Tài liệu Pricing communication networks P14 doc

Auctions, however, can be efficient even when there are asmall number of bidders, although the optimal strategy for some may be not to tell the truth.There are two important and distinct

Auctions

An auction is a sale in which the price of an item is determined by bidding Flowers, wine,antiques, US treasury bonds and land are sold in auctions Takeover battles for companiescan be viewed as auctions (and indeed, the Roman empire was auctioned by the PraetorianGuards in A.D 193) Auctions are commonly used to sell natural resources, such as oildrilling rights, or even the rights to use certain geostationary satellite positions Governmentcontracts are often awarded through procurement auctions There is the advantage that thesale can be performed openly, so that no one can claim that a government official awardedthe contract to the supplier who offers him the greatest bribe In Section 9.4.4 we sawhow instantaneous bandwidth might be sold in a smart market in which the price is set

by auction In recent years, auctions have been used in the communications market to sellparts of the spectrum for mobile telephone licenses Some of these have raised huge sumsfor the government, but others have raised less than expected

An auction can be viewed as a partial information game in which the valuations that eachbidder places on the items for sale is hidden from the auctioneer and the other bidders Thegame’s equilibrium is a function of the auction’s rules, which specify the way bidding oc-curs, the information bidders have about the state of bidding, how the winner is determinedand how much he must pay These rules can affect the revenue obtained by the seller, as well

as how much this varies in successive instances of the auction An auction is economicallyefficient, in terms of maximizing social welfare, if it allocates items to bidders who valuethem most We emphasize that designing an auction for a particular situation is an art There

is no single auctioning mechanism that is provably efficient and can be applied in most ations For example, in spectrum auctions some combinations of spectrum licenses are morevaluable to bidders than others, and so licenses must be sold in packages, using some sort

situ-of combinatorial bidding As we explain in Section 14.2.2, this greatly complicates auctiondesign One can prove important theoretical results about some simple auction mechanisms,(such as the revenue equivalence theorem of Section 14.1.3) They are not easily applied

in many real life situations, but they do provide insights into the problems involved.The purpose of this chapter is to provide the reader with an introduction to auction theoryand some examples of how it can be used in pricing communications services Auctiontheory is now a very well-developed area of research, and we can do no more than give anintroduction and some interesting results We have previously discussed how the mechanism

of tatonnement can be used to maximize social welfare in resource allocation problems(Section 5.4.1) In tatonnement, price is varied in response to excess demand (positive or

Costas Courcoubetis and Richard Weber Copyright  2003 John Wiley & Sons, Ltd.

ISBN: 0-470-85130-9

310 AUCTIONSnegative) until demand exactly matches supply One crucial property of any tatonnementmechanism is that prices should be able to increase or decrease until that point is reached.Auction mechanisms do not usually allow prices to fluctuate in both directions Tatonnementcan take a large number of steps Some auctions take place in just one step, with littleinformation exchange between the buyers and seller In general, auctions are more restrictedthan tatonnement, and do not necessarily maximize social welfare However, they have theadvantage that they can be faster and simpler to implement A second requirement for thetatonnement mechanism to work is that customers should make truthful declarations of theirresource needs for given posted prices This will happen if the market has many customers,with no customer being so large that he can affect the price by the size of his own demand.That is, customers are price takers Auctions, however, can be efficient even when there are asmall number of bidders, although the optimal strategy for some may be not to tell the truth.There are two important and distinct models for the way bidders value items in an auction.

In the private value model , each bidder knows the value that he places on a given item, but

he does not know the valuations of other bidders As bidding takes place, his valuation doesnot change, although he may gain information about other bidders’ valuations when he hears

their bids In the common value model , all bidders estimate their valuation of the item in the

same way, but they have different prior information about that value Suppose, for example,

a jar of coins is to be auctioned Each bidder estimates the value of the coins in the jar, and

as bidding occurs he adjusts his estimate on the basis of what others say For example, ifmost bidders make higher bids than his own, a bidder might feel that he should increase hisestimate of the value of the coins In this case, the winner generally over-estimates the value(since he has the highest estimate), and so it is likely that he pays more than the jar of coins

is worth This is known as the winner’s curse (about which we say more in Section 14.1.7).

Sometimes a bidder’s valuation is a function of both private information and of informationrevealed during the auction For example, suppose an oil-lease is to be auctioned The value

of the lease depends both upon the amount of oil that is in the ground and the efficiencywith which it can be extracted Bidders may have different geological information aboutthe likely amount of oil, and have different extraction efficiencies, and so make differentestimates of the value of the lease During bidding, bidders reveal information about theirestimates and this may be helpful to other bidders

There are many other considerations that come into play when designing auctions The

seller may impose a participation fee, or a minimum reserve price An auction can be oral (bidders hear each other’s bids and make counter-offers) or written (bidders submit closed

sealed-bids in writing) In an oral auction, the number of bidders may be known, but in

a sealed-bid auction the number is often unknown Oral auctions proceed in a progressivemanner, taking many rounds to complete, while sealed-bid auctions may take only a singleround All these things can influence the way bidders compete; by making them competemore fiercely, the seller’s revenue is increased

In Section 14.1 we describe some types of auction and summarize some importanttheoretical results These concern auctions of a single item However, one may wish to sell

more than a single item In a multi-object auction, multiple units of the same or of different items are to be sold Such auctions can be homogeneous or heterogeneous, depending on the items to be sold are identical or not; discriminatory or uniform price, depending on

whether identical items are sold at different or equal prices (this distinction only applies

to homogeneous auctions); individual or combinatorial, depending on whether bids are allowed only for individual items or for combinations of items; sequential or simultaneous,

depending on the whether items are auctioned one at a time or all at once We take up

these issues in Section 14.2 Note, however, that we opt for an informal presentation of themulti-object auction, as there are few rigorous results.

In summary, auctions are mechanisms for allocating resources in situations in which there

is incomplete information and traditional market mechanisms do not provide incentives forparticipants truthfully to declare the missing information Auction design takes account

of this lack of information and can improve the equilibrium properties of the underlyinggames We conclude the chapter in Section 14.3 by summarizing its ideas in the context of

a highspeed link whose bandwidth is put up for sale by auction

14.1 Single item auctions

14.1.1 Take it or leave it Pricing

In this section, we consider the sale of a single item by auction For the purposes ofcomparison, we begin with analysis of a selling mechanism that is not an auction, butwhich could be used under the same conditions of incomplete information that pertainwhen auctions are used

Suppose a seller wishes to sell a single item He does this simply by making a

take-it-or-leave-it offer, at price p If any customers wants to buy the item at that price, then

it is sold; otherwise it is not sold, and the seller obtains zero revenue If more than onecustomer wants the item at the stated price, then there must be a procedure for deciding

who gets it However, the seller still receives revenue of p.

Suppose customers are identical and their private valuations are independent and

identically distributed as a random variable X , with distribution function F x/ D P.X  x/ Given knowledge of this distribution, the seller wants to choose p to maximize his expected revenue Let x p/ denote the probability the item is sold Then

For example, if valuations are uniformly distributed on [0; 1], then F.x/ D x, and

we find that the optimal price is pŁ D .n C 1/1=n The resulting expected revenue is

n n C 1/.nC1/=n For n D 2, the optimal price is pŁ D p1=3 The seller’s expectedrevenue is 2=3/p1=3 (D 0:3849)

Note that, because there is a positive probability that the item is not sold, this method ofselling is not economically efficient We have seen this before in Chapter 6; if a monopolistseeks only to maximize his own revenue then there is often a social welfare loss For the

example above, the maximum valuation is the maximum of n uniform random variables

distributed on [0; 1]; it is a standard result that this has expected value n=.n C 1/ This

is the expected social welfare gain if the item is allocated to the bidder with the highest

valuation For n D 2, this is 2=3 (D 0:6666) However, under take it or leave it pricing, the expected social welfare gain can shown to be 1  pŁ.1  pŁ/.nC1/ =.n C 1/ For n D 2,

312 AUCTIONS

the example with n D 2, he believes that valuations are uniformly distributed on [0; 1] and sets the price optimally at pŁDp1=3 Say, however, he is mistaken: bidders valuations areactually uniformly distributed on [0; 0:5] Then, as pŁ> 0:5, he never sells It would havebeen better if he had auctioned the item, thus ultimately selling it to the highest bidder.Even if the seller does know the distribution of bidders’ valuations, he can do better

by auctioning As we see below, one can design auction rules that increase the expectedrevenue and make auctioning the most profitable selling method One way to do this is to

introduce a minimum price that must be paid by the auction’s winner This reserve price

has the effect of increasing the average price paid by the winner In our example, he couldset a reserve price of 1=2 and would obtain expected revenue of 0:4167 (see Section 14.1.4)

14.1.2 Types of Auction

We now describe some of the most popular types of auction In the ascending price auction (or English auction), the auctioneer asks for increasing bids by raising the price of the item

by small increments, until only one bidder remains Or perhaps bidders place increasing bids

by shouting The item is awarded to the last remaining bidder, at the price of the last bid

at which all other bidders had withdrawn It is clear that in this type of auction the winner

is the bidder with the highest valuation, and he pays a price equal to the second highestvaluation Unique items, such as artworks, tend to be sold in English auction, in order to find

an unknown price Another version of this auction is used in Japan; the price is displayed

on a screen and raised continuously Any bidder who wishes to remain active keeps hisfinger on a button When he releases the button he quits the auction and cannot bid again

In a reverse procedure to the English auction above, the Dutch auction starts by setting

the price at some initial high value A so-called ‘Dutch clock’ displays the price andcontinuously decreases it until some bidder decides to claim the item at the price displayed.Multiple items (such as fish or flowers) tend to be sold in Dutch auctions; this speeds upthe time the sale takes The price is lowered until demand matches supply

In the next two types of auction, bidders submit sealed-bids and the one with the greatest

bid wins The auctions differ in the price charged to the winner Under the first-price bid auction, the winner pays his bid In this auction, the bidder has to decide off-line how

sealed-much he should bid This is equivalent to deciding off-line at what price he would claim theitem in a Dutch auction, since in that auction no information is revealed until the first bid,

at which point the auction also ends Thus, we see that the Dutch auction and first-pricesealed-bid auction are completely equivalent

In the second-price sealed-bid auction, the winner pays the second highest bid This is also known as a Vickrey auction, after its inventor An important property of the Vickrey

auction is that it is optimal for each bidder to bid his true valuation To understand why this

is so, note that a bidder would never wish to bid more than his valuation, since his expectednet benefit would then be negative However, if he reduces his bid below his valuation, hereduces the probability that he wins the auction, but he does not affect the price that hepays if he does win (which is determined by the second highest bidder) Thus, he does best

by bidding his true valuation The winner is the bidder with the greatest valuation and hepays the second greatest valuation But this is exactly what happens in the English auction,

in which a player drops out when the price exceeds by a small margin his valuation, and sothe winner pays the valuation of the second-highest bidder Thus, we see that the Englishand Vickrey auctions are equivalent

Other auctions include the all-pay auction, in which all bidders pay their bid but the highest bidder wins the object, and the k-price auction, in which the winner pays the kth

largest bid Some of these auctions can be easily extended to multiple units For example,

in the two-unit first-price sealed bid auction the participants with the two greatest bids arewinners and pay the third largest bid Multi-unit auctions require bidders to follow muchmore complex strategies We return to multi-unit auctions in Section 14.2

utility function is linear he is said to be risk-neutral His average utility (after repeating

the auction many times) is the same as his utility for the average payment, and hence thevariability of the payment around its mean does not reduce the average utility of the seller

If the utility function is concave then the seller is risk-averse; now the average utility is less

than the utility of the average payment, and this discrepancy increases with the variability

of the payment

Suppose each bidder knows his own valuation of the item, which he keeps secret, andvaluations of the bidders can be modelled as independent and identically distributed randomvariables Some important questions are as follows

1 Which of the four standard auctions of the previous section generates the greatestexpected revenue for the seller?

2 If the seller or the bidders are risk-averse, which auction would they prefer?

3 Which auctions make it harder for the bidders to collude?

4 Can we compare auctions with respect to strategic simplicity?

Let us begin with an intuitive, but important, result

Lemma 1 In any SIPV auction in which (a) the bidders bid optimally, and (b) the item

is awarded to the highest bidder, the order of the bids is the same as the order of thevaluations

Proof Suppose that under an optimal bidding strategy a bidder whose valuation isv bids

so as to win with probability p v/ Let e.p/ be the minimal expected payment that such

a bidder can make if he wants to win the item with probability p Assume v1 andv2 aresuch that v1> v2, but p.v1/ < p.v2/ If this is true, then it is simple algebra to show that,

with pi D p.vi/,

[ p1v2e p1/] C [p2v1e p2/] > [p1v1e p1/] C [p2v2e p2/]

Thus, either p1v2e p1/ > p2v2e p2/, or p2v1e p2/ > p1v1e p1/ In other

words, either it is better to win with probability p1when the valuationv2, or it is better to

win with probability p2 when the valuation isv1, in contradiction to our assumptions We

are forced to conclude that p.v/ is nondecreasing in v By assumption (b) in the lemma

statement, this means that the optimal bid must be nondecreasing in v

We say that two auctions have the same bidder participation if any bidder who finds

it profitable to participate in one auction also finds it profitable to participate in the other.The following is a remarkable result

314 AUCTIONS

Theorem 4 (revenue equivalence theorem) The expected revenue obtained by the seller

is the same for any two SIPV auctions that (a) award the item to the highest bidder, and(b) have the same bidder participation

We say this is a remarkable result because different auctions can have completely differentsets of rules and strategies We might expect them to produce different revenues for theseller Note that revenue equivalence is for the expectation of the revenue and not for itsvariance Indeed, as we see in Section 14.1.5, auctions can have quite different properties

so far as risk is concerned

Proof of the revenue equivalence theorem Suppose there are n participating bidders As

above, let e p/ denote the minimal expected payment that a bidder can make if he wants to win with probability p The bidder’s expected profit is ³.v/ D pv  e.p/, where p D p.v/

is chosen optimally and so, since ³ must be stationary with respect to any change in p,

we must havev  e0.p/ D 0 Hence,

It should be clear that all four auctions described in Section 14.1.2 satisfy the conditions ofthe revenue equivalence theorem Let us work through an example in which the valuations,sayv1; : : : ; vn, are random variables, independent and uniformly distributed on [0; 1] Let

v.k/ denote the kth largest of v1; : : : ; vn (the k-order statistic) A standard result is that E[v.k/ ] D k =.n C 1/ Hence in the Vickrey and English auctions the expected revenue is E[v.n1/] D.n  1/=.n C 1/.

Using this, we can find the optimal bid in the first-price sealed-bid auction By thetheorem the expected revenue in this auction is the same as in the English auction, i.e

.n  1/=.n C 1/ Also, recall that p.v/ D F.v/ n1 Dvn1 Using (14.3), we easily find

e p.v// D n  1/v n =n This must be p.v/ times the optimal bid So a bidder who values

the item atv has an optimal bid of n  1/v=n This is a shaded bid, equal to the expected

value of the second-highest valuation, given thatv is the highest valuation

14.1.4 Optimal Auctions

An important issue for the seller is to design the auction to maximize his revenue We

give revenue-maximizing auctions the name optimal auctions It turns out that a seller who wants to run an optimal auction can increase his revenue by imposing a reserve price or a participation fee This reduces the number of participants, but leads to fiercer competition

and higher bids on the average, which may compensate for the probability that no saletakes place Let us illustrate this with an example

Example 14.1 (Revenue maximization) Consider a seller who wishes to maximize his

revenue from the sale of an object There are two potential buyers, with unknown valuations,

v1,v2, that are independent and uniformly distributed on [0; 1] He considers four ways ofselling the object:

1 A take it or leave it offer

2 A standard English auction

3 An English auction with a participation fee c (which must be paid if a player chooses

to submit a bid) Each bidder must choose whether or not to participate beforeknowing whether the other participates

4 An English auction with a reserve price, p The bidding starts with a minimum bid

c Since a bidder with valuation v0 wins only if the other bidder has a valuation less than

v0, we must have P.winning j v D v0/ D v0, and hencev2

0Dc Thus,v0Dpc.

To compute the expected revenue of the seller, we note that there are two ways thatrevenue can accrue to the seller Either only one bidder participates and the sale price is

zero, but the revenue is c Or both bidders have valuation above v0, in which case the

revenue is 2c plus the sale price of minfv1; v2g The expected revenue is

2v0.1  v0/c C 1  v0/2

[2c Cv0C.1  v0/=3]

Straightforward calculations show that this is maximized for c D 1=4, and takes the value

5=12 (D 0:4167)

Case 4 In the English auction with a reserve price p, there is no sale with probability

p2 The revenue is p with probability 2 p.1  p/ If minfv1; v2g> p, then the sale price is

minfv1; v2g The expected revenue is

316 AUCTIONSThat Cases 3 and 4 in the above example give the same expected revenue is not acoincidence These are similar auctions, in that a bidder participates if and only if hisvaluation exceeds 1=2 Let us consider more generally an auction in which a bidderparticipates only if his valuation exceeds some v0 Suppose that with valuation v it is

optimal to bid so as to win with probability p.v/, and the expected payment is then

e p.v// By a simple generalization of (14.3), we have

We callv0 the optimal reservation price Note that it does not depend upon the number of

bidders For example, if valuations are uniformly distributed on [0; 1], then v0D1=2 This

is consistent with the answers found for Cases 3 and 4 of Example 14.1

If bidders’ valuations are independent, but heterogenous in their distributions, then one

can proceed similarly Let pi v/ be the probability that bidder i wins when his valuation

is v Let ei p/ be the minimum expected amount he can pay if he wants to win with probability p Suppose that bidder i does not participate if his valuation is less than v0i.Just as above, one can show that the seller’s expected revenue is

The term in square brackets can be interpreted as ‘marginal revenue’, in the sense that if a

price p is offered to bidder i , he will accept it with probability xi p/ D 1  Fi p/, and so the expected revenue obtained by this offer is pxi p/ Differentiating this with respect to

Note that the right-hand side of (14.4) is simply E[MRiŁ.viŁ/], where iŁ is the winner

of the auction This can be maximized simply by ensuring that the object is alwaysawarded to the bidder with the greatest marginal revenue, provided that marginal revenue

is positive We can do this provided bidders reveal their true valuations Let us assume that

MR i p/ is increasing in p, for all i Clearly, v 0i should be the leastv such that MRi.v/

is nonnegative Consider the auction rule that always awards the item to the bidder withthe greatest marginal revenue, and then asks him to pay the maximum of v0i and thesmallestv for which he would still remain the bidder with greatest marginal revenue Thishas the character of a second-price auction in which the bidder’s bid does not affect hispayment, given that he wins So bidders will bid their true valuations and (14.4) will bemaximized

Example 14.2 (Optimal auctions) An interesting property of optimal auctions withheterogeneous bidders is that the winner is not always the highest bidder

Consider first the case of homogeneous bidders with valuations uniformly distributed on[0; 1] In this case, MRi.vi/ D vi .1  vi/=1 D 2vi 1 Hence the object is sold to thehighest bidder, but only if 2vi 1> 0, i.e if his valuation exceeds 1=2 The winner payseither 1=2 or the second greatest bid, whichever is greatest In the case of two bidders, withthe seller’s expected revenue is 5=8 This agrees with what we have found previously

Now consider the case of two heterogeneous bidders, say A and B, whose valuations

are uniformly distributed on [0; 1] and [0; 2], respectively So MRA.vA/ D 2vA1, and

MR B.vB/ D 2vB2 Under the bidding rules described above, bidder B wins only if

2vB2 > 2vA1 and 2vB2 > 0, i.e if and only if vBvA > 1=2 and vB > 1; sothe lower bidder can sometimes win For example, if vAD0:8 and vB D1:2, then A wins

and pays 0:7 (which is the smallest v such that MR A.v/ D 2v  1 ½ 2vB2 D 0:4)

14.1.5 Risk Aversion

As we have already mentioned, the participants in an auction can have different attitudes

to risk If a participant’s utility function is linear then he is said to be risk-neutral If his utility function is concave then he is risk-averse; now a seller’s average utility is less than

the utility of his average revenue, and this discrepancy increases with the variability of therevenue Hence a risk-averse seller, depending on his degree of risk-aversion, might choose

an auction that substantially reduces the variance of his revenue, even though this mightreduce his average revenue

The revenue equivalence theorem holds under the assumption that bidders are neutral One can easily see that if bidders are risk-averse, then first-price sealed-bid andDutch auctions give different results from second-price sealed-bid and English auctions.For example, in a first-price auction, a risk-averse bidder prefers to win more frequentlyeven if his average net benefit is less Hence, he will make higher bids than if he wererisk-neutral This reduces his expected net benefit and increases the expected revenue of theseller If the same bidder participates in a second-price auction, then his bids do not affectwhat he pays when he wins, and so his strategy must be to bid his true valuation Hence, afirst-price auction amongst risk-averse bidders produces a greater expected revenue for theseller than does a second-price auction However, it is not clear which type of auction therisk-averse bidders would prefer In general, this type of question is very difficult.The seller may also be risk-averse In such a case, he prefers amongst auctions with thesame expected revenue those with a smaller variance in the sale price Let us compare a firstand second-price auction with respect to this variance Suppose bidders are risk-neutral Let

risk-v.n/andv.n1/be the greatest and second-greatest valuations In a second-price auction, thewinner pays the value of the runner-up’s bid, i.e.v.n1/ In a first-price auction he pays hisbid, which is the conditional expectation of the valuation of the runner-up, conditioned on

his winning the auction, i.e E.v.n1/jv.n/ / Let Y D v .n1/jv.n// and apply the standard

318 AUCTIONSfact that.EY /2EY2 This gives

Subtracting from both sides the square of the expected value of the winner’s bid, i.e

E.v.n1//2, we see that the winner’s bid has a smaller variance in the first-price auction,and so a risk-averse seller would prefer a first-price auction Let us verify this for twobidders whose valuations are uniformly distributed on [0; 1] In the first-price auction, eachbidder bids half his valuation, so the revenue is 1=2/ maxfv1; v2g In the second-priceauction each bids his valuation and the revenue is minfv1; v2g Both have expectation 1=3,but the variances are 1=72 and 1=18, respectively Thus, a risk-averse seller prefers thefirst-price auction

14.1.6 Collusion

It is important when running an auction to take steps to prevent bidders from colluding.Collusion occurs when two or more bidders make arrangements not to bid as high as theirvaluations suggest, and so reduce the seller’s revenue Antique auctions are notorious forthis A number of bidders form a ‘ring’ and agree not to bid against one another and onwhom the winner will be This lowers the winning bid Later, the winner distributes hisgain amongst all the bidders, in proportion to their market power, so that all do betterthan they would have done by not colluding In some spectrum auctions in the US, therehave been instances of bidders using the final four digits of their multimillion dollar bids

to signal to one another the licenses they want to buy Thus, a critical characteristic of anauction is how susceptible it is to collusion This depends upon what incentives there arefor players to stand by the promises they make to one another when agreeing to collude

We can see that an ascending English auction is susceptible to collusion Suppose thebidders meet and determine that bidder 1 has the greatest valuation They agree that bidder

1 should make a low bid and win the object for a payment close to zero No other bidderhas an incentive to bid against bidder 1, since he cannot win without ultimately outbiddingbidder 1; yet if he does so he would incur a loss Thus, the agreement between the bidders

is ‘self-enforcing’ and the auction is susceptible to collusion

In contrast, collusion is difficult in a Dutch auction, or in a first-price sealed-bid auction.There is nothing to stop a ring member bidding higher than was agreed His defectingaction becomes obvious, but the auction is over before anyone can react This is one reasonwhy first-price sealed-bid auctions are often preferred when auctioning large governmentcontracts

There is also a matter of trusting the seller He might want to manipulate the auction toraise prices One way he can do this is by soliciting fake bids In a first-price sealed-bidauction, such bids do not make any sense, since they could prevent the sale of the object(and the seller could anyway use a reserve price) In a second-price auction, fake bids couldbenefit the seller If the seller has approximate knowledge of the highest bidder’s valuation,

he could solicit a ‘phantom’ bid with a slightly smaller value, and hence obtain almost allthe surplus of the bidder

14.1.7 The Winner’s Curse

Thus far we have discussed the private values model In the common values model, i.e.where the item that is auctioned has a common unknown value, the winner is the bidder

who has the most optimistic estimate of the item’s value If bidders’ estimates are unbiased,then the highest estimate will be likely to exceed the item’s actual value, and the winnerwill suffer a loss To remedy this, a bidder should shade his bid to allow for the fact that if

he wins, he has the highest estimate He should find the item’s expected value conditional

on his initial estimate being the highest among all initial estimates of the other bidders, i.e.conditional on his being the winner

To illustrate this, suppose that the item has a random value V Each bidder receives a signal si that is an estimate of the value of the item These signals are independent and

uniformly distributed on [V  ž; V C ž] Since E[V jsi ] D si, a straightforward approach is

for bidder i to bid si But he will suffer the winner’s curse To remedy this, bidder i must

assume that he wins the auction because his estimate is the highest and correct estimate forthe value of the item One can show that

14.1.8 Other Issues

We have mentioned above the issue of strategic simplicity A strong argument in favour

of the second-price sealed-bid auction is that each bidder’s strategy is simple: he just bidshis valuation In contrast, the bidder in a first-price sealed-bid auction must estimate thesecond-highest valuation amongst his competitors, given that his valuation is greatest

It is interesting to ask whether it is advantageous for the seller to disclose the number ofbidders It can be proved that the first-price sealed-bid auction results in more aggressive bid-ding when the number of bidders is unknown, and so the seller may prefer this to be the case

In the SIVP model we assumed bidders are identical If this is not so, then things can

become very complicated Suppose there are two bidders, say A and B, with valuations

dis-tributed uniformly on [0; 1] and [1; 2], respectively In a second-price auction both will bid

their valuations and B will always win, paying A’s valuation However, in a first-price tion A will bid very near his valuation, but B will shade his bid substantially under his valu- ation, since he knows A’s bids are much lower than his Now there is a positive probability that A wins Note that the outcome can be inefficient, in the sense that the object may not

auc-be sold to the bidder who values it most Also, since the item will sometimes sell for more

than A’s valuation the seller’s expected revenue is greater than in the second-price auction.

Suppose the distributions of the bidders’ valuations are correlated, rather than being

independent This is sometimes called affiliation The effects of affiliation are complex to

analyse precisely, but we can give some intuition In the presence of affiliation, it turns outthat ascending auctions lead to greater expected prices than second-price sealed-bid auctions,and these lead to greater expected prices than first-price sealed-bid auctions An intuitiveway to see this is as follows A player’s profit when he is the winner arises from his privateinformation (his ‘information rent’) The less crucial is this information advantage, the lessprofit the player can make In the case of the ascending auction, the sale price dependsupon all other bidders’ information, and because of affiliation, it captures a large part of the