Auction theory

(Redirected from Revenue equivalence theorem)

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.

General idea

Auctions take many forms but always satisfy two conditions:

1. They may be used to sell any item and so are universal, also
2. 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 the amount 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 second-highest bid.
• 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 only actually sold 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 "standard" auction types. However, other auction types have also received some academic study, such as:

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 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 parameters of the probability distribution.

Ex-post equilibrium in a simple auction market.

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).[2][3] One of the earlier published theoretical research addressing properties of auctions among asymmetric bidders is Keith Waehrer's 1999 article.[4] Later published research include Susan Athey's 2001 Econometrica article,[5] as well as Reny and Zamir (2004).[6]

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

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

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

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

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

$w(b)=\Pr {{\{{{b}_{2}}

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

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

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

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

It is not difficult to show that B(v) is the unique symmetric equilibrium. Lebrun (1996)[7] provides a general proof that there are no asymmetric equilibria.

Revenue equivalence

One of the major findings of auction theory is the celebrated 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). In 1979 Riley & Samuelson (1981) 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/auction in which

1. the bidder with the highest type/signal/valuation always wins
2. the bidder with the lowest possible type/valuation/signal expects zero surplus
3. all bidders are risk neutral, and
4. 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).

Revenue equivalence of the open ascending price (or English) auction and sealed first price (or high-bid) auction.

In the open ascending price auction a buyer’s dominant strategy is to remain in the auction until the asking price is equal to his value. Consider the case of two buyers, each with a value that is an independent draw from a distribution with support [0,1], cumulative distribution function F(v) and probability density function f(v). If buyers bid according to their dominant strategies, then a buyer with value v wins if his opponent’s value x is lower. Thus his win probability is

$w=\Pr \{x

and his expected payment is

$C(v)=\int\limits_{0}^{v}{{}}xf(x)dx$

The expected payment conditional upon winning is therefore

$e(v)=\frac{C(v)}{F(v)}=\frac{\int\limits_{0}^{v}{{}}xf(x)dx}{F(v)}$

Multiplying both sides by F(v) and differentiating by v yields the following differential equation for e(v).

${e}'(v)F(v)+e(v)f(v)=vf(v)$.

Rearranging this equation,

${e}'(v)=\frac{f(v)}{F(v)}(v-e(v))$

Let B(v) be the equilibrium bid function in the sealed first-price auction. We establish revenue equivalence by showing that B(v)=e(v), that is, the equilibrium payment by the winner in one auction is equal to the equilibrium expected payment by the winner in the other.

Suppose that a buyer has value v and bids b. His opponent bids according to the equilibrium bidding strategy. The support of the opponent’s bid distribution is [0,B(1)]. Thus any bid of at least B(1) wins with probability 1. Therefore the best bid b lies in the interval [0,B(1)] and so we can write this bid as b = B(x) where x lies in [0,1]. If the opponent has value y he bids B(y). Therefore the win probability is

$w=\Pr \{b.

The buyer’s expected payoff is his win probability times his net gain if he wins, that is,

$U=w(v-B(x))=F(x)(v-B(x))$.

Differentiating, the necessary condition for a maximum is

${U}'(x)=f(x)(v-B(x))-F(x){B}'(x)=F(x)\left(\frac{f(x)}{F(x)}(v-B(x))-{B}'(x)\right)=0$.

That is if B(x) is the buyer’s best response it must satisfy this first order condition. Finally we note that for B(v) to be the equilibrium bid function, the buyer’s best response must be B(v). Thus x=v. Substituting for x in the necessary condition,

$\frac{f(v)}{F(v)}(v-B(v))-{B}'(v)=0$.

Note that this differential equation is identical to that for e(v). Since e(0)=B(0)=0 it follows that B(v)=e(v)

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.[8]

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, 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).

The linkage principle is an important result in auction theory that allows revenue comparisons amongst a fairly general class of auctions with interdependence between bidders' values.

JEL classification

In the Journal of Economic Literature Classification System C7 is the classification for Game Theory and D44 is the classification for Auctions.[9]

Footnotes

1. ^ "The Amsterdam Auction" (PDF). Retrieved 2011-08-27.
2. ^ Milgrom, P., and R. Weber (1982) "A Theory of Auctions and Competitive Bidding," Econometrica Vol. 50 No. 5, pp. 1089–1122.
3. ^ Because bidders in real-world auctions are rarely symmetric, applied scientists began to research auctions with asymmetric value distributions beginning in the late 1980s. Such applied research often depended on numerical solution algorithms to compute an equilibrium and establish its properties. Preston McAfee and John McMillan (1989) simulated bidding for a government contract in which the cost distribution of domestic firms is different from the cost distribution of the foreign firms ("Government Procurement and International Trade," Journal of International Economics, Vol. 26, pp. 291–308.) One of the publications based on the earliest numerical research is Dalkir, S., J. W. Logan, and R. T. Masson, "Mergers in Symmetric and Asymmetric Noncooperative Auction Markets: The Effects on Prices and Efficiency," published in Vol. 18 of The International Journal of Industrial Organization, (2000, pp. 383–413). Other pioneering research include Tschantz, S., P. Crooke, and L. Froeb, "Mergers in Sealed versus Oral Auctions," published in Vol. 7 of The International Journal of the Economics of Business (2000, pp. 201–213).
4. ^ K. Waehrer (1999) "Asymmetric Auctions With Application to Joint Bidding and Mergers," International Journal of Industrial Organization 17: 437–452
5. ^ Athey, S. (2001) "Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information," Econometrica Vol. 69 No. 4, pp. 861-890.
6. ^ Reny, P., and S. Zamir (2004) "On the Existence of Pure Strategy Monotone Equilibria in Asymmetric First-Price Auctions," Econometrica, Vol. 72 No. 4, pp. 1105–1125.
7. ^ Lebrun, Bernard (1996) "Existence of an equilibrium in first price auctions," Economic Theory, Vol. 7 No. 3, pp. 421-443.
8. ^ McAfee, R. Preston; McMillan, John (1987). "Auctions and Bidding". Journal of Economic Literature 25 (2): 699–738. JSTOR 2726107.
9. ^ "Journal of Economic Literature Classification System". American Economic Association. Retrieved 2008-06-25. (D: Microeconomics, D4: Market Structure and Pricing, D44: Auctions)