than would otherwise be the case.^[14]

Winner's curse[]

The winner's curse 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, but the results of the auction will show that the remaining bidders' estimates of the item's value are less than that of the winner, giving the winner the impression that they "bid too much".^[14]

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, described in detail by Richard Thaler.

Optimal reserve prices[]

Myerson (1981) has shown that in the case of independent private values, the optimal reserve price does not depend on the number of bidders.^[23] For example, suppose there is a single potential buyer whose valuation is uniformly distributed on the interval [0,100]. If the seller can make a take-it-or-leave-it price offer, the optimal price is 50. The reason is that the buyer will buy whenever the buyer’s valuation v is at least as large as the price p. Since the probability that v is larger than p is given by 100-p percent, the seller’s expected profit is p·(100-p)/100, which is maximized by p=50. Myerson (1981) proves that the optimal reserve price remains to be 50 in this example, regardless of the number of potential buyers.

Bulow and Klemperer (1996) have shown that an auction with n bidders and an optimally chosen reserve price generates a smaller expected profit for the seller than a standard auction with n+1 bidders (and no reserve price).^[24]

JEL classification[]

In the Journal of Economic Literature Classification System C7 is the classification for game theory and D44 is the classification for auctions.^[25]

Applications to business strategy[]

Scholars of managerial economics have noted some applications of auction theory in business strategy. Namely, auction theory can be applied to preemption games and attrition games.^[26]

Preemption games are a game where entrepreneurs will preempt other firms in entering a market with new technology before it's ready for commercial deployment. The value generated from waiting for the technology to become commercially viable also increases the risk that a competitor will enter the market preemptively. Preemptive games can be modeled as a first-priced sealed auction. Both companies would prefer to enter the market when the technology is ready for commercial deployment; this can be considered the valuation of both companies. However, one firm might hold information stating that technology is viable earlier than the other firm believes. The company with better information would, then, enter the market and bid to enter the market earlier, even as the risk of failure is higher.

Games of attrition are games of preempting other firms to leave the market. This often occurs in the airline industry as these markets are considered highly contestable.^[27] As a new airline enters the market, they will decrease prices to gain market share. This forces the incumbent airline to also decrease prices to avoid losing market share. This creates an auction game. Usually, market entrants will use a strategy of attempting to bankrupt the incumbent. Thus, the auction is measured in how much each firm is willing to lose as they stay in the game of attrition. The firm that lasts the longest in the game wins the market share. This strategy has been used more contemporaneously by entertainment streaming services like Netflix, Hulu, Disney+ and HBOMax who are all loss-making firms attempting to gain market share by bidding on more expansive entertainment content.^[28]

Footnotes[]

Further reading[]

• Cassady, R. (1967). Auctions and auctioneering. University of California Press. 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.
• Klemperer, Paul (2004). Auctions: Theory and Practice. Princeton University Press. ISBN 0-691-11925-2. Draft edition available online
• Krishna, Vijay (2002). Auction theory. New York: Elsevier. ISBN 978-0-12-426297-3. A very good modern textbook on auction theory.
• McAfee, R. P. and J. McMillan (1987). "Auctions and Bidding". Journal of Economic Literature. 25: 708–47.. A survey.
• Myerson, R. (1981). Optimal auction design. Mathematics of Operations Research, 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.
• Parsons, S., Rodriguez-Aguilar, J. A., and Klein, M. (2011). Auctions and bidding: A guide for computer scientists.
• Shoham, Yoav; Leyton-Brown, Kevin (2009). Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. New York: Cambridge University Press. ISBN 978-0-521-89943-7. 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 Economics, vol. I. London: Macmillan.

External links[]

• Auctions on GameTheory.net, also available on the Wayback Machine