An Introduction to Auction Theory, its

Applications and Experimental Evidence

Dirk Engelmann

CES Lectures, July 2010

Overview

• Single-Unit Auctions — Theory and Experiments

• Applications of Auction Theory

• Multi-Unit Auctions — Theory and Experiments

• What Really Matters in Auction Design

Literature

• Krishna, V., 2002. “Auction Theory”, Academic

Press

• Klemperer, P., 2004. “A Survey of Auction The-

ory” (Chapter 1 in “Auctions: Theory and Prac-

tise”) http://paulklemperer.org

• Klemperer, Paul., 2004 “Why Every Economist Should

Learn Some Auction Theory” (Chapter 2 in “Auc-

tions: Theory and Practise”) http://paulklemperer.org

• Klemperer, Paul., 2004 “What Really Matters in

Auction Design” (Chapter 3 in “Auctions: Theory

and Practise”) http://paulklemperer.org

• Goeree, J., Holt, C., Palfrey, T. 2002. “Quantal

Response Equilibrium and Overbidding in Private-

Value Auctions.” Journal of Economic Theory 104(1),

247-272.

• Engelmann, D., Grimm, V. 2009. “Bidding Behav-

ior in Multi-Unit Auctions - An Experimental Inves-

tigation.” Economic Journal 119, 855-882.

Auctions

Introduction

Definition: An Auction is a selling institution that elic-its information from potential buyers in the form of bidsand where the outcome (i.e. who obtains the objectsand who pays how much) is determined solely by thisinformation.

This implies that auctions are universal (i.e. any objectcan be sold by means of an auction) and anonymous(i.e. the identity of the bidders does not matter, henceif the bids of two bidders are exchanged, the allocationand payments are exchanged accordingly and no otherbidder is affected).

Procurement auctions used to buy a good from po-

tential sellers work exactly correspondingly. Hence we

can restrict the discussion to auctions employed to sell

goods.

Auctions are useful when the seller is unsure about the

valuations (i.e. the maximal willingness to pay) of the

buyers. Otherwise he could just offer the good to the

buyer with the highest valuation at a price just below

this valuation.

Auctions have been used for a long time, e.g. govern-

ment bonds, drilling rights. Recent important applica-

tions: privatization, spectrum auctions, internet auc-

tion platforms.

Different auction formats are evaluated on the basis of

revenue (expected selling price) and efficiency (alloca-

tion to the bidder with the highest valuation).

For practical purposes simplicity and the susceptibility

to collusion are further (and possibly more) important

criteria.

Single-Unit Auctions — Theory

Common Single-Unit Auction Forms

1. First-Price auction: All bidders submit a sealed bid,

the highest bidder wins and pays his bid

2. Second-Price auction: All bidders submit a sealed

bid, the highest bidder wins and pays the second

highest bid

3. English Ascending Price auction (Japanese auction):

The auctioneer continuously raises the price until

only one bidder remains active, who obtains the

object at the price where the auction ended (i.e.

where the second to last bidder dropped out).

4. Dutch Descending Price auction: The auctioneer

starts with a high price (presumed to be higher than

the maximal valuation) and continuously lowers the

price until one bidder signals to buy the object at

the current price.

A number of other auctions formats is possible. Some

might appear unusual when thought of as a classi-

cal auction, but less so when seen in another context

(e.g. an all-pay auction, where all bidders pay their bids

seems unusual for selling a painting, but a patent-race,

or lobbying are essentially all-pay auctions). Some auc-

tions may at a first glance not conform to a straight-

forward idea of an auction.

Single-Unit Private-Value Auctions

Definition: Bidders are said to have private values ifeach bidder knows the value of the object to himself(and only to himself) for sure at the time of bidding.Otherwise, if the value of the object to a bidder maydepend on information that other bidders have, valuesare said to be interdependent. An extreme case is thatof a pure common value, where the value of the objectis the same for all bidders, but unknown by the time ofbidding.

Private values do not have to be statistically indepen-dent and in the case of interdependent values the sig-nals of the bidders can still be statistically independent.Independent private values are, however, the standardcase.

Equivalences

For the single unit case, First-Price and Dutch auctionare equivalent in a strong sense, they are strategicallyequivalent (the games have the same normal form).

Available strategies: plainly choose one number, bid inFPA or price where bidder would agree to buy in DA incase it has not been sold yet.

Outcomes derived from strategies in same way: bidderchoosing highest number wins and pays this number.

Bidders do not learn anything in DA, because when theydo learn something the auction is over (that no otherbidder has bought the object yet is not informative,because the strategies condition on this anyway).

Between the Second-Price and the English auction there

is a weaker form of equivalence.

In EA (with more then two bidders) a bidder can learn

something by the drop-out prices of other bidders and

could in principle condition his strategy on his obser-

vations (hence the games are not strategically equiva-

lent).

But in the private value case, the information gathered

from the drop-out prices of other bidders is not infor-

mative and hence the equilibria in both auctions are

identical.

Equilibria in the Symmetric Model

• Single object for sale to N bidders.

• Bidder i assigns value Xi (random variable)

• The Xi are independently and identically distributed

in some interval [0, ω] with distribution function F

• F has continuous density with full support f = F ′.While ω = ∞ is allowed, E[Xi] < ∞ is assumed.

• Bidder i knows the realization xi of Xi but only that

the other bidders’ values are distributed according

to F.

• Except for the realizations of the values, all aspects

of the model, in particular F and N, are common

knowledge.

• Bidders are risk neutral, they try to maximize ex-

pected profits.

• Bidders do not face liquidity constraints, i.e. bidder

i is willing and able to pay up to xi.

• An auction determines a game with the strategies

being bid functions βi : [0, ω] → R+.

• The focus will be on symmetric equilibria, because

bidders are symmetric.

Second-Price and English auction:

Proposition: In SPA and EA it is a weakly dominant

strategy to bid one’s own valuation: βII(x) = x.

Proof:

Assume to the contrary that bidder i bids zi > xi.

This changes outcome only if for p, the highest of the

other bidders’ bids, xi < p < zi.

In this case bidder i now wins the object and makes a

loss.

Hence overbidding is dominated.

Bidding zi < xi only changes outcome if zi < p < xi.

In this case bidder i now misses a profitable deal thathe could have made by bidding xi.

Hence also underbidding is dominated. QED

The argument is even more obvious in EA: it cannotpay to drop out before the price reaches xi and it cannotpay to stay in once the price exceeds xi, so it is (weakly)dominant to drop out at p = xi.

Note that this result depends neither on risk neutralitynor on the symmetry of the bidders (not even on theindependence of the distributions, only on values beingprivate).

Some Notes on Order Statistics: Consider bidder 1.

Let Y1 ≡ Y(N−1)1 denote the highest value of the re-

maining N − 1 bidders, i.e. Y1 is the first order statistic

of X2, . . . , XN . The distribution function of Y1 is given

by G(y) = F (y)N−1, with density g.

First-Price auction: In FPA, each bidder submits a

sealed bid bi and payoffs are

Πi =

{xi − bi if bi > maxj 6=i bj

0 if bi < maxj 6=i bj

A bidder will clearly not submit a bid equal to his val-

uation, because this guarantees a profit of 0. Raising

the bid implies a trade-off. The chance of winning the

object are increased, but so is the expected price.

Assume that there is a symmetric equilibrium with in-creasing, differentiable strategy β. Obviously, biddingb > β(ω) is dominated, and β(0) = 0.

Bidder 1 wins if his bid

b > maxi6=1

β(Xi) = β(maxi6=1

Xi) = β(Y1),

hence if Y1 < β−1(b). His expected payoff is

G(β−1(b)) · (x− b). Maximizing w.r.t b yields

g(β−1(b))

β′(β−1(b))(x− b)−G(β−1(b)) = 0

In a symmetric equilibrium b = β(x) implying

G(x)β′(x) + g(x)β(x) = xg(x)d

dx(G(x)β(x)) = xg(x)

Since β(0) = 0

βI(x) =1

G(x)

∫ x

0yg(y)dy = E[Y1|Y1 < x]

The equilibrium bidding strategy can be rewritten as

βI(x) = E[Y1|Y1 < x] = x−∫ x

0

G(y)

G(x)dy

= x−∫ x

0

[F (y)

F (x)

]N−1

dy

hence the degree of bid shading decreases in the num-ber of bidders.

Example: Values are uniformly distributed over [0,1].Then F (x) = x, G(x) = xN−1 and hence

βI(x) =N − 1

Nx.

Revenue Comparison

The expected payment of a bidder with value x is inSPA

mII(x) = Pr(Win)·E[2nd highest bid|x is the highest bid]

= Pr(Win) · E[2nd highest value|x is the highest value]

= G(x) · E[Y1|Y1 < x]

In FPA it is

mI(x) = Pr(Win) · Amount Bid

= G(x) · E[Y1|Y1 < x] = mII(x)

Hence the expected payment for a bidder with value x

is identical in FPA and SPA.

Thus the ex-ante expected payment of each bidder isthe same and therefore so is the expected revenue of

the seller: E

(Y

(N)2

).

For given values, however, the revenue is usually differ-ent.

Furthermore, the revenues in SPA vary more than inFPA (e.g. for 2 bidders with uniformly distributed valuesin [0,1] maximal revenue in FPA: 1

2, in SPA: 1.)

Distribution of equilibrium prices in SPA is a mean pre-serving spread of distribution of prices in FPA.

Hence a risk-averse seller would prefer FPA (given thatthe bidders are risk-neutral).

The Revenue Equivalence Principle

The observed identity of expected revenues between

SPA and FPA holds in much more generality.

Definition: An auction is standard if the highest bidder

obtains the object.

Proposition: Suppose values are iid and bidders are

risk-neutral. Then any symmetric and increasing equi-

librium of any standard auction, such that the expected

payment of a bidder with value 0 is 0, yields the same

expected revenue to the seller.

Proof: Let mA(x) be the expected payment in the

symmetric equilibrium β of a standard auction A and

let mA(0) = 0.

Assume all bidders except for bidder 1 follow β and that

bidder 1 with value x bids β(z) instead of β(x).

Bidder 1 wins when β(z) > β(Y1) or z > Y1. His expected

payoff is

ΠA(z, x) = G(z)x−mA(z)

Note that mA(z) depends on β and z but not on x.

Maximization yields

∂

∂zΠA(z, x) = g(z)x−

d

dzmA(z) = 0

In equilibrium z = x is optimal, hence for all y

d

dymA(y) = g(y)y

Thus

mA(x) = mA(0) +∫ x

0yg(y)dy

=∫ x

0yg(y)dy = G(x) · E[Y1|Y1 < x]

since by assumption mA(0) = 0.

The right hand side does not depend on A and hence

the expected payment of each bidder for a particular

value does not depend on the particular auction, and

therefore, the ex-ante expected payment and thus the

expected revenue of the seller do not either. QED

Example: Values are uniformly distributed on [0,1]

F (x) = x ⇒ G(x) = xN−1, thus for any standard auction

with mA(0) = 0

mA(x) =N − 1

NxN

E[mA(x)] =N − 1

N(N + 1)

E[RA] = N · E[mA(x)] =N − 1

N + 1

Application: All-Pay Auction

The revenue equivalence principle can be used to de-

rive equilibria in other auctions, e.g. the all-pay auction

where all bidders pay their bid.

Consider an all-pay auction with symmetric, indepen-

dent private values.

Suppose there is a symmetric, increasing equilibrium

such that mAP (0) = 0. The expected payment in an

all-pay auction equals the bid and hence due to the

revenue equivalence principle the equilibrium bid is

βAP (x) = mA(x) =∫ x

0yg(y)dy

Qualifications

1. Risk Aversion

Suppose bidders are risk-averse, all with the sameutility function. In the case of iid private values,FPA yields a higher expected revenue than SPA.

Intuition:

In SPA risk-aversion does not change that biddingone’s value is a dominant strategy (by raising thebid, one can increase the probability of winning, butonly in cases where one does not want to win).

In FPA at the equilibrium bid there is a perfecttrade-off between the probability of winning and theamount won.

A risk-averse bidder will prefer, compared to a risk-

neutral bidder, a smaller gain with a higher proba-

bility and will hence choose a higher bid. By bidding

higher he insures against ending up with 0.

Thus if bidders are risk averse, FPA yields higher

revenues than SPA.

2. Budget constraints

Suppose that bidder i has an absolute budget Wi. In

SPA a bidder is more likely to be constrained than

in FPA and hence FPA leads to higher expected

revenues.

3. Symmetry

Consider the case where bidders are ex-ante asym-

metric, i.e. their valuations are drawn from different

distributions.

In SPA, bidding one’s own value again remains a

dominant strategy and hence SPA allocates effi-

ciently even in case of asymmetries.

In FPA, the weak bidder will bid more aggressively,

because he faces a stochastically higher distribution

of bids.

This more aggressive bidding of the weak bidder

can lead to inefficiencies (if the weak bidder’s value

is slightly smaller than the strong bidder’s), hence

the revenue equivalence principle fails.

Furthermore, there is no definite ranking of rev-

enues between FPA and SPA.

4. Resale and Efficiency

One might argue that, since the auction reveals

the valuations of the bidders, the auctioneer does

not have to worry about efficient allocation because

post-auction transaction will lead to an efficient al-

location.

This, however, is not correct even absent trans-

action costs. If there is the opportunity for post-

auction trade, bidders have an incentive not to com-

pletely reveal their valuation and hence post-auction

bargaining takes place under incomplete informa-

tion which can lead to profitable deals being missed.

Single-Unit Auctions with Interdependent Values

As opposed to the case of private values, with interde-

pendent values, the value of the object to bidder i, Vi is

assumed to depend also on the information that other

bidders have.

Each bidder i has some private information Xi ∈ [0, ωi],

called i’s signal.

Vi is assumed to be given by a function Vi = vi(X1,X2, . . . , XN)

that is non-decreasing in all the arguments, strictly

increasing in Xi and twice continuously differentiable.

Hence the value is completely determined by the sig-

nals.

More general cases where there is some remaining un-certainty can be accommodated by considering the ex-pected value given all signals (bidders are assumed tobe risk neutral)

vi(x1,x2, . . . , xN) = E[Vi|X1 = x1, . . . XN = xN ]

Further assumption are vi(0, . . . ,0) = 0 and E[Vi < ∞].

Extreme cases: private values: vi(X1, . . . , XN) = Xi

pure common value: Vi = V = v(X1, . . . , XN) for all i.

Special case (mineral rights model):

conditional on V = v, the signals Xi of the bidders areiid with E[Xi|V = v] = v.

The Winner’s Curse

If values are not private, the estimate of the value to

one-self has to take information into account that is

obtained during or even after the auctions.

In particular, it brings bad news if one wins the auction:

Bidder 1’s initial estimate of the value upon receiving

the signal x is E[V |X1 = x].

Now if bidders are symmetric and follow the same strat-

egy then winning the auction means that bidder 1 has

the highest signal.

The estimate of the value is then

E[V |X1 = x, Y1 < x] < E[V |X1 = x].

Failure to take this into account, i.e. bidding plainly

according to the initial signal instead of shading the bid

below the initial estimate, leads to the winner’s curse,

paying more than the value.

The Winner’s Curse magnitude increases with the num-

ber of bidders.

Note that the winner’s curse results from the failure

to take the interdependence into account, it does not

occur in equilibrium.

Originally motivated by oil lease auctions to explain low

returns from leases

Also applied to, e.g., book publication rights, corporate

takeover battles, real estate auctions

Experimental Evidence: Kagel, Levin: Common Value

Auctions and the Winner’s Curse, 2002, Princeton Uni-

versity Press)

Equivalences

Dutch and FPA are still strategically equivalent, be-

cause that their normal forms are identical does not

depend at all on the distribution of values and signals.

But English and SPA are not equivalent with interde-

pendent values, since information gathered in the EA

(the drop-out prices of the other bidders) conveys valu-

able information about their signals and hence about

one’s own value.

In the case of only two bidders, however, the auction is

over as soon as one obtains information and hence the

two auctions are equivalent.

Consider again symmetric bidders, i.e. signals are drawnfrom the same distribution, and for bidder i the valua-tion does not change if we swap the signals of biddersj and k.

Let

v(x, y) = E[V1|X1 = x, Y1 = y]

denote the expected value to bidder 1 if his signal is x

and the highest of the other signals is y.

Second-Price Auctions

Proposition: Symmetric equilibrium strategies in thesecond price auction are given by

βII(x) = v(x, x)

English Auctions

In an English auction the bidders learn the prices where

other bidders drop out (in the symmetric model, the

identities of the bidders who dropped out are irrele-

vant).

As this information becomes available, this can (and

should) influence when the remaining bidders plan to

drop out.

Symmetric equilibrium strategies are as follows:

• As long as no bidder has dropped out, drop out ifthe price reaches the expected value assuming thatall bidders have the same signal as yourself.

• If a bidder (say bidder k) drops out before you do,infer his signal xk from the price where he droppedout.

• Then drop out if the price reaches the expectedvalue given xk and assuming that all remainingbidders have the same signal as you.

• Continue in this fashion, i.e. infer the signals of thebidders who drop out and recalculate the expectedvalue.

An interesting implication is that the price at which

you intend to drop out decreases over the course of

the auction.

Experimental evidence: Experimental subjects learn

this in an experiment where they can adjust proxy bids

whenever another bidder drops out (See Engelmann

and Wolfstetter, 2005).

While they do not learn to play the equilibrium, they

learn to lower their drop-out prices when other bidders

drop out.

Single-Unit Auctions — Experimental Evidence

Some well-established results

• Common Value Auctions:

– Winner’s curse frequent

– Even for experienced professional (construction

industry) bidders in lab setting (Dyer, Kagel,

Levin, EJ 1989)

– And if an informed insider is present (Kagel and

Levin, Econometrica 1999)

– But subjects learn to avoid it

– Harrison and List (EJ 2010): no winner’s curse

for experienced sports card dealers. Argue this

is due to experience carrying over, but could just

be because dealers are in habit to bid low since

they want to resell

• Private Value Auctions:

– EA: players learn dominant strategy relatively

quickly.

– SPA: players also learn dominant strategy, but

there is more overbidding than in EA (e.g., Kagel

and Levin, EJ 1993).

– FPA: most experiments find substantial and per-

sistent overbidding of the risk-neutral equilibrium

We will be concerned with the last issue.

See Goeree, Holt, and Palfrey, JET 2002

Two prominent explanations:

• Risk aversion

• “Joy of winning”

First proposed explanation: risk aversion

Critique:

• Rabin (Econometrica 2000): small scale risk aver-sion implies excessive risk aversion for large stakes

• Harrison’s “flat maximum” critique (AER 1989):around the optimum, the profit function is very flat.Hence the incentives are small and deviations notvery informative.

• Problem: loss function is almost symmetric, so it isunclear why the flatness leads to overbidding.

Goeree et al.:

• Incorporate asymmetry of loss functions

• In one treatment losses are higher for overbidding,

in the other for underbidding

• Flatness explanation should lead to overbidding in

one treatment and underbidding in the other

• Overbidding in both treatments indicates risk aver-

sion or joy of winning

• First-price auction, discrete valuations, bids inte-gers.

• Treatment 1: $0,$2,$4,$6,$8,$11, all with sameprobability

• Equilibrium: bid half the valuation (b = $5 for v =$11).

• Overbidding the equilibrium by $1 costs 0.25, butunderbidding only 0.08 (except for v = $11)

• Treatment 2: $0,$3,$5,$7,$9,$12, all with sameprobability

• Equilibrium: bids as above

• Now underbidding the equilibrium by $1 costs 0.25,

but overbidding only 0.08 (except for v = $0,$12)

• So if deviations from equilibrium are just noise, ex-

pect underbidding in treatment 1, but overbidding

in treatment 2.

Results

• Overbidding in both treatments, consistent with

risk aversion and joy of winning

• Overbidding proportional to value, consistent with

risk aversion, but not with joy of winning

• More overbidding in treatment 2, consistent with

intuition for noisy behavior.

Model of noisy behavior: Quantal Response Equilib-

rium

• Probabilistic choice rule: let Ue(i) denote expected

payoff from action i. Then i is chosen with P (i),

P (i) =exp(Ue(i)/µ)∑n

k=1 exp(Ue(k)/µ)

• Alternatively, use Ue(i)1/µ instead of exp(Ue(i)/µ)

• For µ → 0 behavior is noise free, for µ → ∞ it is

random.

• This captures differences between treatments: e.g.

in treatment 2 overbidding is relatively cheaper, so

occurs with higher probability than in treatment 1.

• In Quantal Response Equilibrium beliefs and actions

are consistent. E.g. for µ = 0 it is just the Nash-

equilibrium.

• Players play noisy best response to noisy behavior

of others

• Assume bidders are risk averse, expected utility func-

tion, b: bid, v: value, Pw(b): Probability to win, r

coefficient of relative risk aversion

Ue(b|v) =(v − b)(1−r)

1− rPw(b)

• Can find QRE for each r, µ combination

• Maximum-likelihood estimation used to find best

fitting noise and risk-aversion parameters.

• Fits data well, much better than when imposing

risk-neutrality.

• QRE with joy of winning also performs reasonably

well, but not as good as QRE with risk aversion.

Further Experimental Results on Overbidding

• Filiz and Ozbay (AER, 2007): test of regret theoryas overbidding explanation for FPA

– loser’s regret should lead to overbidding, win-ner’s regret to underbidding

– treatments vary information that winners andlosers receive

– information that allows for loser’s regret leads tosignificant overbidding, whereas with no informa-tion or information that only allows for winner’sregret leads to bidding not significantly differentfrom RNNE

• Garratt, Walker and Wooders (2004) examine over-bidding in SPA with experienced ebay users

– sincere bidding rare (21.2%), in line with exper-iments with standard subjects

– in contrast to previous experiments, underbid-ding as frequent as overbidding

– driven by bidders who also act as sellers on ebay,because to make profit, they need to buy belowvalue

– thus: experienced subjects do not necessarilyperform better in experiment (and can do worse,e.g. wool buyers in double auctions), unless fieldand lab are closely similar

Applications of Auction Theory

There are situations that at a first glance may not look

like auctions, but can be interpreted as auctions, and

where applying the revenue equivalence theorem yields

interesting insights

Suggestion to reform US legal system:

Losing party should pay winning party an amount equal

to the losing party’s expenditures.

Intuition: if spending $1 extra costs me $2, I will spend

less.

But this is wrong: assume that each party has a pri-

vately known value from winning rather than losing

the law-suit, that they independently decide how much

money to spend and that the party that spends more

money wins.

Both the existing system (pay your expenses) and the

suggested new system (pay twice your expenses if you

lose) lead to the party with the higher value winning

(since in equilibrium the expenses increase with the

value) and a party who does not value winning at all

has zero surplus (because it will not spend anything and

will hence lose) and thus RET applies.

Hence the expected total expenses (i.e. the “expected

revenue”) are the same and the new system would not

reduce the legal expenses.

The suggested reform would also not reduce the in-

centives for (and hence number of) lawsuits: while the

legal expenditures for specific values change (they de-

crease for low values and increase for high values), RET

also implies that for any value the expected payment

and hence the incentive to bring a lawsuit are the same

(because the probabilities to win are also the same).

Alternative system: the loser pays a part of the win-

ner’s expenses (common in Europe). RET does not

apply, because a party with value zero of winning the

lawsuit does not have an expected payment of 0. In

fact, expected payments are higher, but incentives to

bring suits are lower.

Queueing: Assume that values from obtaining a ticketare distributed as in RET. Bids correspond to the show-up time. Obviously, the buyers with the highest valueswin and a potential buyer with 0 value will not show up(or show up exactly at the time when the tickets are soldand hence have 0 waiting time) and hence his expectedpayment is 0. Thus the RET applies, implying that anychange to the system (e.g. making the waiting timemore or less comfortable) will not affect the expectedcost from waiting for each potential buyer and hencenot the total social cost (it will affect the waiting time,but not the costs of waiting).

Other types of such all-pay auctions, where auctiontheory can yield useful insights, are lobbying, politicalcampaigns, patent races etc.

Bertrand paradox: it is often claimed that the unique

equilibrium in a Bertrand game with inelastic demand

is to price at marginal costs (assuming that prices are

infinitely divisible).

This, however, is wrong: the Bertrand game corre-

sponds to a sealed-bid auction.

But uniqueness of an equilibrium in an auction requires

that the set of admissable bids is bounded, usually

by constraining bids to be above the minimal possible

value. If negative bids are allowed and if it is common

knowledge that both bidders have a value of 0, then it

is a mixed-strategy equilibrium to bid below a price −p

with probability kp for any fixed non-negative k.

Similarly, in the Bertrand-game with two sellers, it is a

mixed-strategy equilibrium to set prices larger than any

price p with probability kp for any fixed k (and of course,

with probability 1 above any price < k).

To see this, observe that the expected profit from charg-

ing any p ≥ k is then equal to pkp = k and thus each

seller is indifferent between all prices p ≥ k. Hence if de-

mand is perfectly inelastic, there is an equilibrium with

arbitrarily large profits (because k is arbitrary).

Multi-Unit Auctions — Theory

• Multiple objects are being auctioned off, that are ei-

ther identical or close substitutes (the case of com-

plements will not be considered here)

• The seller now has to choose not only between dif-

ferent auction forms but also whether to conduct

multiple sequential auctions or to sell all objects in

a single auction. The latter case is considered here.

Model:

• K identical objects for sale

• N potential buyers, with independent private values

Xi = (Xi1, Xi

2, . . . , XiK),

Xik being the marginal value of the kth object won

to bidder i

• Marginal values are declining: Xi1 ≥ Xi

2 ≥ . . . ≥ XiK

• Bidders are risk neutral

• Bidders are (ex ante) symmetric, each Xi is iid with

density f on

X = {x ∈ [0, ω]K : ∀k, xk ≥ xk+1].

• Special case: limited demand model: each bidder

attaches a positive value to only L < K units, hence

values are drawn from

X (L) = {x ∈ X : ∀k > L, xk = 0].

• For L = 1 this is called single-unit demand.

• In the auctions to be considered, each bidder sub-mits K bids bi

1 ≥ . . . ≥ biK

• Hence a total of N ∗K bids is solicited

• The auctions below are all standard auctions, i.e.they allocate the K units to the K highest bids

• A bidder who has k of the highest K bids will henceobtain k units

• Bidder i’s bid vector bi = (bi1, . . . , bi

K) can be in-verted to obtain i’s demand function

di(p) = max{k : p ≤ bik}

Example:

• Let K = 6, N = 3 and the bid vectors be

b1 = (50,47,40,32,15,5)

b2 = (42,28,20,12,7,3)

b3 = (45,35,24,14,9,6)

• The 6 highest bids are

(b11, b12, b31, b21, b13, b32) = (50,47,45,42,40,35),

• So bidder 1 wins 3 units, bidder 3 wins 2 and bidder

2 wins 1

Sealed-Bid Auctions

Discriminatory Auction, DA

In DA, each bidder pays the sum of his winning bids.

If ki bids of bidder i are among the highest K bids, thenhe pays

∑kik=1 bi

k

This corresponds to perfect price discrimination withrespect to the submitted demand, hence the name.

In the example, bidder 1 would have to pay 50 + 47 +40 = 137

The DA generalizes FPA, for K = 1 it reduces to FPA

It is in general very difficult to derive equilibria for the

discriminatory auction, even in the symmetric case (al-

though it can be shown that a symmetric equilibrium

exists).

Some general properties can, however, be derived.

• There will be bid shading on each unit bik < xi

k,

otherwise the gain from the kth unit would be 0 for

sure.

• Consider K = N = 2 and a symmetric equilibrium

(β1, β2)

– Since β1(x) ≥ β2(x) for all x by definition, the

distribution of β1 stochastically dominates that

of β2

– Now bidder 1 competes with his higher bid with

the lower bid of bidder 2 and vice versa.

– Thus 1 confronts with the lower bid a stochasti-

cally higher distribution of competing bids than

with the higher bid.

– This implies that the lower bid will be more ag-

gressive, i.e. β2(., x) > β1(x, .)

– Thus there will be cases of flat demand, i.e.

β1(x) = β2(x), when x1 − x2 is small.

– On the other hand, when x1−x2 is large, β1(x) >

β2(x)

– This implies inefficiencies.

Proposition: Every equilibrium of the discriminatory

auction is inefficient.

In the case of single-unit demand, the symmetric equi-

librium of FPA generalizes to

β(x) = E[Y (N−1)K |Y (N−1)

K < x]

and this is efficient. Hence the inefficiency is not a

result of multiple units being sold, but of multi-unit

demand.

Uniform-Price Auction, UPS

In UPS all units are sold at a “market clearing price”

where total (expressed) demand equals total supply.

This can be any price between the lowest winning (the

Kth) bid or the highest loosing (the (K + 1)th) bid.

In practice both are used, consider here the latter case.

Denote by c−i the vector of competing bids for bidder

i, that is the highest K bids of the other bidders, in

decreasing order.

Bidder i wins as many units as he defeats competingbids, i.e. he wins 1 unit if bi

1 > c−iK , but bi

2 < c−iK−1, he

wins ki units if

biki > c−i

K−ki+1and bi

ki+1 < c−iK−ki

The highest loosing bid, hence the price is

p = max{biki+1, c−i

K−ki+1} = max

j{bj

kj+1}

and if i wins ki units, he pays kip.

In the example above, bidder 1 faces competing bids

c−1 = (45,42,35,28,24,20)

and his bid vector is

b1 = (50,47,40,32,15,5),

so he wins 3 units and pays 3 ∗ 32 = 96

For K = 1, UPS corresponds to SPA

But it is not its appropriate generalization to the multi-unit case, since it does not share important character-istics, namely truthful bidding and efficiency.

Again, deriving equilibria is difficult even for the sym-metric case, but there are some general features ofequilibria.

It is easy to see that bids do not exceed marginal values.Otherwise, there could be a loss on (at least) the lastunit obtained.

Furthermore, it is a weakly dominant strategy to bidthe marginal value on the first unit bi

1 = xi1 by the

same reasoning as in the second price auction.

Bids on additional units, however, will be lower than

marginal values:

• raising the bid on an additional unit (say for unit k

from bik to bi

k + δ) has two effects:

– it increases the probability of winning a kth unit

– on the other hand it increases the expected price

paid for the 1st to (k − 1)th unit.

• The first effect only matters if bik < c−i

K−k+1 < bik+δ.

The probability for this event is small for δ small.

• Furthermore, the additional gain is xik − c−i

K−k+1which is arbitrarily small for bi

k close to xik.

• But the price for units 1 to k−1 is increased in that

case and also in the case c−iK−k+2 < bi

k+δ < c−iK−k+1.

• So for bik close to xi

k the expected gains from raising

the bid are smaller than the expected profits.

• In extreme cases (e.g. 2 units, 2 bidders) this can

lead to cases of full demand reduction, where bids

are 0 except for the first unit. Full demand reduc-

tion does not occur, however, if N > K

Proposition: Every undominated equilibrium of the

uniform-price auction has the property that the bid on

the first unit is equal to the value of the first unit. Bids

on other units are lower than the respective marginal

values.

Proposition: Every undominated equilibrium of the

uniform price auction is inefficient.

In the case of single unit demand, since it is a domi-

nant strategy to bid the value on the first unit, truthful

bidding is obviously an equilibrium and this is efficient.

Again, the inefficiency results not from multiple units

being offered, but from multi-unit demand.

Vickrey Auction, VA

In VA, if bidder i wins ki units he pays the ki highest

losing bids of the other bidders.

Hence if he wins one unit he pays c−iK and if he wins ki

units he has to pay

ki∑k=1

c−iK+1−k

Note that to win 1 unit, bidder i has to submit bi1 > c−i

K

and to win ki units, he has to submit bids such that

biki > c−i

K−ki+1

Hence he pays exactly all the bids in c−i that he beats.

Think of pik = c−i

K+1−k as the price paid for the kth unit.

In the above example, bidder 1 wins 3 units and hence

pays c−16 + c−1

5 + c−14 = 20 + 24 + 28 = 72

Like UPS, VA reduces to SPA for K = 1

It is, however, the appropriate extension of SPA to themulti-unit case:

Proposition: In a Vickrey auction, it is a weakly dom-inant strategy to bid according to βV (x) = x

Proof:

• Assume bidder i, when bidding according to βV , i.e.by submitting a bid vector xi, wins ki units

• This means that for all k ≤ ki, xik ≥ c−i

K+1−k = pik

and for all k > ki, xik ≤ c−i

K+1−k = pik

• Now if he changes his bids in a way that does not af-fect the number of units won, this does not changethe prices paid and hence does not affect the profit

• If instead the bids are changed such that additionalunits are won, then for the units with k > ki a pricepik ≥ xi

k is paid, leading to losses on these units (or0 profits), but does not affect the prices paid forthe first ki units

• If the bids are changed such that fewer units arewon, then the price on these units, and hence thesurplus, is not affected, but the surplus from addi-tional units won before was positive and is now lost.QED

Note that this result does not depend on symmetry or

risk-neutrality.

An immediate consequence is

Proposition: VA allocates the objects efficiently.

Drawbacks of the Vickrey auction:

• The rules are relatively complicated, and may hence

lead to confusion, resulting, in practice, in lower

efficiency.

• The results might appear unfair. Indeed, if bidder 1

places lower bids than 2 on all units, but both bid-

ders win the same number of units, 1 pays higher

prices (this holds strictly in the case of 2 bidders,

but still weakly for more bidders). Even more ex-

treme, the bidder placing the lower bids may end up

paying more in total for fewer units than the other

bidder.

Open Auctions

Dutch Auction

• In the multi-unit Dutch auction, the auctioneer starts

with a price high enough such that no bidder would

be willing to buy

• The price is lowered until a bidder is willing to buy

one unit at the current price. He obtains the unit

at this price and then the price is lowered further

until all K units are sold.

• This auction is outcome equivalent to the discrim-

inatory auction:

– If all bidders behave as according to the bid vec-

tor bi, i.e. indicating willingness to buy one unit

when the price reaches bi1 and a second unit when

the price reaches bi2, then the results are the same

as if bid vectors bi are placed in the discrimina-

tory auction.

• The Dutch and discriminatory auctions are not strate-

gically equivalent:

– Information becomes available in the course of

the Dutch auction

– In particular, the continuation game is different

if different numbers of units are already sold.

– Example: If there are K units for sale, then if

at a price of p already n units have been bought,

the rest of the game is a Dutch auction with

K − n units, where bidding behavior is obviously

more aggressive the smaller K − n is.

– In addition, in the interdependent value case, the

prices at which bidders buy are informative for

other bidders about their own valuations.

Uniform Price Open (English) Auction, UPO

• In UPO, the price is increased from a low level andbidders indicate their demand at the current price

• As the price increases, this demand naturally de-creases

• The auction ends when the total demand equalsK and the price paid for all units equals the pricewhere total demand decreased from K + 1 to K.

• UPO and UPS are outcome equivalent, but notstrategically equivalent, because information becomesavailable in the course of UPO.

• UPO allows bidders more easily to coordinate on

a (collusive) equilibrium, because, e.g., one bidder

can go ahead and drop out at 0.

Ausubel Auction, AA

• AA is an ascending-price auction that is outcomeequivalent to VA

• As in UPO, as the price is increased the biddersindicate the number of units they are willing to buyat this price

• Let the residual supply s−i(p) of bidder i at price p

be the part of the total supply that is not demandedby other bidders at p :

s−i(p) = max

K −∑j 6=i

dj(p),0

.

• For low prices s−i(p) = 0

• The price is increased until a price p′ such that for

some bidder i, s−i(p′) > 0

• This bidder then receives s−i(p′) units for price p′

• The price is then further increased until a price p′′

such that for some bidder j: s−j(p′′) > s−j(p′)

• This bidder then receives s−j(p′′)− s−j(p′) for p′′

• and so on until all units are sold

• This implements the Vickrey pricing rule

• AA has the advantage of being more transparent

than VA (as the price increases it is quite obvious

that a bidder should reduce demand from K to K−1

units at p if xk = p)

• But it might be more susceptible to collusion, be-

cause it allows for signalling the intention to collude

by dropping out at low prices.

• Also, since its pricing rule is the same as in VA, it

does not eliminated the “unfair” prices of VA.

Revenues

Consider the case of private, i.e. independently drawn,

but possibly asymmetric, i.e. not necessarily identically

distributed, values.

Consider two auctions and fix equilibria in each.

They are said to have the same allocation rule if the

probabilities qik(zi) that i wins the kth unit if he places

bid β(zi) are the same in both auctions

Then as for the single-unit case:

Proposition: The equilibrium payoff (and payment)functions of any bidder in any two multi-unit auctionsthat have the same allocation rule differ at most by anadditive constant.

• In the single-unit demand case DA (in symmetriccase), UPS and VA all allocate efficiently and hencethe revenue equivalence principle implies that theyhave equal revenues.

• For the case of multi-unit demand, the allocationrules differ and no general revenue ranking is possi-ble.

• In some special cases, however, one can argue why

the allocation rules are identical and hence apply

the revenue equivalence principle to gain further in-

sights into the equilibria.

Example:

• Let K = 3, N = 2 and L = 2 (i.e. each bidders

wants at most 2 units). Bidders’ value vectors X =

(X1, X2) are iid with density f on {x ∈ [0,1]2 : x1 ≥x2}.

• In any standard auction, each bidder will win one

unit for sure and will win two units if his second bid

is higher than the second bid of the other bidder.

• Let F1 and F2 denote the marginal distributions ofX1 and X2 with densities f1 and f2.

• Vickrey Auction:

– The Vickrey auction allocates efficiently and sinceit is dominant to bid truthfully, the vector ofcompeting bids for bidder i is c−i = (xj

1, xj2,0)

– The price paid for the first unit is hence 0 andfor the second it is, if won, x

j2

– A bidder’s expected payment is hence

mV (x) = Pr[X2 < x2]E[X2|X2 < x2] =∫ x2

0yf2(y)dy

which is independent of x1

• Uniform-Price Auction:

– It is weakly dominant to bid truthfully on the

first unit

– The bid on the second unit should not depend

on x1 (because the damage done to the profit

from the first unit by a potential price increase

does not depend on x1)

– In a symmetric increasing equilibrium β (β(x1, x2) =

(x1, β(x2)) bidder i will win two units if β(xi2) >

β(xj2) and hence if xi

2 > xj2

– Thus the allocation is efficient

– A bidder’s expected payment is

mU(x) =∫ x2

02β(y)f2(y)dy + (1− F2(x2))β(x2)

• Since both auctions allocate efficiently we can apply

the revenue equivalence theorem and obtain∫ x2

02β(y)f2(y)dy+(1−F2(x2))β(x2) =

∫ x2

0yf2(y)dy.

• From there β can be derived.

• Discriminatory Auction:

– Suppose a bidder bids b1 > b2

– Since he will one unit for sure, lowering b1 (to

a value still ≥ b2) will not affect the number of

units won, but lower the price paid for the first

unit

– Hence he will submit a flat demand b1 = b2

– Furthermore, this bid is completely determined

by x2 (the effect of a change in price on the first

unit does not depend on x1)

– Thus in a symmetric increasing equilibrium β (i.e.

β(x1, x2) = (β(x2), β(x2)), i will win 2 units if

xi2 > x

j2, and hence the allocation is efficient

– The expected payment in the DA is

mD(x) = β(x2) + F2(x2)β(x2)

• Applying the revenue equivalence principle then yields:

β(x2) =1

1 + F2(x2)

∫ x2

0yf2(y)dy

Multi-Unit Auctions — Experimental Evidence

Kagel and Levin (Econometrica, 2001)

• 1 human bidder with flat demand for 2 units

• several computer bidders with single-unit demand,

playing dominant strategy to bid own value

• supply of 2 units

• Compare demand reduction in uniform-price open

and uniform-price sealed-bid and Ausubel auctions

• equilibrium in UPS: b1(v) = v, b2(v) = 0

• equilibrium in UPO: drop on second unit no later

than second-highest computer rival

• equilibrium in AA: bid truthfully

Results:

• substantial demand reduction

• substantially more demand reduction in UPO thanUPS: 30.8% of second-unit bids set market price inUPS, only 11.4% in UPO, few 0-bids in UPS

• substantial overbidding of value (even on secondunit) in UPS, comparable to single-unit SPA

• difference comes from feedback in UPO: clock auc-tion without drop-out information yields similar re-sults as UPS

• bidding in AA near equilibrium

• profits are closest to prediction in AA

• AA works best because equilibrium equals bidding

own value, which bidders often do even in UPO,

so not because bidders adjust to institution but

because institution accommodates bidders’ natural

tendency

Engelmann and Grimm (EJ, 2009)

• simple setting with multiple human bidders with

multiple-unit demand

• 2 units for sale

• 2 bidders with flat demand for 2 units

• compare uniform-price sealed-bid, uniform-price open,

Vickrey, Ausubel, and discriminatory auction

• Equilibria

– VA and AA: truthful

– UPO and UPS: multiple equilibria, from truthful

to full demand reduction (b1(v) = v, b2(v) = 0),

including equilibria with partial demand reduc-

tion (not very plausible)

– DA: b1(v) = b2(v) = v/2

• Design

– fixed matching

– 10 auctions

– in uniform price, follow UPO with UPS or UPS

with UPO to check for learning spill-overs

Results: Vickrey Auction – First-unit bids

60

70

80

90

100ds

0

10

20

30

40

50

0 10 20 30 40 50 60 70 80 90 100

unit1

bid

values

Results: Vickrey Auction – Second-unit bids

100

80

90

100

60

70

80

s

40

50

60

unit2

bid

20

30

u

0

10

0 10 20 30 40 50 60 70 80 90 100values

Results: Ausubel Auction

90

100

70

80

50

60

price

s

40

50

ropo

ut p

20

30dr

0

10

00 10 20 30 40 50 60 70 80 90 100

values

Results: Ausubel Auction – Selected Pairs

90

100

70

80

90

50

60

70

ces

40

50

opou

t pri

c

first dropouts

double dropouts

20

30

dro second dropouts

0

10

0 10 20 30 40 50 60 70 80 90 1000 10 20 30 40 50 60 70 80 90 100values

Results: Uniform-Price Open Auction

90

100

70

80

50

60

price

s

40

50

drop

out

20

30

0

10

0 10 20 30 40 50 60 70 80 90 100values

Results: Uniform-Price Open Auction – Selected

Pairs

60

70

80

90

100

pri

ces

first dropouts

0

10

20

30

40

50

0 10 20 30 40 50 60 70 80 90 100

dro

po

ut

values

first dropouts

double dropouts

second dropouts

Results: Uniform-Price Sealed-Bid Auction – First-

Unit Bids

100

80

90

100

60

70

80

ds

40

50

unit1

bid

20

30

0

10

0 10 20 30 40 50 60 70 80 90 100values

Results: Uniform-Price Sealed-Bid Auction – Second-

Unit Bids

100

80

90

100

60

70

80

ds

40

50

unit2

bid

20

30

0

10

0 10 20 30 40 50 60 70 80 90 100values

Results: Discriminatory Auction – First-Unit Bids

100

80

90

60

70

ds

40

50

unit1

bid

20

30

0

10

0 10 20 30 40 50 60 70 80 90 100values

Results: Discriminatory Auction – Second-Unit Bids

60

70

80

90

100ds

0

10

20

30

40

50

0 10 20 30 40 50 60 70 80 90 100

unit2

bid

values

Results: Summary

• Less overbidding in open than sealed-bid auctions

• More demand reduction in UPO than UPS

• Bids close to equilibrium in AA, but some attempts

to collude (even more demand reduction than in

UPS)

• Substantial efficiency losses in UPO

• little spill-over from UPO to UPS

• Substantial bid-spreading in DA (where bids should

be equal):

– First-unit bids are on average above equilibrium

(in line with risk aversion)

– Second-unit bids are on average below equilib-

rium (suggesting risk seeking)

– Probably myopic “joy of winning”

What Really Matters in Auction Design

There were large differences in revenues from 3rd gen-eration mobile phone license auctions across countriesin 2000, varying between 600 Euros (UK) and 20 Euros(Switzerland) per person. This suggests that some ofthe auctions were poorly designed.

So what is a good auction design?

According to Klemperer one that resembles good com-petition policy: discouraging collusive, entry deterringand predatory behavior.

In contrast, auction theorists tend to assume a fixednumber of competitive bidders.

Hence lessons from auction theory may be misleading.

In a multi-unit simultaneous ascending auction (most

3rd generation license auction were run as such), the

bidders can use early stages to signal intentions to col-

lude

• Example: GSM spectrum auction in Germany 1999:

Mannesmann posted bids that could easily be inter-

preted as a suggestion for tacitly colluding.

Ascending auctions also facilitate collusion by creating

opportunities for punishment

• these opportunities arise within one multi-unit auc-

tion, so repeated-game effects can occur without

repetition, but, of course, repetition makes this

worse.

Prohibiting such activities appears difficult and hence

better auction design appears to be a more successful

approach.

Entry: Why entry is crucial: Bulow and Klemperer

(AER, 1996):

Terminology: an absolute auction is an auction without

a reserve price, i.e. the seller is required to accept the

final bid.

Main Result:

• Expected revenues from an absolute English auc-

tion with N +1 bidders are higher than for optimal

auction with N bidders and all bargaining power on

the side of the seller (under standard assumption in

both private and interdependent values cases)

Hence no amount of bargaining power is as valuable

as an additional bidder. Sellers should rather invest

resources in expanding the market than in gathering

the information necessary for an optimal auction.

Broader perspective: attracting additional entry may

be more important than optimal regulation of an in-

dustry. Furthermore, when selling a company, limiting

the number of bidders to improve seller’s control of the

negotiating process is not warranted.

Simple example: one buyer with value uniformly distrib-

uted on [0,1]. Optimal strategy for seller is to make

take-it-or-leave-it offer with price 12. Expected revenue

is 12 ∗

12 = 1

4. In contrast, if there are two buyers with in-

dependent private values uniformly distributed on [0,1],

the expected revenue from an absolute English auction

is min(v1, v2) = 13.

Theorem 1: Expected revenue from an absolute Eng-

lish auction with N + 1 bidders exceeds expected rev-

enue from an English auction with N bidders followed

by a take-it-or-leave-it offer to the last remaining bidder

if either (i) bidders’ values are private or (ii) bidders’

signals are affiliated.

Theorem 2: With independent signals and N risk-

neutral bidders, an optimal mechanism for a risk-neutral

seller is an English auction followed by an optimally-

chosen take-it-or-leave-it offer to the last remaining

bidder.

Corollary: With independent signals and risk-neutral

bidders, an absolute English auction with N +1 bidders

is more profitable in expectation than any negotiation

with N bidders.

If bidders’ signals are not independent, Theorem 2 does

not hold.

The mechanism of Theorem 2 is, however, still opti-

mal among the mechanisms where losers do not pay

anything and the winner is the bidder with the highest

signal and his payments are weakly increasing in his own

signal for any realization of the other bidders’ signals.

Thus an absolute English auction with N +1 bidders is

better than a standard mechanism with N bidders.

Furthermore, first negotiating with N bidders and re-

serving the right to hold an absolute English auction

with N + 1 bidders yields no benefit.

• The possibility of the auction implies that in the

whole process the object will be sold for sure.

• An optimal mechanism will hence sell to the bidder

with the highest valuation.

• But the highest of the first N bidders is sure to be

the highest bidder overall only if he has the maximal

possible value, which happens with probability 0.

Multiple units: K units, N > K bidders, all with single-

unit demand. Attracting K additional bidders yields

higher expected revenue than optimally negotiating with

N bidders.

Efficient auctions (Ascending versus discriminatory) havethe disadvantage of deterring entry if there are knownasymmetries:

• If a bidder knows that he has the lower value, he hasno reason to enter an ascending auction, becausehe is certain to be outbid later on and hence willnot enter if there are bidding costs.

This problem is amplified by the winner’s course:

• a firm that is almost certain to have a lower valuemust assume that there is something really wrongwhen it wins the auction

Even a small advantage can hence have a major impact

if it deters entry: Klemperer (EER 1998):

wallet-game with two bidders and a bonus for one bid-

der (almost common value):

• in standard wallet game both bidders receive a sig-

nal ti and the value to both is t1 + t2

• The equilibrium in an ascending auction is to bid

up to 2ti

• Now assume that bidder 1 gets a bonus of $1 in

case he wins.

• Then he will always win in equilibrium:

– bidder 1 can bid as if his signal is t1 + 1 now.

– Hence bidder 2, when winning at t2 knows to win

only 2t2 − 1 (his winner’s course is increased).

– Hence he should bid as if his signal is t2 − 1

– This reduces bidder 1’s winner’s course and al-

lows him to bid as if his signal is t1 + 2 and so

on.

If such an advantage is present, entry can be deterred by

demonstrating this advantage clearly and thus threat-

ening competitors with a winner’s course (e.g. Pacific

Telephone’s strategy for California telephone license).

Furthermore, it makes acquiring such a small advan-

tage a successful predatory strategy (e.g. BSkyB’s at-

tempt to buy Manchester United: this would have given

BSkyB a share in revenues from football television rights

and hence allowed it to bid more aggressively).

Further flaws in auction design:

• loopholes (e.g. if there are no costs to defaulting,bidders may place several bids and default if theydo not like the outcome)

• irrationally low reserve prices, that have two disad-vantages:

– They directly lower the revenue in the worst case(the number of bidders drops to number of li-censes as in Switzerland)

– They make collusion more attractive (a strongbidder may be satisfied with a lower share if itcan be acquired at a very low price)

Market structure:

• if it is determined by the auction, this may leadto too few (from the perspective of welfare maxi-mization) winners (because the benefit for remain-ing firms if the number is reduced by 1 is higherthan that firm’s profit)

• but also to too many winners (if they can share themarket at a very low price, but could reduce thenumber of competitors only at very high prices).

• On the other hand, it may lead to higher revenues,since it opens up different levels of competition(German 3rd generation auction).

Solutions:The appropriate auction design depends on

the context:

• If entry is not a problem and collusion only to a mi-

nor extent, it may be sufficient to make an ascend-

ing auction more robust (e.g. making bids anony-

mous and require them to equal round numbers

makes signaling more difficult).

• If insufficient entry is a major problem, then a sealed-

bid auction might be better, because it provides a

chance for a weak bidder to win. But this also im-

plies risks of inefficient allocations.

Anglo-Dutch auction:

• Ascending auction until only K + 1 bidders left

• Then sealed-bid auction at the last stage (some

similarity to auctions on eBay).

• Attracts entry due to chances for weak bidders

• but since information becomes available during course

of the auction, efficient allocation is more likely and

winner’s course problems less pronounced than in

sealed-bid auctions.

Most important might be effective antitrust policy:

several of the strategies employed in auctions would

have lead to legal consequences in other markets (ac-

knowledgment of tacit collusion by T-mobil, public sug-

gestions to share the market by Telekom Austria, threats

and lecturing about the winner’s course by Pacific Tele-

phone). Competition authorities have probably ignored

these actions, because they focus on markets where

consumers are more directly affected.

Failures of some auctions, however, should not be seen

against auctions in general.

Experimental Evidence on Collusion in Auctions

Goswami, Noe, and Rebello (REStud, 1996)

• effect of non-binding pre-play communication

• resembling Treasury bill auctions

• repeated at least 12 times per session

• 100 units for sale, value 20 for each bidder

• 11 bidders, demand up to 100 units

• state demand for prices 10, 15, 20

• uniform-price:

– competitive equilibrium: price 20

– collusive equilibria, e.g.:

– state demand 9 at 20 and 15, 100 at 10.

• discriminatory auction:

– equilibrium: all bid 15 on each unit

• Results:

– no effect of pre-play communication in DA: about

2/3 clear at 15, 1/3 at 20 in each case

– substantial effects in UPA: with communication

36% clear at 10 and 30% at 15, 0% at 10 and

16% at 15 without;

• Hypothesis that UPA is more vulnerable to collusion

is supported

Engelmann and Grimm (2009):

• more cases of complete demand reduction in Ausubelauction than in uniform-price sealed-bid auction,opposed to the theoretical prediction

• thus while auction theory would recommend theAusubel auction due to better properties in equi-librium, the improved chances for tacit collusion re-duce this advantage and may well reverse it in somecases

• for example, while in EG the Ausubel auction stillmaximizes efficiency, it yields lower revenue than allthe sealed-bid auctions

Phillips, Menkhaus, and Coatney (AER, 2003)

• sequential English Auction (like livestock auctions)

• fixed matching (7 auctions)

• 6 or 2 bidders

• treatments with number of units for sale known (19

to 30 per auction)

• treatments with communication via chat

Results:

• collusion increased with experience

• six bidders:

– prices with communication about 2/3 of those

without in last auction

– collusion via bid rotation

– cheating on last units occurred, but did not un-

dermine collusion in following auction

– no impact of information about number of units

• two bidders:

– prices with communication about 60% of those

without in last auction

– information about number of units for sale has

about the same effect as communication

– chats suggest disputes because of profit compar-

isons, which reduced collusion

• presence of a reserve price may have limited effects

of collusion

Corruption in Auctions

In auctions run by mail, bidders tend to prefer first-price

auctions.

This looks surprising:

• in second-price auction optimal bid is a lot easier to

figure out

• in particular if bidders do not know the number of

competing bidders and the distribution of their val-

uations and their risk preferences

Probable reason: it is easier for a seller to cheat in asecond-price auction, in particular if run by mail:

• once the winning bid is determined to be b, justhave someone send in an additional bid of b− ε

• thus the winner pays almost his bid and the sellercaptures the whole surplus

• e.g., in eBay there are some bidders who often re-tract bids and may well be shill bidders operated bysellers (the shill bidder is used to find out the proxybid of the highest bidder, after this is determined,the seller can use another shill bidder to drive upthe price to just below the proxy bid)