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Auctions with Interdependent Valuations Theoretical and Empirical Analysis, in particular of Internet Auctions
Julia Schindler
Vienna University of Economics and Business Administration
January 2003
Abstract
The thesis investigates a number of auction formats both theoretically and empirically. The
effect of different auction rules on the final price and on bidder valuations is analysed. Results
from an experimental sale of real goods, testing revenue equivalence of the open and sealed-
bid second-price auction do not conform to theoretical predictions: the open auction leading to
significantly lower prices than the sealed-bid auction. It turns out that the open auction format
allows bidders to satisfy a tendency to “stick together” with their valuations. The empirical
results motivate a dynamic bidding model of interdependent valuations, bidders being
uncertain about their valuations and learning from the exit-prices of their rivals.
Furthermore, bidding behaviour on the Internet is investigated in the hard close and the
automatically extended auction. Late bidding is shown to be a rational strategy in the hard
close auction, but not in the automatically extended auction. Theoretical results show that the
expected final price and seller revenue is lower in the hard close auction than in the
automatically extended auction, where prestige-concerns can lead to an explosive final price.
Moreover, Yahoo auction data confirms the strong presence of late bidding in the hard-close
auction and the seller’s preference for the automatically extended auction.
2
Introduction ......................................................................................................... 8
PART ONE: Auction Theory.................................................................... 10
1. The Four Standard Auction Formats.......................................................... 10
1.1 English Auction.......................................................................................................... 10
1.2 Dutch Auction ............................................................................................................ 10
1.3 First-Price Sealed-Bid Auction .................................................................................. 11
1.4 Vickrey Auction (Second-Price Sealed-Bid Auction) ............................................... 11
2. The Independent Private Values Model (IPV) ......................................... 11
2.1 Bidding strategies in the IPV Model .......................................................................... 12
2.1.1 First-Price Sealed-Bid Auction ............................................................................... 12
2.1.2 Dutch Auction ......................................................................................................... 15
2.1.3 Second-Price Sealed-Bid Auction........................................................................... 15
2.1.4 English Auction....................................................................................................... 16
2.1.5 Extension: Sealed-Bid Higher kth-Price Auctions................................................... 16
2.2 Results of the Independent Private Values Model ..................................................... 17
2.3 Extension: Uncertainty About the Number of Bidders .............................................. 17
2.3.1 Effect of the Number of Bidders on the Price ......................................................... 18
2.3.2 Effect on the Bidding Strategy ................................................................................ 18
3. Beyond Standard Assumptions.................................................................... 19
3.1 Risk Aversion............................................................................................................. 19
3.2 Asymmetries Between Bidders .................................................................................. 19
3.3 Interdependent Values................................................................................................ 20
3.3.1 The Common Value Model..................................................................................... 20
3.3.2 The Symmetric Model, the Milgrom-Weber Model ............................................... 22
3.3.2.1 Affiliation ............................................................................................................. 23
3.3.2.2 Bidding Strategies in the Milgrom-Weber Model................................................ 24
3.3.2.2.1 Second-Price Sealed-Bid Auction..................................................................... 24
3.3.2.2.2 English Auction................................................................................................. 24
3.3.2.3 Results of the Milgrom-Weber Model ................................................................. 26
3.3.2.3.1 Ranking of Expected Prices .............................................................................. 26
3.3.2.3.2 Linkage principle............................................................................................... 26
4. Revenue Ranking According to Theoretical Predictions .......................... 26
3
5. Experimental Tests of Bidding Behaviour and Auction Revenue ............ 27
5.1 Field Experiments ...................................................................................................... 28
5.1.1 Field Experiments on the Internet ........................................................................... 28
5.2 Laboratory Experiments............................................................................................. 28
6. Theoretical Predictions and Empirical Results.......................................... 30
PART TWO: Dynamic Price Formation in the Japanese Auction............................................................................................................................. 31
1. Introduction ................................................................................................... 32
2. Experiment..................................................................................................... 35
2.1 Experimental Set-Up .................................................................................................. 36
2.2 The Goods .................................................................................................................. 38
3. Results............................................................................................................. 39
3.1 No Revenue-Equivalence........................................................................................... 40
3.2 Lower Bid-Variance in Japanese Auction.................................................................. 40
3.3 Average Bid Not Significantly Different ................................................................... 40
3.4 Reasons for Lower Bid-Variance in Japanese Auction.............................................. 41
3.5. Learning Effects ........................................................................................................ 41
4. Interpretation................................................................................................. 41
5. The Model ...................................................................................................... 43
5.1 The First Round.......................................................................................................... 44
5.1.1 The First Exit........................................................................................................... 45
5.1.2 Common Value Estimation and the Updating Procedure ....................................... 46
5.2 The General Procedure............................................................................................... 48
5.3 Estimation Procedure ................................................................................................. 49
5.4 The Expected Final Price ........................................................................................... 49
5.5 Results of the Model .................................................................................................. 50
6. Conclusion...................................................................................................... 53
7. Appendix ........................................................................................................ 55
PART THREE: Experimental Test of Revenue Equivalence ........ 58
1. Motivation ...................................................................................................... 59
2. Experimental Set-Up..................................................................................... 59
4
3. Revenue Equivalence Between the Second-Price and the Japanese
Auction................................................................................................................ 60
3.1 Breakdown of Revenue Equivalence ......................................................................... 60
3.2 Testing the Effect of Revealing Public Information .................................................. 60
3.2.1 Results ..................................................................................................................... 61
4. English Outcry versus Second-Price Sealed-Bid Auction ......................... 63
4.1 Results ........................................................................................................................ 63
5. Revenue Equivalence: First-Price Sealed-Bid and Dutch Auction .......... 64
5.1 Results ........................................................................................................................ 65
6. Comparison: Second-Price Sealed-Bid, First-Price Sealed-Bid and
Japanese Auction............................................................................................... 66
6.1 Results ........................................................................................................................ 66
6.2 Interpretation .............................................................................................................. 67
7. Six Results of the Experimental Investigation of Revenue Equivalence. 68
8. Conclusion..................................................................................................... 69
PART FOUR: Internet Auctions and their Framework.................. 70
1. Introducing Internet Auctions ..................................................................... 70
1.1 Three Business Models: Ebay, Amazon, and Yahoo ................................................. 70
1.1.1 Revenue................................................................................................................... 72
1.2 Network Effects.......................................................................................................... 73
1.2.1 Loyalty .................................................................................................................... 74
1.3 The Selling Mechanism.............................................................................................. 74
1.3.1 Auctions and Posted Prices ..................................................................................... 74
1.4 The Goods .................................................................................................................. 75
1.4.1 Suitable for Auctions............................................................................................... 75
1.4.2 Goods Sold .............................................................................................................. 75
1.5 The Auction Formats Used ........................................................................................ 77
1.5.1 The Choice of Auction Format by the Auction House............................................ 77
2. Internet-Specific Characteristics ................................................................. 78
2.1 Internet Specific Advantages ..................................................................................... 78
2.2 Internet Specific Problems ......................................................................................... 79
2.3 Outlook....................................................................................................................... 81
5
3. The Internet Auction Rules .......................................................................... 81
3.1 Bid Submission and Procedure .................................................................................. 81
3.2 Bidder and Seller Registration ................................................................................... 82
3.3 Auction-Length .......................................................................................................... 82
3.5 Auction Fees............................................................................................................... 83
3.6 Additional Features .................................................................................................... 83
4. Some Implications of the Auction Rules ..................................................... 85
PART FIVE: Late Bidding Investigation ............................................. 88
1. Introduction ................................................................................................... 89
2. Theoretical Investigation .............................................................................. 90
2.1 Theoretical Investigation of Late bidding .................................................................. 91
2.1.1 The Moral Hazard Incentive ................................................................................... 91
2.1.2 Interdependent Values............................................................................................. 92
2.1.3 General Bidding Model........................................................................................... 92
2.1.3.1 Model of the Hard Close Auction ........................................................................ 93
2.1.3.5 Revenue Comparison: Hard Close and Automatically Extended Auction ........ 109
2.1.4 Prestige Value Model ............................................................................................ 109
2.1.4.1 Symmetrical Case............................................................................................... 110
2.1.4.2 Expert-Amateur Case ......................................................................................... 114
2.1.4.3 Result of Prestige Value Auctions ..................................................................... 117
2.1.4.4 Payoffs in the Automatically Extended Auction................................................ 117
2.1.5 Payoff Comparison: Hard Close and Automatically Extended Auction............... 118
2.1.6 Milgrom-Weber Model ......................................................................................... 118
2.1.6.1 Bidding in the hard close auction....................................................................... 118
2.1.7 General Prediction for Interdependent Valuations................................................ 121
2.1.8 Late bidding with Respect to the Ending-Rule ..................................................... 121
2.1.9 Late Bidding According to Good Type................................................................. 121
2.2 Seller’s Choice of Ending-Rule................................................................................ 122
3. Empirical Investigation............................................................................... 123
3.1 Late Bidding............................................................................................................. 124
3.1.1 Existence of Late bidding...................................................................................... 124
3.1.1.1 Complete Auction Duration ............................................................................... 124
3.1.1.2 Last Twelve Hours ............................................................................................. 126
6
3.1.2 Late bidding: Dependency on Ending-Rule .......................................................... 127
3.1.2.1 Complete Bidding-Path ...................................................................................... 128
3.1.2.2 Last Twelve Hours ............................................................................................. 130
3.1.2.3 Reasons for Late Bidding in Automatically Extended Auctions ....................... 130
3.1.3 Late bidding: According To Type of Good........................................................... 131
3.1.3.1 Art: Strongest Late bidding ................................................................................ 132
3.1.3.2 Computers: Late bidding Similar For Both Ending Rules ................................. 133
3.1.3.3 Late bidding In Car Auctions: Strongly Dependent On Ending-Rule ............... 134
3.1.4 Operational Investigation of Late Bidding............................................................ 135
3.2 Time-Invariance ....................................................................................................... 136
3.3 Winner’s Bidding Behaviour ................................................................................... 139
3.3.1 Entry Time of Winner ........................................................................................... 139
3.3.2 Winning Bid: Single Bid or Proxy Bid?................................................................ 140
3.4 Seller’s Choice of Ending-Rule................................................................................ 142
3.4.1 The Preferred Ending-Rule ................................................................................... 142
3.5 Successful Matchings............................................................................................... 144
3.5.1 Average Number of Bids....................................................................................... 144
3.5.2 Buy-Price............................................................................................................... 145
4. Results of the Empirical Analysis .............................................................. 146
4.1 The Four Main Hypotheses and the Empirical Evidence......................................... 146
4.2 Further Important Results......................................................................................... 147
5. Conclusion.................................................................................................... 147
PART SIX: Conclusion ............................................................................. 148
Literature ......................................................................................................... 149
7
Introduction
What is an auction? A mechanism that determines the price and allocation of goods by
comparing competing bids.
Auctions have been used for selling goods in ancient cultures such as early China, Greece and
the Roman Empire. Herodotus reports auctions of women on the annual marriage market as
early as 500 B.C. in Babylon. Nowadays, auctions are a widely used selling device for diverse
items, such as government bonds, state-owned firms and mineral rights for oil and other
natural resources.
Sotheby’s and Christies, founded in the 18th Century in Great Britain, represent a branch of
traditional auction houses known for selling exquisite items such as art, antiquities and
jewellery, or collectibles, such as coins and stamps to the wider public. Another set of goods
frequently sold in auctions are perishable products, such as flowers (in Holland) and fish (in
Japan).
Due to the Internet and the consequently low transaction costs, auctions have boomed. Ebay,
Amazon and Yahoo auctions enable consumers to buy and sell items on a virtual platform
open to bidders around the world. Supplier contracts are auctioned-off online. Whether
traditional or online, the seller wants to receive the highest possible price for his good. The
question of how buyers form their bids and which auction format realises the highest auction
revenue for the seller needs to be answered given the information technology era and its new
empirical insights.
The dissertation is set-up as follows:
In part one I present an overview of auction theory.
In part two, an experimental test of revenue equivalence between the second-price sealed-bid
and the English auction (also known as the English ascending bid auction) is conducted. The
empirical results do not conform to theoretical predictions: the open English auction yields
significantly higher revenue than the second-price sealed-bid auction. Furthermore, bids are
8
far more narrowly dispersed in the English than in the second-price sealed-bid auction. The
empirical observations call forth a dynamic bidding model of interdependent valuations.
Bidders are uncertain about their valuations and follow a boundedly rational-learning rule to
update their valuations in the course of the auction.
In part three, further tests of revenue-equivalence are conducted. The sealed-bid format is
compared to the open format for the two pairs of strategically equivalent auctions: First-price
sealed-bid and Dutch auction, English outcry and second-price sealed-bid auction. The results
are tested under the effect of revealing public information.
In part four the rules and framework of the currently existing Internet auctions are presented.
In part five bidding behaviour in Internet auctions is analysed using two models of
interdependent valuations: a general and a prestige value model. Late bidding is found to be a
rational bidding strategy in the hard-closing auction, lowering the price and seller revenue. On
the other hand prestige-effects can lead to exorbitant seller revenue in the automatically
extended auction. The theoretical predictions are tested using Yahoo auction data with respect
to two ending rules (hard close) and three categories of goods (cars, computers and paintings).
Late bidding is found to be strongly present both in terms of the dynamic and operational
bidding path as well as the winning bidder’s entry time. The proposed seller’s preference for
automatically extended auctions is empirically confirmed.
In part six I conclude.
9
PART ONE: Auction Theory
Auction rules can be chosen with respect to two goals. One goal is maximization of the seller
revenue; the second goal is efficiency; efficiency meaning that the good is allocated to the
highest-valuing bidder. Efficiency and revenue-maximisation do not necessarily conflict. Here
we focus on private goods, where a seller is usually concerned with finding an auction
mechanism to maximise his revenue. In the best case, the seller could charge the highest
valuing bidder a price exactly equal to this valuation. But, the seller does not know the
bidders‘ valuations. The goal of the bidder is to maximise his utility, which is the difference
between the valuation of the good and the price he has to pay. Thus, the bidder has no interest
in revealing his valuation to the seller.
No auction mechanism can determine prices directly in terms of bidder preferences and
information. The seller must choose auction rules that reveal information about the bidders’
preferences. There are a large number of rules a seller could choose when designing his
personal auction selling device. Auction theory works with the following four auction
formats:
1. The Four Standard Auction Formats
1.1 English Auction
The English auction is an open, ascending bid auction. The price is raised sequentially until
only one active bidder remains. The good is allocated to the highest bidder, who has to pay a
price equal to the second-highest bid.
1.2 Dutch Auction
The Dutch auction is an open, descending bid auction. A counter showing the current price is
lowered continuously until the first bidder cries: „halt“. The Dutch auction is for example
used in Holland for selling flowers.
10
1.3 First-Price Sealed-Bid Auction
The first-price sealed-bid auction is a closed auction. Every bidder enters a private bid. The
good is awarded to the highest bidder at the price of his own bid.
1.4 Vickrey Auction (Second-Price Sealed-Bid Auction)
The second-price sealed-bid auction is a closed auction. The good is awarded to the highest
bidder at the price of the second-highest bid.
2. The Independent Private Values Model (IPV)
In order to analyse which auction format is the revenue maximising choice, we need to make
some assumptions about the way bidders form their valuation of the good. One frequently
chosen set of assumptions is the independent private values model:
Assumptions:
- Risk-neutrality: All bidders are risk-neutral, maximising their expected profits.
- Independence: The bidders’ values are private and independently distributed.
- Symmetry: The values of the bidders are distributed according to the same distribution
function.
- No budget constraint: Bidders have the ability to pay up to their respective values.
A risk-neutral seller wants to sell an indivisible object that he himself values with zero1. There
are n bidders. Bidder i (i = 1,.., n) draws his valuation xi from the distribution function Fi(xi)
independently and identically distributed on the interval ], xx[ with the density function fi(xi)
fi(x) = f(x) for all x ∈ ], xx[ .
1 This is equivalent to assuming the good has already been produced and the seller’s utility from using it is zero.
Jehle and Reny (2001), p.374.
11
Every bidder knows his valuation, but cannot observe the private valuations of the other
bidders. The seller does not know the bidder valuations, but he and all bidders know the
distribution of the bidder valuations and the number of bidders.
A bidder’s valuation is independent and private, denoting differences in taste. The value of
the good depends only upon personal preferences; a bidder’s value is unaffected by the
valuations of the other bidders (even if he knew them, his valuation for the good would
remain unchanged).
The winning bidder receives the good and has to pay a price p. His payoff is given by: xi – p.
If he does not win the good, his payoff is zero.
2.1 Bidding strategies in the IPV Model
Bidding behaviour in an IPV auction is a non-cooperative game. Bidders devise a strategy; i.e.
a bidding function +ℜ→],0[: ωβ that maps every possible value bidder i could draw into a
non-negative bid. Bidders search for the bidding function that leads to the most desirable
outcome, given that all other bidders also form their bid according to that same bidding
function.
2.1.1 First-Price Sealed-Bid Auction
In a first-price sealed-bid auction the bidder with the highest bid wins and pays a price equal
to his bid.
−
−=Π
,/)(0
kbx
bx
ii
ii
i }maxarg{: j
jbk =
ififif
jiji
jiji
jiji
bbbbbb
≠
≠
≠
=<<
maxmaxmax
In the first-price sealed-bid auction bidders shade their bids; i.e. bid less than their valuation.
If a bidder bid his true valuation, he would have to pay a price equal to his valuation in case of
winning and receive a payoff of zero.
12
Bidders face a trade-off; shading the bid downwards means lowering the probability of
winning but also means increasing the expected gain (when being the winning bidder).
G…distribution function of Y1, where Y1 is the second-highest private signal.
g…the density of Y1, g = G’
The expected payoff of the winning bidder is given by: G )))((( 1 bxb −−β
Maximising this with respect to b, yields the following first-order condition:
0))(()())(('))(( 1
1
1
=−− −−
−
bGbxbbg β
βββ
In a symmetric equilibrium )(xb β= and yields the following differential equation:
)()()()()( ' xxgxxgxxG =+ ββ
or equivalently,
)())()(( xxgxxGdxd
=β
since ,0)0( =β
][
)()(
1)(
11
0
xYYE
dyyygxG
xx
<=
= ∫β
The symmetric equilibrium strategy in a first-price auction is: ][)( 11 xYYExF <=β
Proof:
Only strictly increasing bidding functions are considered; it is assumed that bidders with
higher valuations make higher bids.
z denotes the value for which b is the equilibrium bid, , so that )(1 bz −= β bz =)(β .
13
Bidder 1’s payoff from bidding )(zβ when his value is x is given by:
∫
∫
∫
+−=
+−=
−=
<−=
−=Π
z
z
z
dyyGzxzG
dyyGzzGxzG
dyyygxzG
zYYEzGxzGzxzGxb
0
0
0
11
)())((
)()()(
)()(
][)()()]()[(),( β
It follows that: 0)())(()),(()),(( ≥−−=Π−Π ∫z
x
dyyGxzzGxzxx ββ
If all bidders follow the strategy β , a bidder with a value of x will be best off bidding )(xβ ;
thus β is a symmetric equilibrium strategy.
The equilibrium bid can be written as: ∫−=x
F dyxGyGxx
0 )()()(β
This is the symmetric Nash equilibrium of a first-price sealed-bid auction. The bidding
function is strictly increasing in x and offers a unique solution.
As can be seen from the expression above, bidders in a first-price auction bid less than their
valuation. The degree of bid shading depends on the number of rival bidders, because 1
)()(
)()(
−
=
N
xFyF
xGyG
As the number of bidders increases, the equilibrium bid approaches x. )(xFβ
In the case of uniformly distributed valuations on [0,1]:
F(x) = x, then G(x)=xN-1 and xN
NxF 1)( −=β
The expected seller revenue, i.e. expected price is:11][
+−
=NNRE F
The expected utility of the winning bidder with signal xN is:NxN
14
2.1.2 Dutch Auction
In the Dutch auction a bidder needs to decide at what price to cry “halt”. The winning bidder,
has to pay a price equal to his “bid”. The Dutch auction is strategically equivalent to the first-
price auction.
2.1.3 Second-Price Sealed-Bid Auction
In the second-price sealed-bid auction, the price the winner has to pay is determined by the
second-highest bid and is thus independent of the winner’s bid.
−
−=Π
,/)(0
kbx
bx
ii
ii
i }maxarg{: j
jbk =
if if
if
jiji
jiji
jiji
bbbbbb
≠
≠
≠
=<<
maxmaxmax
It is a unique weakly dominant strategy to bid one’s own valuation: xxS =)(β
Proof:
•By bidding above his valuation, a bidder runs the risk of winning the auction, in cases where
he would make a loss.
Assume that bidder 1 has a valuation and that the highest competing bid is: .
By bidding , bidder 1 will win if b and does not win if b
1x jij bp ≠= max1
11 xb = 11 p> 11 p< .
In case bidder 1 bids an amount higher than his valuation: . If , then bidder
1 wins with a payoff: . This is the same payoff he would have received from bidding an
amount equal to his valuation. If , bidder 1 loses. If , bidder 1 wins
but makes a loss equal to , whereas by bidding an amount equal to his valuation he
would not have made a loss. It follows that it is never profitable for bidder 1 to bid above his
valuation, as this never increases his profit, but may actually decrease his profit.
11 xb > 111 pxb ≥>
11 xp >>
11 px −
x
111 xbp ≥>
1
1b
1 p−
• By bidding below his valuation, a bidder lowers his chances of winning: he does not win in
cases where he could have received a positive payoff. Thus, the pay-off maximising strategy
for bidder i is to bid his valuation.
15
Assume bidder 1 has a valuation of and that the highest competing bid is .
By bidding , bidder 1 will win if b and does not win if b
1x jij bp ≠= max1
11 xb = 11 p> 11 p< .
In case bidder 1 bids an amount smaller than his valuation: b 11 x< . If , then
bidder 1 wins with a payoff: . This is the same payoff he would have received from
bidding an amount equal to his valuation. If , bidder 1 loses. If ,
bidder 1 loses, but could have won by bidding b
111 pbx ≥>
11 px >
11 px −
111 bxp ≥>
11 x
1b>
= . It follows that it is never profitable for
bidder 1 to bid below his valuation, because this may decrease his profit.
2.1.4 English Auction
In an English auction a bidder has to decide when to drop out of the auction. The English
auction differs from the sealed-bid auctions in that bidders observe the exit-prices of the
others and have the possibility to revise their valuation – as long as they are active
participants.
In an English auction truthful bidding is a weakly dominant strategy. If weakly dominated
strategies are eliminated, the bidder with the highest valuation wins and pays a price equal to
the second highest valuation.
2.1.5 Extension: Sealed-Bid Higher kth-Price Auctions2
Theoretically it is possible to conduct an auction, where it is neither the highest bid that
determines the price the winner has to pay, nor the second-highest bid, but instead the third-
highest or fourth-highest bid. Using higher k-th price auctions leads to the following results:
1.) Bids are higher than valuations.
2.) Equilibrium bids increase as k increases.
3.) Equilibrium bids decrease as the number of bidders is increased.
The reason why third-price auctions and higher are generally not found in practice, is because
they expose the seller to higher risk than the standard auction formats.
2 Higher k-th price auctions, meaning higher than second-price auctions.
16
2.2 Results of the Independent Private Values Model3
Result 1: The Dutch auction is strategically equivalent to the first-price sealed-bid auction.
Two auction formats are strategically equivalent, when the expected seller revenue is equal
and an identical bidder would choose the same strategy under both auction formats.
Result 2: The English auction is strategically equivalent to the second-price sealed-bid
auction, but in a weaker form than the strategic equivalence of the Dutch and first-price
sealed-bid auction - the latter holding even when bidders are uncertain about their valuation.
Result 3: The second price and the English auction lead to an efficient allocation, a Pareto-
optimal outcome. The Dutch and first-price sealed-bid auction also lead to an efficient
outcome as long as the bidders’ valuations are drawn from a symmetric distribution.
Result 4: The expected seller revenue is equal to the expected value of the second highest
bidder.
Result 5: The seller’s expected revenue is equally high in all four auction formats. This is the
famous revenue equivalence theorem by Vickrey (1961).
Result 6: The four standard auction forms can be designed so as to produce an optimal
outcome by using entry fees or reserve prices. This result is true for many common sample
distributions, including the normal, exponential, and uniform distribution.
Result 7: When the seller, the bidders or both are risk-averse, the seller strictly prefers the
Dutch or first-price sealed-bid auction to the English or second-price auction.
2.3 Extension: Uncertainty About the Number of Bidders
3 As noted in Milgrom and Weber (1982).
17
It is generally assumed in auction theory that the number of participating bidders is known. In
reality bidders often face uncertainty with respect to the number of bidders, for example in
Internet auctions, where bidders are allowed to enter until the very last moment.
2.3.1 Effect of the Number of Bidders on the Price
In the independent private values model, as the number of bidders increases, the second
highest valuation approaches the upper limit of the distribution of valuations, and thus the
price tends to the highest possible valuation (Holt 1979). As long as the number of bidders is
finite, the price the winning bidder has to pay is smaller than his valuation. A higher number
of bidders raises the seller revenue and lowers the bidder revenue in all four auction formats.
2.3.2 Effect on the Bidding Strategy
The number of bidders does not influence the bidding strategy in the second-price auction, but
does influence it in the first-price auction.
The bidding strategy in the second-price auction is given by: xx =)(β . Uncertainty about the
number of participating bidders, and consequently under- or overestimating the number of
participating bidders has no effect in the second-price auction.
When values are uniformly distributed x ~U the bidding strategy in the first-price
auction is determined by:
]1,0[
xN
Nx 1)( −=β . The expected seller revenue in the first-price
auction is equal to 11
+−
NN and the expected payoff of the winning bidder (with the private
signal ) is equal toNxNxN .
Under- or overestimating the number of participating bidders in the first-price auction affects
a bidder’s probability of winning and his expected revenue. Underestimating the number of
participating bidders reduces the individual bidder’s probability of winning. If all bidders
underestimate the number of participating bidders, the seller’s expected revenue falls.
Overestimating the number of bidders increases the individual bidder’s probability of
winning, but lowers his expected revenue. The seller’s expected revenue rises when all
bidders overestimate the number of participating bidders.
18
3. Beyond Standard Assumptions
The Revenue Equivalence Theorem does not always hold when assumptions of the
independent private value model are relaxed.
3.1 Risk Aversion
When either the seller or the buyers are risk-averse, the first-price auctions lead to higher
seller revenue than the second-price auctions.
Under risk-aversion the equilibrium bidding strategy in the second-price auction remains
unchanged, but changes in the first-price auction. In a first-price auction bidders shade their
bid whether they are risk-neutral or risk-averse. Risk-averse bidders in a first-price auction
shade their reservation price more heavily than when they are risk-neutral. Risk-neutral
bidders shade their bid less, because the increase in expected-payment due to a marginal
increase of the bid, is less costly than the reduced probability of not winning the auction due
to the lower bid. This raises the seller’s expected revenue and lowers the bidder’s expected
payoff4.
With constant absolute risk-aversion the first-price auction produces higher expected revenues
than the second-price auction.
3.2 Asymmetries Between Bidders
If the assumption of symmetrical bidder valuations is removed, the first-price auction does not
always create an efficient outcome (the good is not always awarded to the highest valuation
bidder).
Roughly speaking, the sealed-bid auction generates more revenue than the open auction when
bidders have distributions with the same shape (but different supports). In contrast the open
4 Riley and Samuelson (1981).
19
auction generates more revenue than the sealed-bid auction when distributions have different
shapes but approximately the same support.
Ex ante asymmetries can discourage participation by lower valuing bidders. Small
asymmetries can lead to highly asymmetric equilibria that result in low seller revenues5.
3.3 Interdependent Values
The private value model is often unrealistic, because there are many goods where bidders are
uncertain about their valuation and are influenced by the valuations of the other bidders. In
the following section the private values assumption is relaxed and instead bidders are assumed
to have interdependent values. Interdependent values imply that every bidder has some private
information in form of a signal, but a bidder does not perfectly know his valuation for the
object. It may now be the case that other bidders possess information that would - if known to
the bidder - affect his valuation. This can be due to resale or prestige considerations: a buyer
of an old-timer might want to resell the car after some time, thus he will let his valuation be
somewhat dependent on the other bidder’s valuations.
Interdependent values do not imply anything about the distribution of the bidders’ signals:
signals can be independently distributed or correlated. The best-known model of
interdependent values is that of Milgrom and Weber (1982). They assume that bidders’
signals are affiliated, which is a special form of positive correlation (see Part Two Chapter
2.3.1 below).
Milgrom and Weber’s general symmetric model of interdependent valuations can account for
the case of strictly private valuations (see above for the IPV model), for the intermediate cases
and for strictly common valuations (see below).
3.3.1 The Common Value Model
A good having a single objective value is offered for sale. The bidders do not know the value
of the object, but every bidder has access to some information on its value, each bidder
5 See Klemperer (1998), p.764.
20
making a different estimate of the good’s value. V is the true value of the good, drawn
randomly from a probability distribution (here: a uniform distribution) on the interval: ], xx[ .
Each bidder receives a private signal xi, i =1,..,N. The private signals are independent draws
from the uniform distribution on [V-ε, V+ε]. A first-price auction is considered here:
Bidders do not know the true value V and try to estimate the correct expected value. The
expected value of the item conditional on signal xi is: ii xXVE =][ . In this case every bidder
would take his private signal to be the best estimate of the good’s value, knowing that the
expected mean signal is equal to the site’s true value. But if every bidder bids his private
signal, the winning bidder will be the one with the highest private signal. He will have
overbid the true value V most highly, making a loss in turn. This is known as the winner’s
curse.
Foreseeing that a bidder will only win when his signal is the highest signal, he bids the
expected value conditional on being the high bidder:
11][ max +
−−==
NNxxXVE ii ε
The expected value conditional on being the high bidder is lower than the expected value
conditional on the private signal:
][][ max iiii xXVExxXVE =>== 1>Nfor
The winner’s curse can be measured as the difference in the two expected values. Avoiding
the winner’s curse requires considerable discounting of bids relative to the signal values. The
size of the discount is an increasing function of the number of bidders N and the dispersion of
the signals around the true value ε. Raising the number of bidders or lowering the precision of
the signals leads to a higher winner’s curse (when bidders ignore their judgemental failure).
The symmetric equilibrium bid function is equal to:
Yxxb ii +−= ε)( where ( )
+−
−
+= )(
2exp
12 ε
εε xxN
N iY
21
Expected profits for the high bidder are equal to: YN
−+1
2ε
Y diminishes rapidly as xi moves beyond x + ε. Ignoring Y, the bidding function is
approximately equal to ε−= ii xxb )( and the high bidder’s profit equal to 1
2+Nε .
If bidders ignore the winner’s curse, the bid function under risk neutrality is:
NY
Nxxb ii
s +−=ε2)(
The model predicts that the high signal holder always wins the auction. This is because all
bidders use the same bid function, their only difference being their private information xi
regarding the value of the item.
3.3.2 The Symmetric Model, the Milgrom-Weber Model
As introduced in Section 3.3 the most prominent model of interdependent valuations is the
Milgrom-Weber model (1982). Bidders have some uncertainty about their valuation, due to
resale or prestige considerations. Being a general model it can take account of the various
degrees of uncertainty ranging from the purely private value model to the purely common
value model. For all intermediate cases they assume that private signals are positively
correlated by affiliation:
There are n bidders. Bidder i’s value of the object is Vi = u i (S, X). The bidder’s valuation
does not only depend upon his private signal, but also upon the other bidders’ private signals.
S = (S1,..., Sm ) is a vector of variables measuring the good’s quality, which influence the
value of the object to the bidders. The bidders cannot observe S, but the seller can observe
some or all components of S.
X = (X1 ,...,Xn) is a vector of value signals observed by the individual bidders. Let Y1,…,Yn-1
represent the largest to the smallest estimates from among X2,…,Xn. Every bidder observes a
private signal about the value of the good. Bidder i, i = 1,..,n observes the private signal Xi
about the value of the good.
Bidder 1’s value is: V1 = u1 (S1, …Sm, X1, Y1,…Yn –1)
22
Assumption 1: There is a function u such that for all i, u i (S,X) = u(S,X i,{X j}j ≠i). Thus, all of
the bidders’ valuations depend on S in the same way, and each bidder’s valuation is a
symmetric function of the other bidders’ signals.
Assumption 2: u is nonnegative, continuous and non-decreasing in its variables.
Assumption 3: For each i, E [Vi] < ∞
When Vi = Xi for all i , the model is reduced to the independent private value model.
When Vi = S1 for all i , the model is reduced to the common value model.
Bidders are risk-neutral and their valuations are in monetary units, so that when bidder i
receives the object and has to pay p, his payoff is Vi – p.
f(s,x) is the joint probability density of the random elements of the model. Two assumptions
are made about the joint distribution of S and X:
Assumption 4: f is symmetric in its n last arguments.
Assumption 5: The variables S1,..., Sm, X1 ,..., Xn are affiliated.
E [V1|X1= X, Y1 = Y1…, Yn-1 = Yn-1] is non-decreasing in x.
3.3.2.1 Affiliation
Every bidder has some private information about the value of the good. This private
information is expressed in the signal he draws. In case of affiliation it is assumed that the
bidders’ signals are positively affiliated. Affiliation is a strong form of positive
correlation and means that if a subset of the take on large values, then the remaining
also take on large values. Variables are affiliated if large values for some of the variables
make the other variables more likely to be large than small. A high value of one bidder’s
estimate makes high values of the other bidders’ estimates more likely.
nXXX ,...,, 21
iX jX
For variables with densities, affiliation can be defined as such:
Let z ∨ z’ denote the component-wise maximum of n+m dimensional vectors z and z’ and let
z ∧ z’ denote the component-wise minimum. Variables are affiliated if, for all z and z’,
f(z ∨ z’)f(z ∧ z’) ≥ f(z)f(z’).
23
Three implications of affiliation6:
1.) Y1,…,Yn-1 are the largest to the smallest estimates from among X2,…, Xn. If the variables
are affiliated, then the variables are also affiliated. nXXX ,...,, 21 111 ,...,, −nYYX
2.) )( xG ⋅ denotes the distribution of Y1 conditional on X1 = x. and Y being affiliated
implies that if x’>x, then
1X 1
)'( xG ⋅ dominates )( xG ⋅ in terms of the reverse hazard rate, that is,
for all y, )()(
)'()'(
xyGxyg
xyGxyg
≥
3.) If γ is any increasing function, then x’>x implies that:
])([]')([ 1111 xXYExXYE =≥= γγ
3.3.2.2 Bidding Strategies in the Milgrom-Weber Model
3.3.2.2.1 Second-Price Sealed-Bid Auction
Bidder 1’s decision problem in the second-price sealed-bid auction is to choose a bid b that
maximises the expected value minus the price conditional on bidder 1’s signal (when this is
the highest signal).
The equilibrium strategy of every bidder is to bid ),()( xxvxS =β
],[:),( 111 yYxXVEyxv === . v is non-decreasing.
In the case of private values (where v(x,x)=x) the equilibrium strategy is weakly dominant.
With general interdependent values however is not a dominant strategy. Sβ
3.3.2.2.2 English Auction
The English auction in the Milgrom-Weber Model is modelled as a Japanese Auction. The
auction begins at a price of zero, at which all bidders are active. The auctioneer raises the
price and bidders can quit the auction by depressing a button. Bidders who have quit the
auction cannot return at a later point in time. Every time a bidder quits the auction, the exit-
price is revealed to all remaining bidders.
6 Taken from Krishna (2002), p.86.
24
A bidding strategy in the English Auction must specify for each of his possible valuations,
whether he will be active at any given price level, as a function of the bidding activity
observed until then.
The exit-prices can be ordered as follows: p1§ … § pk
Bidder i’s strategy can be described by a function bik(xi | p1, …, pk), specifying the price at
which bidder i will quit if, at that point, k other bidders have left at the prices p1, …, pk.
Because the price can only rise, bik(xi | p1, …, pk) has to be greater or equal to pk.
The symmetric equilibrium is:
[ ]
2,...,1],)(
,...,),...,(
,...[),...,(
...)(
11*0
11*
1
11111*
1111*0
−==
=
=====
=====
−
−−−
−
−
nkpY
pppY
xYYXVEppx
xYYXVEx
n
kkknk
nkk
n
β
β
β
β
)(*0 xβ denotes the optimal bid when all bidders are active,
),...,( 1*
kk ppxβ the equilibrium bid after k bidders have quit the auction.
3.3.2.2.3 First-Price and Dutch Auction
The first-price sealed-bid and the Dutch auction are strategically equivalent and can be treated
equally. The equilibrium strategy for a bidder is:
∫=x
F xydLyyvx0
)(),()(β
where, =)( xyL
− ∫
x
y
dtttGttg)()(
exp
)( xG ⋅ is the distribution function of Y , i.e. the second highest signal; under the condition that
the highest signal is equal to x.
1
)x(g ⋅ is the conditional density function of Y . 1
Milgrom and Weber further prove that the expected seller revenue of the second-price sealed-
bid auction is greater or equal to that of the first-price sealed-bid auction.
25
3.3.2.3 Results of the Milgrom-Weber Model
1.) The Dutch and the first-price sealed-bid auction are strategically equivalent.
2.) When bidders are uncertain about their value estimates, the English and the second-price
sealed-bid auction are not equivalent. The English auction leads to higher expected prices
due to the linkage principle (see below).
3.) When bidders’ value estimates are statistically dependent, the second-price sealed-bid
auction generates higher average prices than does the first-price sealed-bid auction.
4.) If the seller has access to a private source of information, his best policy is to commit
himself to honesty (always reporting all information completely). This is true for the first-
price sealed-bid, Dutch, second-price sealed-bid and the English auction.
3.3.2.3.1 Ranking of Expected Prices
English > Second-price sealed-bid > Dutch = First-price sealed-bid
3.3.2.3.2 Linkage principle
The linkage principle explains why the English auction yields higher revenue than the second-
price sealed-bid auction under affiliation and why revealing public information raises the
price. Every bidder receives indirect information about the valuations of the other bidders
from their publicised exit-prices. Observing bidding behaviour by others makes bidders more
confident and lets them bid higher on average. All measures that increase the information of
bidders, for example quality guarantees, are price increasing and advantageous to the seller.
The price is linked to the valuations of the non-winning bidders and the winning bidder. The
auction prices depend on the reports the bidders make and on the seller’s information.
The Dutch and first-price sealed-bid auction with no linkages to the other bidders’ estimates,
yield the lowest expected price. The English auction with linkages to any of the estimates of
the non-winning bidders yields the highest expected price. Revealing public information
raises the price in all three auctions, by adding a linkage.
4. Revenue Ranking According to Theoretical Predictions
The table below summarises the theoretical predictions about seller-revenue for the
independent private values and the affiliated values model (for private and interdependent
26
valuations). As can be seen the first-price sealed-bid and Dutch auction always yield equal
revenue. The second-price sealed-bid and the English auction yield equal revenue under the
private values assumption, but not when values are interdependent:
Model Revenue Ranking
IPV and risk neutral bidders All equal
IPV and risk averse bidders Dutch = 1st Price > 2nd Price = English
Affiliated, privately known values and risk
neutral bidders
Dutch = 1st Price < 2nd Price = English
Affiliated, privately unknown values7 and risk
neutral bidders
Dutch = 1st Price < 2nd Price < English
Table 1: taken from Lucking-Reiley (1999), p.1065
Returning to our original question of finding the auction format that leads to the highest seller
revenue out of the four standard formats, we arrive at the result that when bidders are risk-
neutral the revenue of the English auction is greater (affiliated values) or equal (private
values) to that of the other auction formats. Under risk-aversion the first-price auctions yield
highest seller-revenue.
If the theoretical predictions were trustworthy and the seller were able to know what risk-
attitude the buyers have and how the bidder valuations are distributed, then he could choose
the auction format accordingly. In the next chapter empirical tests of the theoretical
predictions are presented.
5. Experimental Tests of Bidding Behaviour and Auction Revenue
Experimental tests of auction revenue can either be conducted through controlled laboratory
experiments or field studies. There are far more laboratory tests comparing auction-revenue
than field-studies. One disadvantage of using field-studies for revenue comparisons is that the
7 The bidder is uncertain about his valuation only having received a noisy signal about his value.
27
theoretical revenue predictions rely on assumptions about bidder valuations, however it is
difficult (impossible) to control field-data for the type of bidder valuations.
5.1 Field Experiments
There is very little field data comparing auction formats, because real auctions tend to be
conducted according to one pre-determined mechanism.
One example of data making an empirical comparison of two auction formats possible, is the
U.S. Forest Service auction for timber harvesting rights in the Pacific Northwest. Due to a
change in federal law, the U.S. Forest Service conducted some of its auctions by a first-price
sealed-bid auction and the others by an English auction. Mead (1967) and Johnson (1979)
used this data in an empirical study and found that the first-price sealed-bid auction raises
significantly higher revenue than the English auction. However, Hansen (1985, 1986) finds
that after correcting the data for a bias in the selection method for the timber lots, the lower
revenue of the English auction is no longer statistically significant.
The striking part of the results is that timber sales are likely to have strong common value or
at least correlated private value elements, which in theory should lead to higher prices in the
English auction.
Tenorio (1993) studies multi-unit auctions by using data from Zambian currency auctions.
Tenorio finds that multi-unit auctions yield higher revenue when the price is determined by a
discriminatory rule than when it is determined by a uniform-pricing rule.
5.1.1 Field Experiments on the Internet
Lucking-Reiley (1999) conducted auctions of “Magic Cards” on Ebay to empirically test
revenue- equivalence on the Internet. He finds that the Dutch auction leads to thirty percent
higher revenue than the English auction and that the second-price sealed-bid and the English
auction are roughly revenue-equivalent.
5.2 Laboratory Experiments
Most empirical revenue-comparisons are carried out by controlled laboratory experiments.
Bidders are assigned valuations distributed according to the assumptions of the theoretical
28
model tested. Under the assumption of private valuations, the participants in the experiment
are told that their valuation for the good is exactly x monetary units. The experiment tests
whether the participants can “guess” the rational bidding strategy, in the case of independent
private values, whether they realise that they are supposed to bid their valuation in the second-
price auctions and are supposed to shade their bid in the first-price auctions.
Laboratory experiments that test the existence of the winner’s curse in the common value
model, test whether the participants realise that they are not supposed to bid their private
signal, but are supposed to shade their bid to discount for the strategic error of overbidding
due to the winner’s curse.
Laboratory experiments testing the private-values assumption show that bids tend to be higher
in the sealed-bid than in the open auctions:
Coppinger et al (1980) and Cox et al (1982,1983) find that revenue in the first-price sealed-
bid auction is significantly higher than theoretically predicted by the risk neutral Nash
equilibrium strategy (RNNE); revenue in the Dutch auction is approximately equal to or
slightly below the RNNE prediction.
Kagel et al (1987) and Kagel and Levin (1993) find that revenue in the second-price sealed-
bid auction is higher than in the English auction format: bidders bid their valuations in the
English auction but bid above their valuation in the second-price format. Results were tested
with respect to bidder experience, but the breakdown of revenue-equivalence remains.
Experiment Results
Coppinger et al (1980) 1st price > Dutch
Cox et al (1982, 1983) 1st price > Dutch
Kagel et al (1987) 2nd price > English
Kagel and Levin (1993) 2nd price above theoretical predictions Table 2: taken from Lucking-Reiley (1999), p.1066
29
6. Theoretical Predictions and Empirical Results
The theoretical predictions of auction theory strongly rely on assumptions about the
distribution of bidder valuations. Auction theory expects revenue-equivalence in the case of
private values and expects the English auction to yield highest auction revenue in the case of
affiliated values. Contrary to theoretical predictions, experimental laboratory results show that
the sealed-bid format leads to higher revenue than the open auction format. Possible
explanations for this discrepancy may include important aspects being neglected in auction-
models or experiments being carried according to unsuitable methods.
30
PART TWO: Dynamic Price Formation in the Japanese
Auction
Overview
A dynamic bidding model is presented in which bidders are uncertain about their own
valuation. Bidders learn about their private valuation from the exit prices observed. As
a result, the second-price sealed-bid auction produces significantly higher revenue than
the Japanese auction: moreover, bids in the Japanese auction are far more narrowly
spread than in the second-price sealed-bid auction. The model explains this result by
showing that bidders are able to satisfy a tendency to “stick together” in the open
Japanese auction, whereas the secret second-price sealed-bid auction offers no such
opportunity. Furthermore, the model can explain the results of an experimental sale of
real goods.
31
1. Introduction
Sellers want to use the auction mechanism that maximises their expected revenue. In this
section we compare two auction formats: the Japanese8 and the second-price sealed-bid
auction9. Turning to auction theory, the seller has to make an assumption about the bidders’
valuations, whether bidders have purely private, purely common, or interdependent
valuations.
Auction theory predicts that when bidders have private values, a good yields equivalent
expected revenue whether sold by an English or second-price sealed-bid auction. This is a
result of William Vickrey’s fundamental Revenue Equivalence Theorem10. The second-price
sealed-bid auction and the English auction are not only revenue equivalent, but are also
strategically equivalent. The dominant strategy of both auction formats is to bid an amount
equal to one’s private valuation11.
The assumptions underlying the private value model are stringent and maybe unrealistic, as
they impose that bidders value the good independently of the valuations of all other bidders.
In many instances bidders are influenced by the values that their rivals assign to a good.
Imagine for example art, second-hand objects or collector items where bidders are often
subject to reputational concerns. Bidders partly base their valuation on other bidders’ value
judgements, believing the good to be more precious when others value the good highly and
less valuable when others do not care much for the good.
An important instance when bidders do not act according to the predictions of the private
value model can be observed in Internet auctions. Goods - loosely classifiable as private-
value goods - hardly receive bids for days until only some hours or minutes before the
planned auction end, when all of a sudden bidding activity rises incomparably. Late bidding
32
8 The Japanese auction is a sub-variant of the English auction and is also called ascending-clock auction. The English auction is an open, ascending-bid auction. 9 The high-bidder wins, but pays only the second highest bid. 10 It states that all four standard auction formats (first-price, second-price, English and Dutch auction) lead to equally high expected seller-revenue under the assumption of independent private valuations. 11 Furthermore, the dominant strategy is unaffected when bidders have private affiliated values, see Kagel and Roth (1995), p.508.
occurs despite bidders having the possibility to use a proxy-bidding agent12. When bidders
have private values, they are expected to have no incentive to hold back their valuation. A
possible explanation could be that the high-valuing bidder believes that by bidding early and
publicising his high bid, he will cause low valuing bidders to revise their valuation upwards,
raising the price the winner has to pay. Prestige considerations and uncertainty about the true
quality of the good could be causes of this behaviour.
At the other extreme of independent private values lie purely common values: the good
having an unknown but common value to all bidders. Only few goods are pure common
value goods, such as for example oil fields or gold nuggets. Many goods, however, have
some uncertainty surrounding their true quality, making them irreconcilable with both the
purely independent private value model and the purely common value model.
For most goods it is realistic to relax the private values assumption and instead to assume that
values are interdependent13. Interdependent values can be of many a kind, but auction theory
focuses almost exclusively on the Milgrom and Weber model of interdependent values. In
their general model of symmetric interdependent values they assume that the bidders’ private
signals are affiliated14, i.e. positively correlated, and predict that the Japanese auction yields
higher expected revenue than the second-price sealed-bid auction.
Laboratory experiments test theoretical predictions based on the models’ assumptions.
Theoretical predictions concerning auction revenue strongly rely on assumptions about the
distribution of the bidders’ valuations. There are many laboratory experiments testing the
private values predictions15 (both for affiliated and independent private values), experiments
33
12 A bidder can submit his maximum-willingness-to-pay to the proxy-bidding agent, who will then bid on his part, raising the current high-bid by a minimum-increment until he appears as the high-bidder. The proxy-bidding agent will stop bidding once the maximum-willingness to pay is reached. 13 See Part One Chapter 3.3. Interdependent values: Bidders have some uncertainty about their values, their value partly being influenced by private information held by the other bidders. 14Affiliation: Bidders know the value of the item to themselves with certainty, but a higher value of the item for one bidder makes higher values for the other bidders more likely (private values are positively correlated relative to the set of possible valuations)”. Kagel and Roth (1995), p.517. 15 Kagel, Harstad and Levin (1987) test revenue equivalence for affiliated private values. Empirical results show failure of the theoretically predicted strategic-equivalence between the second-price sealed-bid and the Japanese auction.
testing the existence of the winner’s curse for common values, but there is a lack of
experiments comparing revenue for interdependent valuations16.
Laboratory experiments are conducted by assigning a private value or value estimate to every
bidder and observing whether bids and revenue correspond to dominant strategy predictions.
This method has the drawback that viewed critically it is merely a test of a bidder’s cognitive
ability of guessing the dominant bidding strategy. A seller wanting to know which of the two
auction formats yields higher expected revenue, might prefer an experiment that is less
controlled but has a set-up that makes its results more meaningful to the practical
undertaking.
We designed an experiment to test revenue equivalence of the Japanese and second-price
sealed-bid auction in a realistic setting: in the experimental sale of real consumption goods.
The experimental results in Chapter Three show that bidding behaviour differs in the two
auction formats examined, specifically bids in the open auction being far more clustered than
under the sealed-bid format. As a result, the final price in the second-price sealed-bid is
higher than in the Japanese auction. The results suggest that bidders do not solely base their
valuation on their private value estimate, but instead partly base their reservation price on the
other bidders’ valuation of the good. Motivated by the experimental observations, a bidding
model is presented in Chapter Five.
This model differs from Milgrom and Weber’s general model in a number of respects.
Milgrom and Weber assume valuations are exogenous and affiliated. In our boundedly
rational model bidders are uncertain about their own valuation and partly base their own
valuation on other bidders’ private information, i.e. independent signals. Bidders update their
valuation using the information revealed through the exit prices of the other bidders. The
final price is reached in a dynamic process, bidders forming their valuation adaptively.
34
16 An exception being Kirchkamp and Moldovanu (2001), who conduct laboratory experiments for a simple model of interdependent values testing efficiency of the Japanese and second-price sealed-bid auction. Their empirical results with respect to revenue are consistent with the theoretical predictions: they find that seller revenue is equal under both formats and bidder-payoff higher in the Japanese-auction.
The model presented in this paper is a general model applicable to all kinds of goods,
provides predictions on expected price and revenue and insight on the updating-procedure and
price formation.
2. Experiment
There have been many laboratory experiments carried out testing revenue-equivalence
between the Japanese and the second-price sealed-bid auction. The following result is
characteristic of the results obtained in laboratory experiments (e.g. Kagel, Harstad and Levin
(1987)): bids in the second-price sealed-bid auction are above Nash equilibrium predictions;
bids in the Japanese auction correspond to Nash equilibrium predictions.
Risk aversion does not qualify as a possible explanation for this result, because it is a
dominant strategy in a second-price sealed-bid auction to bid one’s private valuation –
independent of the risk-attitude or number of rivals.
Kagel, Harstad and Levin explain overbidding in the second-price sealed-bid auction by a
bidder’s lack of ability to understand that bidding above his valuation does not increase his
chances of winning, except in cases where this is not desirable. They further explain that
overbidding is not found in the Japanese auction, because the open auction format makes it
clear that once the counter reaches a bidder’s private valuation, no gains can be made by
remaining active.
Returning to the original problem of finding the format that yields highest expected seller
revenue, empirical results turn out not to conform to the theoretical predictions. Theory
predicts that the English auction leads to greater or equal revenue than the second-price
sealed-bid auction; experiments show that the second-price sealed-bid auction leads to higher
revenue than the English auction. A seller now faces the decision of either trusting the
theoretical prediction or the empirical result, in which case he would choose the second-price
sealed-bid auction and hope that his bidders will make the mistake of bidding above the
equilibrium bid.
35
Below we present an experiment that differs from standard experiments in that it is designed
to test how revenue and bids in the second-price sealed-bid and Japanese auction compare in
practice, in the sale of real goods:
2.1 Experimental Set-Up
The experiment was conducted with undergraduate business and economics students at the
University of Vienna and Vienna University of Economics and Business Administration in
May and June 2001. Students were not paid for participating in the experiment. Half of the
experiments were conducted with unpaid volunteers, and half of the experiments as an
alternative to the “normal” course program. After the experiment participants were debriefed
on the hypothesis tested and theoretical predictions.
The thirty experiments were divided into four sessions. Seven experiments were conducted
with 5 participants, seven with 7 participants, eight with 12 participants and eight with 17
participants.
In a series of experiments a single good was sold twice, first by means of a second-price
sealed-bid auction and successively by an Japanese auction without announcing the final price
obtained in the second-price sealed-bid auction. The second-price sealed-bid auction was
conducted first because the secret format of the second-price sealed-bid auction discloses no
information to bidders, does not allow them to draw conclusions on the other bidders’
valuations and final price. At the beginning of the Japanese auction, the bidders were thus in
the same informational-state as before the second-price auction.
The Japanese auction was conducted as follows:
The auctioneer calls out the current price starting at zero17, i.e. zero Austrian Schillings18
(ATS), increasing the price by the increment of 1 ATS. The price was raised by the minimum
increment after about one second. A bidder was considered to be an active bidder until he
raised his hand to signal his exit. At any point in time, the number of active bidders and the
36
17 The auction started at a price of zero ATS, so that bidders were not be forced to pay for a good that they did not want. 18 1 Euro is equivalent to 13,76 Austrian Schillings.
exit prices of the bidders who had already quit were publicly known. The auction ended when
the before-last bidder quit the auction, only one more bidder remaining active. The remaining
bidder is the winner of the Japanese auction and in case he turns out to be the overall-winner
of both runs (explained below) has to pay a price equal to the highest exit price, i.e. the exit
price of the second highest bidder19.
Since one and the same good is auctioned off in two auction runs, the good can only be
allocated to either the winner of the second-price or the winner of the Japanese auction. It may
be the case that the same person wins the good in both runs and at the same final price, but in
case this situation does not apply, the winner of the good is determined by throw of the die at
the end of the two runs20. The probability that the second-price or alternatively the Japanese
auction turns out to be the auction relevant for the transfer of the good is thus one half.
This method has the advantage of allowing a good to be sold twice, while holding all
conditions equal (same group of bidders, same good, same informational structure), except the
auction rules. If two units of the same good were sold, valuations would be different from the
one unit case due to automatically decreased demand by the winner of the first round
(amongst others).
The winner receives the good and has to pay the relevant final price21.
37
19 If the last and before-last bidder exit simultaneously, the final price is determined by the bid of the before-last bidder, which in this case constitutes the last exit price. 20 The probabilistic determination of the run, that is relevant for pay was also used by e.g. Kagel and Levin (1986), p.897 (in their case by tossing a coin). 21 As (winning) bidders carried unobservable amounts of money in their wallet, they were not obliged to pay the final price immediately, but had one week of time within which they could pay the final price. Bidders knew this before the auction commenced.
2.2 The Goods
The following goods were sold:
- Toblerone (a mixed version of dark and white chocolate22)
- Turron (a Spanish sweet)
- French wine
- Posters and T-shirts (bought in the USA with US specific design)
- Handmade pottery
The goods sold in the experiment do not fall into the category of purely private value goods,
because there was uncertainty about various aspects of the quality of the good. None of the
auction goods were offered for sale in Austria. The food and drinks sold are experience goods.
Because they were not offered for sale in Austria, there was uncertainty about the quality, in
terms of the taste of the product. Some bidders had perhaps been to Spain and had tasted the
Turron and thus had better knowledge about their preference for the good.
Bidders also had uncertainty about the value of the good in terms of the purchase price. The
goods were unique goods in the eyes of the bidders, because it was not possible to get a
second unit in Austria, but bidders were aware of the fact, that the goods were readily
available in the country of purchase. These are reasons why the goods cannot be viewed as a
purely private value good.
The goods had purchase prices in the range of one to twelve Euro, most of them in the range
of four to six Euro.
22 Pure dark and pure white Toblerone can be bought in Austria, but not the mixed version.
38
3. Results
The results of the experiment:
Final Price Diff. Diff. Higher Participants
2nd Jap (in %) Final Price
1 1 Turron 30 25 -5 -17 2nd Price 17
2 Wine 90 111 21 23 Japanese 17
3 Praline de Café 35 30 -5 -14 2nd Price 17
4 Pottery 10 5 -5 -50 2nd Price 17
5 Book 50 42 -8 -16 2nd Price 17
6 Ameretti 45 33 -12 -27 2nd Price 17
7 Wine 30 23 -7 -23 2nd Price 17
8 Toblerone 20 18 -2 -10 2nd Price 17
2 1 Crema C. 30 25 -5 -17 2nd Price 5
2 Pottery 21 20 -1 -5 2nd Price 5
3 Turron 60 55 -5 -8 2nd Price 5
4 T-Shirt 121 120 -1 -1 2nd Price 5
5 Cap 30 25 -5 -17 2nd Price 5
6 Biscuits 40 39 -1 -3 2nd Price 5
7 Toblerone 25 20 -5 -20 2nd Price 5
3 1 Duck 33 32 -1 -3 2nd Price 12
2 Wine 75 95 20 27 Japanese 12
3 SF Badge 20 19 -1 -5 2nd Price 12
4 Turron 25 25 0 0 Equal 12
5 Ireland Badge 25 10 -15 -60 2nd Price 12
6 Toblerone 25 21 -4 -16 2nd Price 12
7 Ameretti 37 36 -1 -3 2nd Price 12
8 Chocolate 20 24 4 20 Japanese 12
4 1 Turron 34 25 -9 -26 2nd Price 7
2 Wine 48 39 -9 -19 2nd Price 7
3 Key-Chain 28 12 -16 -57 2nd Price 7
4 T-Shirt 50 34 -16 -32 2nd Price 7
5 Toblerone 16 26 10 63 Japanese 7
6 Book 30 30 0 0 Equal 7
7 Poster 25 7 -18 -72 2nd Price 7
Table 3
39
On average the final price of the Japanese auction was 13% below the final price of the
second-price sealed-bid auction. The bids of all bidders can be found in the appendix.
3.1 No Revenue-Equivalence
Thirty experiments were conducted in order to test revenue-equivalence between the second-
price sealed-bid and the Japanese auction. In twenty-four of the thirty experiments the final
price of the second-price auction was higher than that of the Japanese auction; in four
experiments the final price of the Japanese auction was higher and in two experiments the
final prices of the two auctions were equal. On average the final price in the Japanese was
14% below that of the second-price sealed-bid auction.
3.2 Lower Bid-Variance in Japanese Auction
The bids in the Japanese auction were far less dispersed than in the second-price sealed-bid
auction. In two-thirds of all experiments bid-variance was lower in the Japanese than in the
second-price sealed-bid auction23. To test whether there was a significant difference in the bid
variance, F-Tests were conducted. In 63% of the experiments, the Japanese auction had a
significantly lower bid-variance than the Japanese auction (according to the F-Tests).
3.3 Average Bid Not Significantly Different
Comparing the mean bid with that of the second-price auction (for every good sold) the t-
tests show no significant difference (for any of the thirty goods sold). On average the mean
bid in the Japanese auction was 0.6% above that of the second-price sealed-bid auction.
Thus, the mean bid of the Japanese auction was little influenced, especially considering that
the highest bid was not taken into consideration in the statistic, because the Japanese auction
does not reveal the highest bid and thus in order to compare the two formats the highest bid
in the second-price sealed-bid auction was also left out.
23 See Appendix.
40
3.4 Reasons for Lower Bid-Variance in Japanese Auction
Comparing the change in bid-level of every bidder in the two runs shows that originally low-
valuing bidders revised their bids upward, while originally high-valuing bidders revised their
bid downwards in the Japanese auction. Bidders who made a below average bid in the
second-price sealed-bid auction, revised their bid upwards by 62% on average (median:
30,3%) in the succeeding Japanese auction. Bidders who made an above average bid in the
second-price sealed-bid auction revised their bid downward by 30% on average (median:
19%) in the succeeding Japanese auction. As stated in Chapter 3.3 the mean bid for each
good was fairly equal in the two auction runs. The change in variance was caused by a
change in the upper and lower bound of the interval.
3.5. Learning Effects
In previous literature learning effects were observed in experiments, so that bids in the
Japanese auction adjusted to dominant strategy predictions after a few rounds. In the
experiments above, the difference in final prices between the second-price sealed-bid and the
Japanese auction did not change in the course of the auction. This is not surprising because in
standard laboratory experiments, bidders are paid the difference between their valuation and
the final price, thereby receiving feedback on their bidding behaviour. Bidders in this
experiment do not receive feedback on whether they overbid, in fact one cannot tell whether a
bidder overbid (by mistake) or revised his valuation intentionally.
4. Interpretation
Experimental results show that bids are far more clustered together in the Japanese than in the
second-price sealed-bid auction. The secret format of the second-price sealed-bid auction
implies that bidders receive no other information than their private signal. The open format of
the Japanese auction in contrast reveals information about the valuations of the other bidders,
by publicising all exit prices. The lower final price in the Japanese auction and the clustering
of bids could indicate that bidders try to take other bidders’ valuations into account when
41
forming their own valuation of the good, this however only being possible under the open
auction format.
The discussion in Chapter Two and the empirical findings presented above all imply that
bidders have some form of interdependent valuations. Auction theory (almost) exclusively
focuses on the Milgrom-Weber model when treating interdependent valuations. My empirical
results (as well as previous laboratory experiments), however, do not correspond to the
theoretical predictions of the Milgrom-Weber’s model: the Japanese auction leads to lower
revenue than the second-price sealed-bid auction in practice, theory predicting the opposite.
Regarding bid dispersion, my experiment shows that the open Japanese auction leads to an ex-
post affiliation of bids: bidders with a high valuation in the sealed-bid auction lowering their
bid in the open auction and those with a low valuation in the sealed-bid auction raising their
bid in the open auction. Milgrom-Weber in contrast assume that private signals are ex-ante
affiliated leading to an upward clustering of bids in the open English auction.
The surprising experimental results and the fact that they do not conform to the Milgrom-
Weber model calls forth the need for a different model of interdependent valuations. In the
next section a general model of interdependent valuations is presented. Bidders are uncertain
about their valuation and learn from the exit prices of the other bidders, updating their
valuation every time new information (in form of exit prices) is revealed. Bidders have
interdependent values, but independently distributed signals, this perhaps being a less
restrictive assumption than Milgrom-Weber’s assumption of correlated signals.
The bidding strategy modelled is boundedly rational – it is not calculated whether it
represents a Nash-equilibrium. Real bidders are not perfectly rational and do not have the
ability to calculate their expected valuation conditional on the valuations of their rivals. The
updating procedure presented here has the advantage of being empirically motivated and is
close to being rational in that bidders satisfy their target-function by a suitable estimation
method.
A further advantage of the model presented below is that the updating-procedure is not only
implied but also modelled in particular. The procedure and result is thus better 42
comprehensible to the reader. Learning and updating in the Japanese auction is not limited to
the highest two bidders, but instead all active bidders engage in the updating-procedure. The
degree of uncertainty about one’s own valuation specified by lambda can be adapted and thus
applied to a wide variety of goods.
5. The Model
A single seller wishes to sell a single indivisible object to one of n buyers participating in a
Japanese auction.
The Japanese auction: Every bidder knows how many bidders are participating in the auction.
There is a counter counting upwards starting with zero. A bidder is an active participant in
the auction until he exits. A bidder exits the auction once his reservation price24 is reached.
Once a bidder quits the auction, he cannot return. When a bidder quits the auction, all
remaining bidders are informed about the price at which the particular bidder quit the
auction. At every point in time the active bidders know how many bidders are still active and
know the exit prices of all bidders who have already quit. The auction ends when the before-
last bidder quits the auction, which is when there is only one more active bidder remaining.
This last recorded exit price is the auction’s final price.
At the outset of the auction, bidder i receives a signal si about his valuation of the good. The
private signal is not the sole factor determining bidder i’s reservation price. It is also
dependent on the other bidders’ valuation of the good. Bidders do not enter the auction with
perfect knowledge about their own valuation, but learn from the valuations of the other
bidders. This can be due to prestige reasons; bidders do not want to possess a good if no one
else thinks it is valuable, on the other hand, a good becomes more valuable to a bidder when
the other bidders value the good highly.
43
24 The reservation price is the maximum willingness to pay. It is the price at which a bidder is indifferent between purchasing the good and not purchasing it.
A second reason why bidders decide to incorporate the private valuation of each of the other
bidders into their own reservation price is because bidders believe that they can extract
information about the quality of the good from the valuation of the other bidders. Imagine for
example a classic common value good such as an oil field that has not been drilled yet. Every
bidder will use all data material and research reports accessible to him in order to estimate
the field’s value. A bidder knows that a multitude of research reports exist, but he himself
does not possess knowledge of all of them. Knowing that his own information is limited, a
bidder does not let his reservation price equal his private value estimate, but instead only
denotes a certain weight to his private value-estimate. His total valuation (the reservation
price) is composed of the weighted sum of his private value-estimate and a measure for the
other bidders’ private value estimates. The model is thus applicable both for goods for which
bidders have reputational concerns, as well as for classic common value goods.
5.1 The First Round
n... number of bidders at the beginning of the auction
k… number of bidders who have already quit the auction
λ... measure of how important the common value is to a bidder, 0 ≤ λ ≤ 1, every bidder has
the same λ
si... private signal of bidder i, the si are uniformly distributed ∼ [0,1]
ek... kth observed exit price
pi … reservation price (valuation) of bidder i
C... common value
At the outset of the auction every bidder draws a private signal from a uniform distribution
U∼[0,1]. Every bidder knows his private signal, but not the private signals of the other
bidders. Every bidder knows that the signals si are uniformly distributed on the interval [0,b],
but does not know the upper bound.
A bidder’s reservation price in round k is the weighted sum of the private signal si and the
common value of this round. iki sCkp *)1(*)1( λλ −+=+ Every bidder knows how the
reservation price is formed and knows in which way the valuation of the other bidders
44
changes from round to round. The common value is weighted withλ , λ taking on values
between zero and one. λ denotes how important the valuation of the other bidders is to
bidder i, that is how relevant bidder i believes the information included in the valuation of the
others to be. Every bidder has the same λ .
iCn
20+s
=
Bidders update their valuation in every round, the common value denoting the estimate of the
mean private signal of all other bidders. Because bidders cannot observe the private signals
of the other bidders, they estimate the mean private-signal from the exit prices observed (the
private signals of the bidders having already quit the auction can be inferred from the exit
prices observed).
Bidders estimate the average private signal by the Maximum-Likelihood method, using the
number of remaining active bidders and the exit prices observed so far for the estimation.
Bidders want to estimate the mean private signal of all other bidders, ∑≠=−
=n
jj
ij
s11
1 .
Bidders form their valuation adaptively by following a complicated learning rule; complicated
because bidders have to maximise. The method is explained in the section below.
In the first round no exit price has yet been publicised. Every bidder knows that signals are
uniformly distributed on the interval [0,b] and he knows his private signal, which was one
sample draw from the distribution. In order to estimate the interval mean, a bidder needs to
estimate the interval’s upper bound. He can do this using the maximum likelihood method, in
which case the best estimate of the upper bound is the bidder’s private signal si. It follows
that the best estimate of bidder i for the common value is 22
ˆˆ iii
sabC =+
= .
5.1.1 The First Exit
The first bidder who exits the auction is the bidder with the lowest private signal. The
expected lowest signal drawn from the signals uniformly distributed on [0,1] is: 1
1+n
45
When X1, X2, …Xn is a sample of n independent random variables, each having the same
probability distribution function F(x) with density f(x) = F’(x) and the realizations of these
random variables are ranked in increasing order25: nk vvvv xxxx ≤≤≤≤≤ ......
21, the kth order
statistic is the function X(k) that assigns to each realization of the series (X1,…Xn) the kth
smallest value xvk. Altogether, there are n order statistics X(1), X(2) , …, X(n). Each order
statistic is a random variable. If X is uniformly distributed on the domain [0,1],
[ ]1)( +
=n
kXE k
5.1.2 Common Value Estimation and the Updating Procedure
At the outset of the auction, a bidder’s best estimate of C is half his private signal. Once the
first bidder exits the auction, bidders use this information to estimate the most likely mean of
the uniformly distributed private signals by maximising the following likelihood function:
Prob (lowest private signal = s1) =
− −
1))(1)(( 1
11
nsFsf n
The maximum likelihood method is used to estimate the unknown parameters of a population
of which an observable sample was drawn. The maximum likelihood method is used in this
model because we are confronted with a unique auction good and a unique set of bidders,
presenting small data from which to make inferences (see Chapter 5.3 for further
justifications).
The likelihood function above symbolises the probability that the first bidder has a private
signal equal to s1, times the probability that n-1 bidders have a private-signal greater than s1,
times a combinatorial constant. Realisations of private-signals are ranked, s1 denoting the first
private signal, s2 the second, s1< s2<…<sn.
The combinatorial constant reflects the fact that there are a number of possible combinations
when choosing one bidder out of a group of n bidders, having a private-signal below or equal
to s1.
The density function of a uniform distribution [a, b]: ab
xf−
=1)(
The probability function of a uniform distribution [a, b]:abaxxF
−−
=)(
25Section taken from Wolfstetter (1999), p.344.
46
Assuming that all bidders use the maximum likelihood method to calculate their valuation in
the first round, bidders can infer from . 1s 1e
In the first round the likelihood function (given the first private signal is s1 and (n-1) active
bidders remain) is:
1
maxb
L:= Prob (1st private signal = s1, n-1 bidders still active) =
−−
−−
−
111
1
1
1
1
nabas
ab
n
The b that maximises the above expression is found to be nsb *11 = , in the first round, where
a is equal to zero.
The expected first exit, ][)1(][][ 1)1(1 sECEeE λλ −+=
][)1(2 11 sEsE λλ −+
=
1
1)1()1(2
1+
−++
=nn
λλ
1
maxb
L:= Prob (1st private-signal = s1, n-1 bidders still active) =
+
−−
1)1(111
1
11
nnbb
n
And, 11
11 +
=+
=n
nn
nb
b1 constitutes the upper bound of the distribution of private signals estimated in round one.
The lower bound of the private-signal distribution (a) is zero by definition.
Bidders use b1 to calculate the common value of the next round. Bidder i, who still is active,
defines the common value as the mean private signal of all other bidders. It is calculated by
taking the average of all inferred private signals of the bidders who have quit the auction
already, k, and the interval mean of the signal distribution of the remaining active bidders
times the number of active bidders excluding himself, n-k-1.
( )
+
−−+−
=2
11
1 1112
sbknsn
C
The second exit price is: 222 )1( sCe λλ −+=
47
5.2 The General Procedure
A bidder carries out the following three steps in every round, estimating bk (the upper
bound), calculating Ck (the common value) and his reservation price:
+
−−+++−
=2
)1(...1
11
kkkk
sbknss
nC
iki sCkp )1()1( λλ −+=+
The bidder with the lowest valuation in this round will be the next bidder to exit the auction.
All remaining active bidders infer the private signal of the exit price observed in this round
by: λλ−−
=1
kkk
Ces .
Given the updating-procedure described above, bidders treat the likelihood function as if it
were a constant. Ck being a constant, pi(k) is an affine transformation of the uniformly
distributed si. The pi(k) therefore are also uniformly distributed.
The current exit price is the lowest reservation price of all active bidders in this round. This
new exit price is publicised and again bidders calculate their new reservation price by
maximising the likelihood function:
bmax L:=
− −
kn
sFsfsfsf knkk ))(1)(()...()( 21
whereby,
k...Number of bidders who have already quit the auction
n...Number of bidders who were participating at the beginning of the auction
Strictly speaking, the probability that a bidder exits in the price-interval [p, p+∆] is,
∫∆+
∆≈=−∆+p
p
pfdxxfpFpF )()()()(
This approximation is correct only for small values of ∆.
48
The ∆s are constant and do not influence the Maximum-Likelihood Estimation. Every time a
bidder exits the auction, all remaining active bidders calculate the common value a new with
the updated information, to form their new reservation price. This iteration is carried out until
there is only one more active bidder left. Once n-k is equal to one, the auction ends.
5.3 Estimation Procedure
The estimation procedure used above is the maximum likelihood method.
Ideally an estimator should be unbiased and consistent. An estimator is unbiased when it does
not consistently under- or overestimate the true value. In this case, the maximum-likelihood
estimator slightly underestimates the true value. The estimate of b, kk skn
=b̂ .
][]ˆ[ kk sEknbE = , whereby
1][
+=
nksE k , so that
1]ˆ[
+=
nnbE k
An estimator is consistent when it converges more closely to the true value as the sample size
increases. The maximum likelihood estimator fulfils the desirable property of being
consistent: as the sample size (here: the number of bidders) increases, the estimator converges
to its true value: kk sknb =ˆ and
1]ˆ[
+=
nnbE k
If , ∞→n 1]ˆ[ == bbE k
5.4 The Expected Final Price
The expected final price is equal to the expected exit price of the (n-1)th bidder and is
determined by first calculating the expected common value after k bidders, that is n-2 bidders
have quit the auction:
49
[ ]
)1(2
)1)(1(2)1(
)1)(1(2
)1)(1(22224
)1)(1(222
)1)(1(1
)1)(1(2)1(
)1)(1(222
)1)(1(
211
2
)2(12...
12
11
11
2
2
2
1
2
+=
+−−
=
−+−
=
−+−++−+
=
−+−
+−+
−+−
−++
=
−+−
+−+
=
+
++−
−++−
+++
++−
=
∑−
=
−
nn
nnnn
nnnn
nnnnnn
nnn
nnnn
nnnn
nnn
nn
i
nn
nn
nnn
nnnCE
n
i
n
The expected exit price of the (n-1)th bidder: 11)1(
)1(2][ 1 +
−−+
+
=− nn
nneE n λλ
5.5 Results of the Model
The expected final prices calculated for the ten-bidder model:
Expected Final Prices
00,10,20,30,40,50,60,70,80,9
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Fina
l Pric
e
λ
Figure 1: λ=0 represents the purely private value case, λ=1 represents the purely common value case
The final price of the Japanese auction (which is also the seller-revenue) is determined by the
last recorded exit price, which corresponds to the exit price of the second highest bidder:
50
11)1(
)1(2][ 1 +
−−+
+
=− nn
nneE n λλ . As can be seen from the diagram above, the auction’s
expected final price is largest when 0=λ . In this case the expected final price is equal to
)1(2 +nn . The expected final price decreases, as λ gets larger. This is because the common
value, C, is a constant (as are the estimates of C) equal to )1(2 +n
n and 11
)1(2 +−
<+ n
nnn for
. The final price determined by the combination of the second highest private signal
and the estimated common value,
2≥n
12 )1()1 −−( −+=− nn si np C λλ , results in smaller final
prices as λ gets larger. When a common value element is taken into account in the
reservation price, the final price is always lower than in the private value case.
In order to compare the results of the model with the final prices observed in the second-price
sealed-bid and Japanese auction to the experimental results (Chapter 3), we need to calculate
bidding behaviour in the second-price sealed-bid auction. The information-structure of the
second-price sealed-bid auction corresponds to the first round of the Japanese auction, when
no exit price has yet been publicised. A bidder only knows that signals are uniformly
distributed on [0,b] and knows his private signal. In order to estimate the common value, he
tries to infer the interval mean, by using the maximum likelihood method to estimate the
upper bound of the signals. As signals are uniformly distributed, a bidder takes his private
signal to be the best estimate of the upper bound of the signal distribution. Bidder i’s best
estimate of the interval mean, knowing that the lower bound is zero, is thus 2
is. By using the
maximum likelihood method for estimating the interval mean, bidders avoid the winner’s
curse, because the maximum likelihood method tends to underestimate the common value
when the sample is small (one in this case).
We conclude that the second-price sealed-bid auction results in an expected final price
greater than that of the Japanese auction for λ > 0 and a final price equal to that of the
Japanese auction forλ = 0. This result is in accordance with the experimental results. The
second-price sealed-bid auction does not offer the possibility to learn about the valuations of
the others, bidders can only use their private signal for estimating the common value. In the
Japanese auction bidders can learn about the valuation of the others. Being better informed
51
about the other bidders valuations, bidders are able to form a reservation price that reflects
their preferences more accurately.
Another interesting feature of the Japanese auction model is the fact that bidders with low
private signals revise their bids upwards and bidders with high private signals revise their
bids downwards (for λ > 0). This result also corresponds to the experimental observations.
The chart below shows the standard deviation of the expected exit prices in the ten-bidder
model:
Bid Dispersion
00,050,1
0,15
0,20,250,3
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Stan
dard
Dev
iatio
n
λ
Figure 2
The standard deviation of the exit prices is largest for λ = 0. The diagram above shows that
bids are more narrowly dispersed when the common value element is of greater importance.
The results of the model correspond very well with the results from the experimental sale of
real consumption goods described above in Chapter Three. (The mean λ for the goods sold
in the experimental auction was 0.28; the median λ was 0.27).
The model thus provides an explanation for both the lower final price in the Japanese in
comparison to that of the second-price sealed-bid auction and for the observation that bids
were more narrowly distributed, closer “stuck together” in the Japanese than in the second-
price sealed-bid auction.
λ , the factor of importance of the other bidders’ valuations, can be varied, being able to
account for all cases ranging from purely private (λ=0) to purely common valuations (λ=1).
52
One advantage of this model is its wide-reaching application to all kinds of goods.
Furthermore, the adverse selection bias encountered in the classic common value model does
not appear in this model. The maximum-likelihood method allows bidders to estimate the
correct common value directly without having to disclose information on the estimation
error, upper- and lower bound (as in the classic common value model). The approach
presented here provides readers and bidders with a clear and understandable bidding model
that explains how bidders use information publicised in the bidding course to update their
valuation due to prestige concerns or in order to update their information about the true value
of the good.
6. Conclusion
Revenue equivalence between the second-price sealed-bid and Japanese auction is tested in
the experimental sale of real consumption goods. A series of experiments shows that the final
price of the second-price sealed-bid auction lies significantly above that of the Japanese
auction. Furthermore, bids were much more narrowly dispersed in the Japanese than in the
second-price sealed-bid auction.
The empirical observations motivated a general bidding model applicable to the purely
private value, the purely common value and all intermediate cases. Bidders face uncertainty
about their own valuation and include the private value estimate of all other bidders in their
own valuation of the good. It turns out that the Japanese auction offers bidders the
opportunity to learn about the valuation of other bidders from the publicly observable exit
prices. Bidders update their reservation price every time someone quits the auction, the final
price thus being determined iteratively.
The model reaches the same two striking results attained in the experimental sale of real
goods: First, that the expected final price of the Japanese auction is lower than that of the
second-price sealed-bid auction and second, that bids are more “stuck together”, more
narrowly dispersed, in the Japanese than in the second-price sealed-bid auction. The model
can thus show and explain the real observations.
53
Finally, the model could be extended to allow some bidders to have high and others low λ
values. In this case lowλ bidders will revise their bids more strongly than the high λ
bidders.
54
7. Appendix
7.1.Bids in session one (17 bidders):
1 2 3 4 5 6 7 8
Turron Wine Praliné Pottery Book Ameretti Wine Toblerone
2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l
1 0 0 0 0 0 0 1 0 0 0 2 1 4 0 1 1 0 0
2 0 0 0 10 13 0 1 0 0 0 7 0 5 1 5 1 0 0
3 1 2 0 25 30 0 2 0 0 0 8 0 5 1 6 0 4 0
4 3 3 0 28 45 1 3 0 0 0 8 1 6 0 9 1 5 0
5 3 5 0 40 85 3 3 1 0 0 15 0 9 0 10 0 5 0
6 5 5 0 50 51 0 5 1 1 0 15 1 10 1 10 1 7 0
7 6 7 0 50 101 10 8 0 1 0 16 2 12 0 11 0 9 0
8 6 10 1 50 80 6 9 2 4 -7 19 1 14 0 11 0 9 0
9 10 10 0 50 102 10 11 2 4 0 20 2 15 1 15 1 10 1
10 10 9 -1 60 55 1 14 -9 5 3 21 1 25 0 17 0 10 2
11 11 11 0 70 95 -2 15 -5 5 0 27 2 25 0 20 0 15 1
12 15 10 2 70 100 -2 15 -1 5 1 28 0 25 0 20 0 15 0
13 15 14 0 80 80 0 16 -4 6 2 31 0 30 0 22 0 15 0
14 19 15 1 85 103 -1 16 -1 10 5 1 35 0 30 0 30 1 15 0
15 30 25 0 90 103 -0 20 -1 10 5 1 36 0 40 1 30 1 20 0
16 35 25 0 101 111 -0 35 30 0 11 1 50 42 0 45 33 1 30 18 1 20 23 0
17 30 win 81 win 48 win 11 win 55 win 50 win 35 win 27 win
Table 4: Individual bids were not recorded in six out of the Japanese auctions.
7.2 Bids in session two (5 bidders):
1 2 3 4 5 6 7
Crema Cat. Pottery Turron T-Shirt Cap Biscuits Toblerone
2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l
1 20 21 0,1 5 21 0,8 20 26 0,3 1 1 0,0 1 10 0,4 20 20 0,0 2 11 0,5
2 20 15 -0,4 15 21 0,5 25 19 -0,3 10 1 -0,2 15 15 0,0 22 20 -0,2 12 10 -0,3
3 30 25 -2,5 20 21 0,2 25 26 0,1 50 56 0,4 25 25 0,0 25 22 -0,4 15 15 0,0
4 30 20 -5,0 20 20 0,0 60 55 0,3 121 120 0,0 30 16 2,5 40 39 0,1 25 20 0,8
5 60 win 70 win 90 win 150 win 51 win 56 win 40 win
Table 5
55
7.3 Bids in session three (13 bidders):
1 2 3 4 5 6 7 8
Duck Wine SF Badge Turron Ireland Badge Toblerone Ameretti Chocolate
2 nd Jap l l 2 nd Jap 2 nd Jap l l l 2nd Jap 2nd Jap 2nd Jap l 2 nd Jap l l nd2 Jap
1 0 0 0,0 0 0 0,0 0 1 0,1 5 7 0,2 1 2 0,1 0 3 0,2 0 0 0,0 0 1 0,1
2 5 19 0,9 25 25 0,0 1 1 0,0 10 10 0,0 3 3 0,0 5 17 1,1 5 5 0,0 2 2 0,0
3 5 5 0,0 26 36 0,7 3 1 -0,5 10 10 0,0 3 3 0,0 11 13 0,4 17 17 0,0 3 1 -0,2
4 12 17 0,6 30 37 0,6 3 7 1,0 11 17 1,1 4 5 0,2 12 9 -0,7 18 6 -4,1 10 10 0,0
5 13 13 0,0 30 41 1,0 4 2 -0,7 13 13 0,0 5 5 0,0 12 14 0,5 19 12 -3,7 10 10 0,0
6 22 26 -2,7 35 50 2,5 4 4 0,0 15 7 -6,5 5 6 0,2 15 16 0,9 20 25 5,5 11 13 6,5
7 24 22 0,6 45 45 0,0 5 5 0,0 15 17 1,6 5 5 20 11 2,3 23 0,0 25 -1,0 12 12 0,0
8 28 26 0,3 45 60 -3,8 7 6 6,0 15 15 0,0 10 8 2,4 20 18 0,5 24 33 -2,9 12 15 -4,3
9 30 25 0,5 50 50 0,0 7 7 0,0 16 16 0,0 12 9 1,1 20 17 0,8 28 30 -0,3 12 11 1,4
10 30 20 1,1 57 71 -0,9 10 10 0,0 21 21 0,0 12 9 1,1 21 21 0,0 30 36 -0,7 13 14 -0,6
11 30 32 -0,2 74 74 0,0 18 19 -0,1 22 22 0,0 25 10 0,9 21 20 0,2 30 25 0,6 20 15 0,6
12 35 16 1,3 75 95 -0,6 20 3 1,3 25 24 0,1 25 9 1,0 25 9 1,8 37 36 0,1 20 18 0,2
13 33 win 85 win 25 win 33 win 35 win 28 win 43 win 22 win
Table 6
7.4 Bids in session four (7 bidders):
1 2 3 4 5 6 7
Turron Wine Key Chain T-Shirt Toblerone Book Poster
2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l 2nd Jap l
1 9 7 -0,2 10 30 0,8 5 5 0,0 12 17 0,3 7 11 1,0 5 7 0,2 2 4 0,2
2 12 12 0,0 20 20 0,0 5 9 0,6 25 28 0,5 9 9 0,0 10 10 0,0 10 9 -0,5
3 15 10 -0,9 20 20 0,0 5 11 0,9 30 29 -0,7 9 13 2,0 13 28 4,3 12 13 5,0
4 15 21 1,1 30 25 -1,0 10 10 0,0 40 34 0,7 12 16 -4,0 17 16 2,0 12 13 5,0
5 25 25 0,0 36 36 0,0 15 11 1,1 50 33 0,9 13 13 0,0 24 28 -0,5 25 7 1,4
6 35 12 1,6 80 39 0,9 28 12 1,0 59 32 1,0 16 26 -2,0 30 30 0,0 25 9 1,3
7 34 win 48 win 38 win 30 win 30 win 38 win 44 win
Table 7
56
7.5 F-tests and T-tests:
critical critical
Bidders F-tests p-value f-value sign t-statistic p-value t-value l
1 1 1 Turron 17 1,8 0,1 2,4 n 0,4 0,4 1,7 0,2
2 2 Wine 17 0,7 0,2 0,4 y -1,6 0,1 1,7 1,6
3 3 Praline de Cafe 17 no data no data no data no data no data no data no data
4 4 Pottery 17 no data no data no data no data no data no data no data
5 5 Book 17 no data no data no data no data no data no data no data
6 6 Ameretti 17 no data no data no data no data no data no data no data
7 7 Wine 17 no data no data no data no data no data no data no data
8 8 Toblerone 17 no data no data no data no data no data no data no data
1 2 1 Crema C. 5 2,0 0,3 9,3 n 1,3 0,1 2,0 -2,0
2 2 Pottery 5 200,0 0,0 9,3 y -1,6 0,1 2,4 0,4
3 3 Turron Almond 5 200,0 0,0 9,3 y 0,1 0,5 1,9 0,1
4 4 T-Shirt 5 0,9 0,5 0,1 y 0,0 0,5 1,9 0,1
5 5 Cap 5 4,2 0,1 9,3 n 0,2 0,4 2,1 0,7
6 6 Biscuits 5 1,0 0,5 0,1 y 0,4 1,9 0,8 -0,1
7 7 Toblerone 5 4,3 0,1 9,3 n -0,1 0,5 2,1 0,3
1 3 1 Turron 7 2,0 0,2 5,1 n 0,8 0,2 1,8 0,3
2 2 Wine 7 9,5 0,0 5,1 y 0,4 0,3 1,9 0,1
3 3 Key-Chain 7 13,2 0,0 5,1 y 0,4 0,3 1,9 0,6
4 4 T-Shirt 7 7,6 0,0 5,1 y 1,0 0,2 1,9 0,4
5 5 Toblerone 7 0,3 0,1 0,2 y -1,4 0,1 1,9 -0,5
6 6 Book 7 0,8 0,4 0,2 y -0,6 0,3 1,8 1,0
7 7 Poster 7 6,7 0,0 5,1 y 1,3 0,1 1,9 2,1
1 4 1 Duck 13 1,7 0,2 2,8 n 0,2 0,4 1,7 0,2
2 2 Wine 13 0,8 0,3 0,4 y -0,8 0,2 1,7 0,0
3 3 SF Badge 13 1,5 0,3 2,8 n 0,6 0,3 1,7 0,6
4 4 Turron 13 1,0 0,5 0,4 y 0,0 0,5 1,7 -0,3
5 5 Ireland Badge 13 8,8 0,0 2,8 y 1,2 0,1 1,8 0,6
6 6 Toblerone 13 2,0 0,1 2,8 n 0,4 0,3 1,7 0,7
7 7 Ameretti 13 0,7 0,3 0,4 y 0,0 0,5 1,7 -0,5
8 8 Baci 13 1,2 0,4 2,8 n 0,1 0,5 1,7 0,3
Table 8: In the first session there is no data for 3-8, because in these experiments erroneously only the final
prices, but not the individual bids in the Japanese auctions, were recorded.
57
PART THREE: Experimental Test of Revenue Equivalence
Overview
In this chapter further experimental tests of revenue equivalence are conducted. Revenue
equivalence of the second-price sealed-bid and the Japanese auction is tested when the price
attained in the sealed-bid auction is publicised before conducting the Japanese auction.
Revenue equivalence of the first-price sealed-bid, second-price sealed-bid and the Dutch
auction is tested. And revenue-equivalence between the second-price sealed-bid and the
English-outcry auction is tested.
58
1. Motivation
As explained above in Part One, theoretical predictions and experimental laboratory results
differ significantly with respect to revenue predictions for the standard auction formats.
Whereas theory predicts that both independent and affiliated private values26 and
interdependent valuations with independently distributed private signals lead to revenue
equivalence, laboratory experiments show that sealed-bid auctions lead to significantly higher
revenue than their strategically equivalent counterpart27. It seems that important elements of
auctions are ignored in the theoretical literature.
Due to the Internet, auctions have become a widely used selling mechanism. Sellers want to
achieve the highest possible price and can choose auction rules accordingly. Due to the non-
conforming theoretical and laboratory results, experiments are conducted to test revenue
equivalence in the sale of real goods.
Theoretical revenue predictions comparing all four standard auction formats28 to one another
differ with respect to the assumptions about bidder valuations. The goods sold in the
experimental auctions were goods with uncertain value and uncertain quality (see Part One
Chapter 3.3).
2. Experimental Set-Up
The experiments were conducted (as explained in Part Two) with undergraduate business and
economics students at the University of Vienna and Vienna University of Economics and
Business Administration in May and June 2001. Students were not paid for participation in
the experiment. Half of the experiments were conducted with unpaid volunteers, and half of
the experiments were conducted as an alternative to the “normal” course program. After the
experiment, participants were debriefed on the hypothesis tested and theoretical predictions.
26 Valuations are private and satisfy the criterion of strict positive affiliation (Milgrom and Weber, 1982). 27 Two auction formats are strategically equivalent, when the expected seller revenue is equal and an identical bidder would choose the same strategy in both auction forms. The first-price sealed-bid auction is strategically equivalent to the Dutch auction and the second-price sealed-bid auction is strategically equivalent to the Japanese auction under the (independent) private value assumption. 28 First-price sealed-bid, second-price sealed-bid, Dutch, and Japanese auction .
59
3. Revenue Equivalence Between the Second-Price and the Japanese
Auction
In a series of thirty experiments a single good was sold twice, first by means of a second-price
sealed-bid auction and successively by an Japanese auction without announcing the final price
obtained in the second-price sealed-bid auction. The second-price sealed-bid auction was
conducted first because the secret format of the second-price sealed-bid auction discloses no
information to bidders, does not allow them to draw conclusions on the other bidders’
valuations and the final price. At the beginning of the Japanese auction, the bidders were thus
in the same informational state as before the second-price auction.
3.1 Breakdown of Revenue Equivalence
Thirty experiments were conducted in order to test revenue-equivalence between the second-
price sealed-bid and the Japanese auction. In twenty-five of the thirty experiments the final
price of the second-price auction was higher than that of the Japanese auction, in three
experiments the final price of the second-price sealed-bid auction was higher and in two
experiments the final prices of the two auctions were equal. On average the final price in the
Japanese was 13% below that of the second-price sealed-bid auction.
3.2 Testing the Effect of Revealing Public Information
In the following experiment the results of Part One Chapter Two regarding revenue-
equivalence of the second-price sealed-bid versus the Japanese auction are tested when
revealing the final price attained in the second-price sealed-bid auction before carrying out the
Japanese auction.
Hypothesis 1: Revealing Public Information Influences Bids
Does it make a difference whether the final price of the second-price sealed-bid auction is
revealed before carrying out the Japanese auction? Fourteen experiments were carried out
with a group of twelve participants, seven revealing the final price attained in the second-
price sealed-bid auction and seven without revealing this information. 60
Final Price 2nd Price
Revealed or Not 2nd Jap Diff Diff in %
1 Ireland Revealed 33 31 -2 -6,1
2 Toblerone Revealed 28 27 -1 -3,6
3 Ameretti Revealed 20 21 1 5,0
4 Chocolate Revealed 35 37 2 5,7
5 T-Shirt Revealed 50 51 1 2,0
6 Biscuits Revealed 35 36 1 2,9
7 Pottery Revealed 25 24 -1 -4,0
1 Teddy Not Revealed 25 10 -15 -60,0
2 Wine Not Revealed 73 90 17 23,3
3 SF Badge Not Revealed 18 18 0 0,0
4 Turron Not Revealed 26 22 -4 -15,4
5 Australia Not Revealed 34 33 -1 -2,9
6 Crema Cat. Not Revealed 29 28 -1 -3,4
7 Poster Not Revealed 55 52 -3 -5,5
Table 9
3.2.1 Results
When the final price of the second-price sealed-bid auction was revealed before carrying out
the Japanese auction, the final price of the Japanese auction was roughly equal to that of the
second-price sealed-bid auction. The final price of the Japanese auction was on average 0.3%
above that of the second-price sealed-bid auction.
In the experiments where the final price of the second-price sealed-bid auction was not
revealed, the final price of the Japanese auction was on average 9 % below that of the
second-price sealed-bid auction.
We can conclude that revealing the second-highest bid of the second-price sealed-bid auction
leads to a much higher final price in the Japanese auction than not doing so. In fact, revealing
the final price of the second-price sealed-bid auction leads to approximate revenue-
equivalence between the Japanese and the second-price sealed-bid auction.
This result coincides with Milgrom and Weber (1982), revealing public information increases
seller revenue. 61
Data: Testing Effect of Revealing Information
1 2 3 4 5 6 7
Ireland Toblerone Ameretti Chocolate T-Shirt Biscuits Pottery
2nd Jap 2nd Jap 2nd Jap 2nd Jap 2nd Jap 2nd Jap 2nd Jap
1 0 0 5 7 0 1 0 0 0 0 0 5 0 0
2 5 19 10 10 1 1 5 5 0 5 1 7 0 0
3 5 5 10 10 3 1 17 17 0 7 10 10 0 1
4 12 17 11 17 3 7 18 6 5 10 14 14 1 1
5 13 13 13 13 4 2 19 12 5 22 15 17 2 2
6 22 26 15 7 4 4 20 25 10 25 15 16 7 7
7 24 22 15 17 5 5 23 25 15 30 16 16 8 7
8 30 25 15 15 7 1 24 win 20 32 17 15 9 11
9 30 20 21 21 7 7 28 33 25 28 18 19 15 14
10 30 31 22 22 10 10 30 30 30 40 21 25 22 19
11 33 win 25 25 18 21 30 32 50 42 29 26 23 20
12 33 16 28 27 20 win 35 37 50 win 35 36 25 24
13 35 27 33 win 25 3 43 31 55 51 37 win 27 win
Table 10: Results when the final price of the second-price auction was revealed
1 2 3 4 5 6 7
Teddy Wine SF Badge Turron Australia Crema Cat Poster
2nd Jap 2nd Jap 2nd Jap 2nd Jap 2nd Jap 2nd Jap 2nd Jap
1 1 2 0 15 2 3 0 3 0 0 0 0 8 12
2 3 3 19 19 3 3 5 17 1 1 0 0 10 13
3 3 3 20 20 3 4 11 13 2 3 5 5 10 10
4 4 1 34 35 5 5 12 9 3 1 11 12 13 15
5 5 5 38 38 7 5 12 14 3 9 15 13 14 15
6 5 1 40 35 8 10 20 11 4 8 15 12 15 16
7 5 5 48 37 9 12 20 18 5 16 19 17 18 19
8 10 5 50 52 10 15 20 17 7 12 22 17 24 25
9 12 5 50 53 15 15 21 21 14 19 23 24 25 25
10 12 6 55 65 15 17 21 20 17 24 24 24 35 32
11 25 10 70 75 15 18 25 9 34 28 28 28 45 43
12 25 win 73 90 18 18 26 win 38 33 29 27 55 52
13 35 3 100 win 26 win 28 22 22 win 36 win 62 win
Table 11: Experiments where the final price of the second-price sealed-bid auction was not revealed
62
4. English Outcry versus Second-Price Sealed-Bid Auction
Most experimental tests of revenue equivalence between the second-price sealed-bid auction
and the English auction model the English auction as an ascending-clock (i.e. Japanese)
auction. In the experiment described below the second-price sealed-bid auction was
compared to the English outcry auction, carried out as follows: bids could be made, whereby
a subsequent bid always had to be at least one currency-unit29 higher than the current bid.
When no more bids are made the auctioneer counts three hammer-hits. If no bid is made
within these three hammer-hits, the auction ends.
Hypothesis 2: The English outcry auction is revenue equivalent to the second-price sealed-
bid auction
Bidders 2nd Price English Outcry Difference Difference
(in %)
Wine 25 100 100 0 0
Toblerone 25 37 41 4 11
Pasta Nero 25 45 56 11 24
Wine 25 105 180 75 71
Wine 19 95 70 -25 -26
Turron 19 45 80 35 78
Toblerone 19 20 21 1 5
Pasta Nero 19 25 31 6 24
Table 12
4.1 Results
Out of the eight auctions conducted, the second-price sealed-bid auction led to a higher price
in six experiments, the English-outcry auction led to a higher price in one experiment and the
two auction formats were tied once. On average the English-outcry auction led to a 23.4 %
higher final price than the second-price sealed-bid auction.
29 1 Austrian Schilling (i.e. about € 0.73) higher than the previous bid
63
It can be concluded that the English auction does not produce the same revenue, when
conducted as an English-outcry and when conducted as a Japanese auction. Whereas the
Japanese auction yields lower seller revenue than the second-price sealed-bid auction, the
English outcry auction yields higher seller revenue than the second-price sealed-bid auction.
Bid dispersion cannot be analysed in this experiment, because the format of the English
outcry auction does not release information on the exit prices of any of the non-winning
bidders.
5. Revenue Equivalence: First-Price Sealed-Bid and Dutch Auction
Experimental tests of revenue equivalence generally compare a subset of the four standard
auction formats to one another. Most often, the two pairs of strategically equivalent auctions
(the English and second-price sealed-bid; or the Dutch and first-price sealed-bid auction) are
compared to one another in the private values setting. In this section the Dutch and first-price
sealed-bid auction are compared to one another.
Theory predicts that the Dutch and the first-price sealed-bid auction are revenue equivalent in
case of independent (or affiliated) private valuations.
Laboratory experiments indicate that there is no revenue equivalence between the Dutch and
first-price sealed-bid auction. Instead, for example Coppinger et al (1980) and Cox et al
(1982) find that prices are significantly higher in the first-price sealed-bid than in the Dutch
auction.
64
5.1 Results
1st Price Dutch Difference Difference
(in %)
Pottery 45 42 3 7
Poster 153 120 33 22
Toblerone 25 21 4 16
Turron 37 35 2 5
Wine 73 68 5 7
Ameretti 32 29 3 9
Crema Cat 31 33 -2 -6
Table 13
Results show no revenue equivalence between the Dutch and first-price sealed-bid auction.
The Dutch auction leads to higher prices than the first-price sealed-bid auction. The final
price in the Dutch auction was on average 8% lower than that in the first-price sealed-bid
auction. My experimental results conform to the experimental laboratory results. Cox et al
(1982) offer two explanations for this phenomenon. Their first explanation is based on the
existence of a positive utility of suspense during the price-countdown in the Dutch auction.
The positive utility of suspense is additive with respect to the expected utility of income from
the auction.
Their second explanation “the probability miscalculation model” involves bidders engaging in
an updating procedure in the Dutch auction. Seeing that their rivals have not called the Dutch
auction to a halt yet, they lower the estimate of their rival’s valuations mistakenly.
Cox et al (1983) test the two hypotheses and find no evidence of the “suspense model”. Their
results do show evidence in favour of the “probability miscalculation model”, however further
tests are necessary for validation.
No results can be given concerning the individual bids because the Dutch auction only reveals
one bid, the bid of the winning bidder.
65
6. Comparison: Second-Price Sealed-Bid, First-Price Sealed-Bid and
Japanese Auction
In Chapter 3.1 we found that the secret second-price sealed-bid auction leads to higher seller-
revenue than the open Japanese auction. In Chapter 3.3 we found that the secret first-price
sealed-bid auction leads to higher seller-revenue than the open Dutch auction. Our
experimental results conformed to the results of laboratory experiments testing revenue-
equivalence in the independent and affiliated private value setting. Neither of the two
conforms to the theoretical predictions.
In order to complete the experimental investigation of revenue equivalence, a comparison
between the first-price sealed-bid, second-price sealed-bid and the Japanese ascending-clock
auction is undertaken in the following section.
Fifteen experiments were carried out, testing revenue-equivalence of the first-price sealed-
bid, second-price sealed-bid and Japanese auction. Eight experiments were conducted in the
session with seventeen bidders and seven experiments were conducted in the session with
five bidders. The auctions were carried out in the following order: first-price sealed-bid, then
second-price sealed-bid and last the Japanese auction. The purpose and advantage of this
order is to have the same informational structure at the outset of each of the three auctions.
This is possible because the first-price sealed-bid and second-price sealed-bid auction can be
carried out without revealing any information. Bids are submitted secretly and neither bids
nor final prices are publicised. The Japanese auction reveals information due to its open
format and thus is carried out after the other two auctions.
6.1 Results
Table 14
17 bidders 5 bidders
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7
Turron Wine Praliné Pottery Book Ameretti Wine Tobler-
one Crema Cat.
Pottery Turron T-Shirt Cap Biscuits Tobler-one
1st 27 85 42 10 50 35 50 29 50 50 75 121 50 120 20
2nd 30 90 35 10 50 45 30 20 30 21 60 121 30 40 25
Jap 25 111 30 5 42 33 23 18 31 21 55 120 25 39 20
66
In eight out of the fifteen experiments, the first-price sealed-bid auction led to the highest
final price, in three experiments the first-price sealed-bid auction was tied with another
auction for having the highest final price and in four experiments the first-price sealed-bid
auction was not the auction yielding the highest final price.
On average the first-price sealed-bid auction led to 16% higher final prices than the second-
price sealed-bid auction. The first-price sealed-bid auction led to 27% higher final prices than
the Japanese auction on average.
6.2 Interpretation
Clearly we do not find revenue-equivalence between the three auction formats. The final
price of the first-price sealed-bid auction was higher on average than that of the second-price
sealed-bid and Japanese auction. In case of independent and affiliated private valuations this
empirical result would contradict the theoretical prediction.
Theoretical prediction for revenue ranking in case of affiliation under risk-neutrality:
According to Milgrom and Weber (1982), affiliation leads to the following ranking of
expected final prices under risk-neutrality:
Dutch = first-price sealed-bid auction § second-price sealed-bid auction § Japanese auction
The experimental results do not correspond to theoretical predictions for the case of affiliated
valuations, because the second-price sealed-bid auction produced higher revenue than the
Japanese auction. The experimental results also do not correspond to theoretical predictions
for the purely common value model, because theory would predict lower prices in the first-
price sealed-bid than the second-price sealed bid or Japanese auction.
One possible explanation for why the first-price sealed-bid auction led to higher seller-
revenue than the second-price sealed-bid and the Japanese auction is risk-aversion. Risk
aversion does not affect the bidding strategy in the second-price sealed-bid and Japanese
auction, but leads to higher seller-revenue in the first-price auctions.
67
In case of affiliated valuations and risk-averse bidders, Milgrom and Weber reach the
following result with respect to revenue-ranking: Dutch = first-price sealed-bid < second-
price sealed-bid § Japanese auction.
In laboratory experiments the first-price auction yields consistently higher revenue than the
second-price sealed-bid auction. See for example: Coppinger et al (1980), Cox et al (1982),
Kagel and Levin (1983). Our experimental results conform to the laboratory results.
Field studies also show slightly higher prices for the sealed-bid auction than the Japanese
auction. Walter Mead (1967) analysed data from U.S. timber auctions and found that sealed-
bid auctions led to about 10% higher revenue than Japanese auctions. Hansen (1985b, 1986)
criticised these results claiming that data selection was biased. After correcting for the bias, he
concluded that sealed-bid auctions did lead to higher prices but that the difference was
statistically insignificant and the revenue-equivalence hypothesis could not be rejected.
Our experiments conform to the empirical results obtained in field and laboratory studies, but
not to the theoretical predictions.
7. Six Results of the Experimental Investigation of Revenue Equivalence
Second-Price Sealed-Bid and the Japanese Auction:
1.) The second-price sealed-bid auction leads to a higher final price than the Japanese
auction.
2.) The bid dispersion is smaller in the Japanese auction than in the second-price sealed-bid
auction.
3.) Revealing public information (revealing the final price of the second-price sealed-bid
auction) weakens the effect above. Revealing public information is to the advantage of
the seller.
68
Second-Price Sealed-Bid and English outcry Auction:
4.) The English outcry auction yields higher seller revenue than the second-price sealed-bid
auction. The English outcry auction leads to higher revenue than the Japanese auction.
First-Price Sealed-Bid and the Dutch Auction:
5.) The first-price sealed-bid auction leads to higher revenue than the Dutch auction.
First-Price Sealed-Bid, Second-Price Sealed-Bid and Japanese Auction:
6.) The first-price sealed-bid auction leads to higher revenue than the second-price sealed-bid
auction and the Japanese ascending-clock auction.
8. Conclusion
Revenue equivalence is tested for the four standard auction formats in the sale of real
consumption goods. The sealed-bid auctions produce higher revenue than their strategically
equivalent open auction counterparts; the first-price sealed-bid leads to higher revenue than
the Dutch auction, the second-price sealed-bid auction leads to higher revenue than the
Japanese auction. The theoretically expected strategic equivalence between the two auction
pairs is robust with respect to risk-aversion and thus presents a puzzle. Furthermore, the first-
price sealed-bid auction yields higher revenue than both the second-price sealed-bid and the
Japanese auction. The results found in the sale of real goods are confirmed by numerous
laboratory experiments, however stand in contrast to theoretical literature.
69
PART FOUR: Internet Auctions and their Framework
1. Introducing Internet Auctions
The break-through of the Internet has led to a huge boom of auctions. The best-known
Internet auction houses in the Consumer-to-Consumer segment are currently Ebay, Amazon
and Yahoo. These three auction houses will be the ones dealt with in the following chapters.
We aim to investigate the rules, success and problems of the current Internet Auctions, when
success is defined in terms of maximisation of seller-revenue. Economists do not generally
evaluate the entire framework an auction operates under such as psychological influences,
legal and structural constraints – even though these are also revenue determinants. The aim of
the economist is to find the mechanism (the bundle of rules) under which the expected seller
revenue is maximised.
The success of an auction can be determined by the rules chosen or by the auction
environment as a whole. In case the current auctions turn out not to deliver the seller revenue
hoped for, it is necessary to determine the source. In this Chapter we want to look at the big
picture, at the general framework Internet Auctions are embedded in.
1.1 Three Business Models: Ebay, Amazon, and Yahoo
In order to describe the framework of Internet Auctions let us begin with the three auction
houses: Ebay, Amazon, and Yahoo.
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Ebay, with headquarters in San Jose, California, started out in September 1995 and had its
initial public offering in September 1998. Ebay is the best–known auction house today and
was one of the first, but not the first consumer-to-consumer auction-house: Onsale had come
earlier starting its business in May 1995. Yahoo followed October 1998, Amazon in March
1999.
Ebay controls more than 80% of the online-auction market, with Amazon and Yahoo lagging
far behind30. Unlike Amazon and Yahoo, Ebay is not a retailer and can thus avoid a number of
complexities and costs31. Ebay collects fees (about seven to eight percent of the sales price)
for being an intermediary and for providing the auction platform. The autos category accounts
for 30% of Ebay’s gross sales32.
Yahoo is a portal, has 3260 employees33 and 185 million visitors worldwide34. Yahoo offers a
lot of services, users spending on average one and a half hours per day35 on the website. This
has given Yahoo the possibility of tracing the customers’ steps through the web pages and
learning about their preferences. Yahoo uses targeted ads which sell for thirty to sixty times
the price of untargeted ads, but is dependent on dot.com advertising with sixty-percent of its
advertising revenue coming from dot.com companies36.
Much of Yahoo’s growth comes from overseas. Nevertheless Yahoo closed five of its
European auction services (UK& Ireland, Germany, France, Italy and Spain) in June 2002. It
is still very successful in Japan, where Ebay shut down its respective auction service in March
200237.
Amazon is not only an auction-house but acquires, stores and delivers brand name products. It
has seven huge distribution centres in the USA, one each in Britain, France, Germany, and
Japan. Amazon built a warehouse in every country it entered and is hoping for an
30 “Ebay’s Bid to Conquer All” by Adam Cohen, Time (New York), Feb 5th 2001, p.48-51. 31 “Plugged in Yahoo Blues: So much for the new economy” by Mark Veverka, Barron’s (Chicopee), March 12th
2001. 32 In the year 2000 Ebay started the Ebay Motors category in form of a partnership with the used car dealer Auto
Trader. “Margaret Whitman”, Business Week, Jan 8th 2001, p.68. 33 “Is there life in E-Commerce?”, The Economist, Vol.358 Issue 8207, Feb 3rd 2001, p.19-20 34 Figures for the year 2000. “Yahoo revises estimates, loses CEO as online ads drop” InfoWorld (Framingham),
March 12th 2001, p.23 35 “Internet Pioneers: We have lift-off”, The Economist, Feb 3rd 2001, p.69-72. 36 “Internet Pioneers: We have lift-off”, The Economist, Feb 3rd 2001, p.69-72. 37 “Yahoo shutters European Auction Sites” by Troy Wolverton on Cnet News.com, June 28th, 2002
(http://news.com.com/2100-1017-940580.html).
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improvement in the efficiency of Europe’s transportation systems38. Due to its fast-moving
centralised inventory and relatively fixed handling costs, Amazon has a competitive
advantage in business areas such as consumer electronics.
1.1.1 Revenue
Registered Users (in millions)
Gross Merchandise Sales per day (in $)
Ebay 22,5 12 Million
Yahoo 236 500 000
Amazon 29 200 000
Table 15: Figures taken from Goldsborough and Veverka39:
Auctions closing per day Revenues per month (in $)
Ebay 340 000 190 000 000(18 000 000)
Yahoo 88 000 19 000 000 (7 900 000)
Amazon 10 000 2 000 000 (620 000)
Table 16: Size Estimates Summer 1999, estimated standard errors in parenthesis40
Monthly Volume ($) Number of Sites 10,000 or less 83
10,001 to 100,000 27
100,001 to 1,000,000 21
1,000,000 or more 7
Table 17: Size Estimates November 199841
As can be seen from the tables above, Ebay’s monthly revenue is ten times as large as that of
Yahoo, which likewise is ten times as large as that of Amazon. Causes are discussed below.
38 “Yahoo revises estimates, loses CEO as online ads drop”, March 12th 2001. 39 “Internet Auctions Examined” by Reid Goldsborough , Link-up (Medford) Nov/Dec 2000, p.24 and “Plugged
in Yahoo Blues: So much for the new economy” by Mark Veverka, Barron’s (Chicopee), March 12th 2001. 40 Taken from: Lucking-Reiley (2000), p.248. The size estimates were computed by choosing a day in June or July to
visit each site, observing the number of auctions on that day and taking a sample of closed auctions to estimate
the average revenue per auction closing. 41 Lucking-Reiley (2000), p.230
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1.2 Network Effects
This chapter shows which network effects are present in Internet Auctions and how they
work.
Internet auctions are characterised by network externalities, i.e. external demand side scale
economies. Network externalities are conditions that give each user benefits as the set of users
expands. External demand side economies of scale imply that when a new user subscribes to
the service, the other users benefit: the benefit thus being external to the new user42.
The platform with the highest number of users (potential bidders) attracts the highest number
of sellers. The bigger the auction platform, the higher the utility for each buyer, because they
benefit from a reduction in search costs, a richer variety, and a better comparison due to more
items per category.
A higher number of bidders directly affects the expected return of the seller positively. As the
number of bidders increases, the valuation of the second highest bidder approaches that of the
highest bidder (in the independent private values model).
Logging onto an auction generates costs for the buyer. Once a buyer is registered as an
auction user, he does not need to enter his personal information and credit card information
again, when he next wishes to buy an item via an auction. More importantly, a larger variety
of goods produces higher utility for the bidder. Shoppers shop where they expect to find
something that they will want to buy.
Logging onto a new website also produces costs for the seller, having to learn the specifics of
the auction-house and having to enter information about himself and his good. He can reduce
these costs - except for the specific information about the good, description and picture of the
item - by using the same auction-house for all his sales. The fact that a buyer or seller faces
costs when conducting his transactions through a new and different auction-house shows the
existence of switching costs and leads to lock-in.
42 Rohlfs (2001), p.14.
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It is assumed that network externalities and switching costs help the largest player, Ebay in
growing larger, pushing Amazon and Yahoo into the background. Internet auctions also show
evidence of the first-mover advantage at work. Ebay being the first-mover in Europe was
probably partly responsible for Yahoo shutting down five of its European auction services in
June 2002. In contrast, Yahoo was the first-mover in Japan, where Ebay decided to shut down
its services in March 2002.
1.2.1 Loyalty
The registration effort required for participation in the auction is an exit barrier; in order not
to incur these costs again, bidders have an incentive to stick with the first auction house they
deal with. The feedback rating system is another feature that promotes loyalty. On Ebay the
successful bidder and the seller receives plus one, zero or minus one points, after the deal is
closed, the payment made and the good shipped - from their respective transaction partner.
The rating of a bidder or seller is displayed next to his name on the website. Sellers and
bidders who have a rating above zero have an incentive to keep on conducting transaction
with that auction house, because the feedback rating serves as information to other
participants about the trustworthiness.
The feedback-rating system acts as a lock-in for bidders and provides an incentive for bidders
to remain loyal to their auction house and to conduct all auction purchases through the same
auction house. The switching costs benefit the first-mover, who will further benefit from
network-effects and positive feedback (i.e. the process with which an increase in customer-
base leads to a further increase in customer-base and so forth).
1.3 The Selling Mechanism
1.3.1 Auctions and Posted Prices
There are a number of ways a good can be sold, using posted prices, bargaining or auctions.
When selling a single-unit item the seller ideally wants to achieve a price equal to the highest
bidder’s reservation price. If the seller were perfectly informed he would simply charge a
price equal to the highest bidder’s reservation price. Auction theory assumes that there is
asymmetrical information, that the seller does not know the bidders’ valuations. The seller
searches for an auction mechanism that extracts the maximum amount of consumer surplus
from the buyer. The rules of the English auction give bidders the chance to post bids. A buyer
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will find it useful to bid (up to his reservation price) because this is the only way he can
possibly win. Valuations will thus be revealed naturally. This is in contrast to a posted-price,
where no revelation mechanism is used. Given asymmetrical information about the bidders
valuations, it is highly unlikely that the seller will be able to set a posted price that precisely
corresponds to the highest valuation.
The four standard auction formats discussed in the theoretical literature are presented in Part
One. However, none of these auction mechanisms claim to be able to extract the reservation
price from the highest valuing bidder. Instead, as shown by William Vickrey’s Revenue
Equivalence Principle, the expected revenue is equal to the second highest bidder valuation in
all four standard auction formats (under specific assumptions).
The main advantage of auctions in comparison to posted prices (when selling multi-units) is
that auctions offer the possibility to price-discriminate. Wang (1993) shows that the seller
prefers auctions to posted-price selling when bidder valuations are more strongly dispersed.
1.4 The Goods
1.4.1 Suitable for Auctions
The theoretical literature recommends auctions under the following circumstances:
- The seller is a monopolist
- Demand is unknown to the seller, high uncertainty about demand
- Unique goods, goods in limited supply
- Perishable goods
1.4.2 Goods Sold
- Currently fashionable items (signed Harry Potter books, Pokemons)
- Collectibles (60% of all Internet Auctions sell collectibles, e.g. coins, stamps.43)
Empirically one finds a large amount of collectibles sold through Internet Auctions.
- Antiques and art
- Second hand goods
- Small, easy to ship (most deliveries take place through parcel delivery) 43 David Lucking-Reiley (2000), p. 231.
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- Cheap. Collectibles traded on Ebay have median prices well below $100 with almost no
items above $100044.
- Used cars
- Perishable goods, such as plane tickets, hotel rooms, last-minute holiday-packages
Category Sites, that list this category
Sites that are specialised in this category
Collectibles 90 56 Antiquities 40 10 Star Memorabilia 16 7 Stamps 11 5 Coins 17 2 Toys 17 0 Playing Cards 14 0 Electronics and Computers 48 9 Jewellery 17 1 Computer Software 16 0 2nd hand Items 15 7 Sports Items 13 4 Travel Services 7 5 Real Estate 4 2 Wine 3 2 Table 18: Taken from Lucking-Reiley (2000), p.233.
Suitable Goods Sold in Internet-Auctions
- Perishable Goods
- Travel: Airline Tickets, Bed and Breakfast, Hotel, Packages
- Tickets: Concerts, Musicals, Sporting Events
- Collectibles: Stamps, Coins
Goods Sold But Not Recommended By Theory
- Consumer Durables: Computers, Photo Equipment, Clothes
- Real Estate: Commercial, Residential, Land
44 Lucking-Reiley (2000), p.232.
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1.5 The Auction Formats Used
According to a survey of 142 auction sites by Lucking-Reiley (2000, p.237), the following
auction formats were used:
121 Japanese
15 first-price sealed-bid
5 second-price sealed-bid
3 Dutch
4 continuous-trading double auctions45
6 used more than one auction format46
The auction format used most often is clearly the Japanese Auction. Part One presents the
four standard auction formats and an introduction to Auction Theory. Given the figures above,
we would assume that the Japanese auction has a quality that makes it most attractive for
auction houses.
1.5.1 The Choice of Auction Format by the Auction House
An auction house is a firm that maximises its profit. Two of its sources of revenue are the
insertion and commission fee that sellers have to pay. The commission fee is dependent on the
final price reached, the seller therefore striving to maximize his sales (measured in monetary
units). A sale will only be successful when an item is matched with a buyer, this being more
probable the larger the customer base.
Buyers prefer an auction format that maximises their expected revenue and sellers prefer an
auction format that maximises their expected seller revenue. As the seller revenue is equal to
the price paid by the buyer, there is a trade-off between the buyer and seller revenue.
The seller’s choice of auction format is only based on his expected revenue, but bidders do
not base their decision of participating in an auction exclusively on their expected revenue.
Instead they also care about the availability of interesting goods, which in turn depends on the
auction format the sellers prefer. The choice of auction format is characterised by a strong 45 Double auctions allow continuous updating of seller offers and buyer bids 46 The sum is more than 142, because some sites offer more than one auction format
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interdependency of preferences. Empirical results about seller preferences can be found in
Part Five.
2. Internet-Specific Characteristics
2.1 Internet Specific Advantages
There are a number of advantages in using the Internet as a platform to conduct auctions on.
Accessibility, Reach
The Internet provides access to a huge number of people.
Cheaper Technology
Auctions become cheaper relative to other pricing mechanism due to the lower-cost Internet
technology. Therefore a higher number of items can profitably be sold. The only true costs of
Internet auctions are auction fees and transportation costs, dependent on transportation
distance and weight of item.
Capacity
A traditional auctioneer like Sotheby’s can only perform a certain number of auctions per day
because of the limited capacity of its rooms and staff. Online auctions are able to deal with
huge capacity and are able to benefit from economies of scale47.
Lower Fees
Fees are much lower for Internet Auctions than for traditional auction houses. Ebay’s fees
amount to about five to seven percent of the final bid; Sotheby’s charged a buyer premium of
15% over the final bid price and a standard seller’s commission fee of 20% of the bid price48.
47 “Internet Pioneers: We have lift-off”, The Economist, Feb 3rd 2001, p.69-72. 48 Hildesley, C.H. (1997), „The Complete Guide to Buying and Selling at Auctions“, W.W.Norton, New York
quoted in Lucking-Reiley (2000), p235.
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Lower Costs
The bidders face lower costs due to lower fees, no travel costs, no time loss of travelling to
and attending the auction. The costs of logging on to the Internet, registering for the auction,
bidding and keeping track of the auction are very low in comparison to actually attending a
traditional auction or sending an agent.
2.2 Internet Specific Problems
Anonymity of the Internet
The anonymity of the Internet creates various sources of uncertainty for the buyer and the
seller.
Lack of Information
The number of successful transactions, that is transactions where a good is sold to a buyer at a
price acceptable to the seller, is dependent on the size of the market, the number of sellers and
buyers conducting transactions on the website. The fact that only a small number of goods
find buyers49 can be partly caused by an informational problem. Search engines and product
notification emails for subscribed categories are useful in calling auctions to the attention of
buyers and matching sellers with buyers.
Uncertainty About the Nature of the Good
Buying an item through the Internet means that one does not get the chance to see the item
before purchasing it. The buyer has to base his purchase decision on the description of the
object. This problem is fundamental to the Internet, but photos and accurate descriptions by
the seller can prevent unwanted purchases.
Uncertainty About the Enforceability of the Contract
One reason why some potentially interested bidders decide not participate in Internet auctions
is the uncertainty with respect to the observance of the contract, due to a lack of legal security.
There is lack of legal security ensuring the enforcement of contracts, leaving both the seller
and buyer no other option but to trust the other party. The seller is uncertain whether he will
ever receive the money; the buyer is uncertain whether he will ever receive the good.
49 See Part Five Chapter 3.5
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Auction houses have however taken measures to increase information about the
trustworthiness of the sellers and buyers, through the implementation of a rating system,
where sellers and buyers are awarded points by their transaction partners. Uncertainty about
the arrival of the good and payment can be mitigated by using escrow services, agents that
inform the seller upon receipt of payment by the buyer, in order for sellers to be able to then
safely send the good to the buyer. Intermediaries of this kind already exist, although they are
costly, they also reduce a great deal of uncertainty.
Fraud
Two types of fraudulent behaviour are particular of Internet auctions, shilling and shielding.
Shilling: The seller enters an (artificially) high bid, when only one or few bidders are left, in
order to drive up the price
Bid Shielding: A bidder who is truly interested in the item, enters a low bid and asks a friend
(collaborator) to enter a very high bid. This high bid has the aim of discouraging bidders who
have reservation prices between the low first bid and the shielding high bid. Just before the
auction ends the high bidder retracts his bid.
Measures to prevent fraudulent behaviour include a feedback and rating systems for both
sellers and buyers; escrow services where the buyer sends his payment to the escrow agent,
who then informs the seller that he can now send the item to the buyer; and co-operations with
prosecutors to encourage defrauded buyers and sellers to sue.
Transportation Costs
Transportation costs are responsible for favoring light (weight-wise), small items and short-
distance transactions as auction-goods. Ebay and Amazon have both started regional auction-
sites, such as ebay.de or amazon.de directed in the German market. Regional auctions are not
only advantageous with respect to transportation resulting in cost-savings and less risk of
breakage; they also provide the possibility of inspecting the good prior to purchase.
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2.3 Outlook
Some of the problems discussed above could be solved or at least weakened. The problems
related to the enforceability of the contract can be partly solved by the use of intermediaries,
who supervise the transaction by waiting for the receipt of the money until they advise the
seller to send the good to the buyer. Uncertainty will also be reduced once the legal
framework for e-commerce is clearly set out.
The current structural framework definitely accounts for a lack of bids and transactions, but
the auction rules might also not be perfect to stimulate an exciting auction. Nevertheless the
Internet can be seen as an amazing opportunity for determining the price and value of a good,
when nearly the entire world is available as a potential bidder. The Internet provides the
opportunity for reaching a huge number of people, who could bid quickly, comfortably (from
their own computer), and without global limits.
3. The Internet Auction Rules
3.1 Bid Submission and Procedure
There are two ways in which bidders can submit bids on Internet auctions, by a straight bid or
a proxy bid.
Straight Bid
A bid can be submitted as a straight bid. When a straight bid constitutes the winning bid, the
winning bidder has to pay a price equal to his bid. When a straight bid is outbid, Yahoo,
Amazon and Ebay provide “outbid notification” by email.
Proxy Bid
A bid can be submitted as a proxy bid. The bidder enters a bid denoted as his “maximum
willingness to pay”. This automated proxy-bidding mechanism bids automatically, placing a
bid one minimum-increment above the current high bid on behalf of the bidder, until the
bidder’s “maximum willingness to pay” is reached.
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3.2 Bidder and Seller Registration
Yahoo: In order to be able to sell an item or submit a bid, the seller / bidder must enter his
name, email address, birth date, gender, zip code, occupation, industry, and personal interests.
Ebay: In order to be able to sell an item or submit a bid, the seller / bidder must enter his
name, Ebay user ID or email address, billing address, credit card number and expiration date.
Bidders have to let Ebay verify their credit card before they are allowed to place bids of more
than $15.
Amazon:
A bidder and seller must enter his name, address, phone number, email address, and valid
credit card information. He also has to register with Amazon.com Payments, who require a
credit card issued by a U.S. bank.
3.3 Auction-Length
The length of Internet-auctions varies between two and fourteen days. A length of seven days
is most common50. The seller can generally choose the auction length within set limits.
Yahoo: Sellers choose a length from two to fourteen days.
Ebay: Sellers choose a length of three, five, seven, or ten days
Amazon: Sellers can choose a length of one, three, five, seven, or fourteen days.
Ending Rule
Yahoo: The auction ends at a specified end date, except if the option “automatic extension” is
chosen. In this case, if bids are placed less than five minutes before the (tentative) auction
end, the auction continues for five more minutes.
Ebay: The auction ends at a time specified by the seller.
50 Lucking-Reiley (2000).
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Amazon: The auction ends at a specified end date, with the additional requirement, that there
must be no bids within the last ten minutes of the auction. If bids are placed less than ten
minutes before the auction end, the auction is extended until the last ten minutes are bid-free.
3.5 Auction Fees51
Yahoo:
Listing Fee: between $0.2 and $1.5
Final Value Fee: between 1% to 4% of the final sale price
Reserve Price Fee: The reserve fee lies between $0.4 and $0.75, which is refunded if the
auction closes successfully.
Ebay:
Insertion Fee: between $0.3 and $3.3
Final Value Fee: between 1.25% to 5% of the final sale price
Reserve Price Auction Fee: between $0.5 and $1, which is fully refunded if the item sells.
Amazon:
Listing Fee: $0.1 per listing
Closing Fee (required only if the auction is a success): between 5% of final price and $25.63
plus 1.25% for any amount above $1000
3.6 Additional Features
Reserve Price
The seller has to choose a starting price for his auction. The starting price is the lowest
possible price at which bids are accepted.
Yahoo, Ebay, Amazon: Additionally the seller can choose to use a reserve price, which is then
the relevant minimum price at which the seller is obliged to sell the item.
Sellers can choose to use a secret reserve price. In this case the reserve price is not revealed.
The website indicates that there is a secret reserve price as long as the reserve price has not 51 Data from November 2001
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been met. Amazon and Ebay do not reveal bidder identities and bid amounts as long as the
reserve price has not been met.
Buy Price
Yahoo: The seller has the possibility of setting a buy price. When a buyer bids the buy price,
the auction ends immediately and the buyer receives the good at this price.
Early Close
Yahoo, Ebay: Sellers can choose the Early Close option, which enables them to close the
auction at any time during the auction and sell the item to the current winning bid.
Amazon: The same option is offered as a “Take-It Price”.
First Bidder Discount
Amazon gives sellers the option of a first bidder discount. The winning-bidder gets a discount
of ten percent on the final price if he was also the first bidder.
Cancelling Bids
Yahoo: Bids can only be cancelled at the seller’s discretion.
Bidding Increments
Yahoo: $0.1 to $10
Ebay and Amazon: $0.05 to $100
Payment Method
Yahoo:
Credit Card, Money Order, Check
Yahoo PayDirect! Sending and Receiving money by email, when both parties have a Yahoo
PayDirect! Account.
Escrow Service: A third party acts as an agent, holding the buyers' money until there is proof
that the item was shipped and received.
Ebay:
Credit Card
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Online Payments with Billpoint
Amazon:
Credit Card, Money Order, Check, C.O.D.
Online Payments with Amazon.com Payments
Ratings and Feedback
Buyers and Sellers rate each other.
Ebay, Yahoo:
+1: is added for each positive comment or praise.
0: is given for neutral comments.
-1: is given for each negative comment or praise.
Amazon: Five stars is the highest possible rating.
4. Some Implications of the Auction Rules
One major difference in the auction rules of Ebay, Amazon, and Yahoo is the ending rule.
Ebay uses the hard-closing ending, Amazon the automatically extended ending and Yahoo
offers the seller the choice.
Uncertainty about the Number of Bidders
Bidders in Internet auctions are uncertain about the total number of participating bidders.
There is no requirement to post a bid early or to pre-register for an auction, therefore bidders
can join the auction until the very last minutes of the auction duration. The auction sites
publicise the number of bidders who have posted bids so far, but new bidders can join
anytime until the last moments of the auction.
When all bidders have independent private valuations, not knowing the number of bidders
does not influence bidding behaviour in the Japanese and second-price sealed-bid auction, but
does influence bidding behaviour in the Dutch and first-price sealed-bid auction (see Part One
Chapter 1.5).
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In the case of interdependent valuations, the number of participating bidders affects the bid
and final price. If all bidders have the same assumption about the number of bidders then the
final price will be equal to the case when the number of bidders is publicly known.
Effect on Profitability
Ebay’s monthly revenue is far higher than that of Amazon52 and Yahoo. This could be due to
a first-mover advantage and network effects. However, Ebay using „more profitable“ rules
might be an important cause too. The main difference between Amazon and Ebay is the
ending rule used. Yahoo in contrast offers sellers the possibility to choose their auction end.
Whether the ending rule affects profitability and which auction end the seller prefers will be
investigated in Part Five.
Effect on the Timing of Bids
On average Amazon auctions have fewer last-minute bids than Ebay auctions53. The existence
of late bidding and the effect of rules on the timing of the bid are discussed extensively in Part
Five.
Auction Fees
Fees increase the auction house’s revenue (ceteris paribus) and have the advantage of
inducing commitment on part of the seller. When a seller can post goods on a website at no
cost, he has an incentive to choose a high reservation price54: if he does not find a seller at the
high price, he can lower his reservation price and try selling the good again just a few days
later. However, when the seller has to incur fees for every good he offers on the website, he
has an incentive to post a reservation price that he believes has reasonable chances of being
met. This allows the auction house to save on costs and raises its percentage of successful
auctions.
52 See Part Four Chapter 1.1.1. 53 Roth and Ockenfels (2000). 54 The economic term “reservation price” corresponds to the term “minimum-price” used by the auction-houses.
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Out of the three auction houses studied, Yahoo was the only one where sellers could offer
goods without having to pay a listing fee. However, in February 2001, Yahoo also decided to
introduce auction fees55.
55 Exact date: February 9, 2001. Taken from “Yahoo shutters European Auction Sites” by Troy Wolverton on
Cnet News.com 28. June 2002 (http://news.com.com/2100-1017-940580.html).
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PART FIVE: Late Bidding Investigation
Overview
Internet auctions are analysed with respect to two different ending rules (hard close and
automatic extension) and three different product categories (cars, computers and paintings).
We find that Internet auctions are characterised by late bidding with a strong price-increase in
the last hundredth of the auction. This effect is stronger for art than for car or computer
auctions. Furthermore, the winner enters the auction at a very late point in time: on average in
the last hundredth of the auction. This is true for art, car, and computer auctions. The late
entry of the winning bidder is caused by the fear that lower-valuing bidders with
interdependent valuations will revise their bid upwards due to common value or prestige
effects.
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1. Introduction
Internet auctions are a relatively new phenomenon. In the past years many new consumers
have been attracted to Internet auctions for various reasons, for example because Internet-
access has become the norm and because trust in the security of payment via the Internet has
grown.
Observing Internet-auctions, an interesting new phenomenon is found: late bidding. In order
to investigate whether late bidding is a rational bidding strategy I present two models of
interdependent valuations. The first model is a general model of interdependent valuations;
the second model a model with reputational effects. The bidding strategy and expected final
prices are calculated for the symmetric and asymmetric (Expert-Amateur) case in both
models. The effects of interdependencies on bidding-behaviour and price are compared to
those of the automatically extended auction.
In order to empirically test the theoretical predictions, data is used from hundreds of Yahoo
auctions. We want to find out whether bidders wait to submit bids until the very last moment
and how this affects the price path over the complete auction duration; whether there is a late
bidding effect in the sense that the price increases over-proportionately at the end of the
auction. Roth and Ockenfels (2000) were the first, to my knowledge, to publish an empirical
study on bidding on the Internet. They investigate auctions with regard to the number of bids
submitted in the last twelve hours and find that many bids are submitted in the last few
hours56. In this work price-formation is studied during the complete auction duration.
The theoretical prediction that late bidding is dependent on the ending-rule is tested
empirically57. The theoretical models presented in Chapter Two predict that sellers prefer the
automatically extended auction to the hard close auction. Yahoo auction data is used to verify
the prediction because Yahoo lets the seller choose between the hard close and automatically
extended ending. Late bidding is studied for three categories of goods: computers, art and
cars, different goods implying different valuations and bidding behaviour.
56 The average auction length of seven days translates into 168 hours. Roth and Ockenfels only analyse the last
12 hours, thereby ignoring 156 hours (on average). 57 Roth and Ockenfels (2000) find a higher number of late-bids in Ebay than in Amazon auctions. Ebay has a
hard close end and Amazon an automatically extended auction end.
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2. Theoretical Investigation
Auction theory analyses four standard auction-formats with respect to their revenue
implications. The Ebay, Amazon, and Yahoo auctions all fit the definition of an English
auction, i.e. an open ascending bid auction. In order to differentiate between the different
English auction formats on the Internet, the standard model for the English auction has to be
modified for the specific rules. The main difference in the rules of Ebay, Amazon and Yahoo
is the ending rule. The Ebay auction has a fixed ending. The Amazon auction has an ending
rule specifying that when bids are submitted in the last ten minutes before the tentative end,
the auction be extended until the last ten minutes are free of bids. Yahoo lets the seller choose
the ending rule, offering a hard close or a tentative end with 5-minute automatic extension.
One big difference between the English ascending-bid auction and the current Internet
auctions is that there is uncertainty about the number of bidders. As discussed in Part One
Chapter 2.3, uncertainty about the number of bidders is of no relevance in second-price
auctions with independent private valuations. Internet auctions, like most other auctions, are
characterised by some form of interdependent valuations. The Milgrom-Weber model
represents a general model of interdependent valuations, using affiliated interdependent
valuations. In the model presented in Part Three bidders have interdependent valuations with
independently distributed signals.
Below I present four different forms of interdependent valuations: the symmetric model of
interdependent valuations used in Part Two, the model from Part Two but with asymmetric
valuations (an Expert-Amateur Model) and a second Expert-Amateur Model where the
amateur has a prestige value. Finally I present an Internet auction model for the general
Milgrom-Weber framework. The asymmetric models of interdependent valuations reflect the
fact that bidders in Internet auctions often have very different informational structures –
participants in Internet auctions for antiques include professional dealers as well as “innocent”
house-wives.
Two ending rules are compared to one another: hard close and automatic extension. An
auction format representative for the hard close auction is presented. The price and seller
revenue of the hard close auction is compared to that of the automatically extended auction.
90
2.1 Theoretical Investigation of Late bidding
An increase in the number of bids submitted toward the end of Internet auctions has been
noticed and conjectured by bidders, Internet auction houses, as well as academics (e.g. Roth
and Ockenfels).
Some economists have already started to work on finding rational explanations for late
bidding. Roth and Ockenfels (2000) present a model in which technical problems create a
positive probability that the auction house does not receive a late bid. This can induce late
bidding as a collusive attempt to keep the price low. Rasmussen (2001) presents an Internet
auction model with asymmetrically informed bidders. After a certain point in time, the
uninformed bidder has the possibility to discover his private but unknown valuation at a
certain cost. In turn, late bidding can be a rational strategy.
2.1.1 The Moral Hazard Incentive
One intuitive explanation that might account for late bidding is the moral-hazard incentive of
the auction house. Auction houses earn profit by the commission they charge buyers and/or
sellers based on the final selling price. The auction house would like to earn as high
commission payments as possible: a greater number of successful sales increase the
commission as well as higher selling prices.
A bidder can bid by making a single bid (if the winning bid is a single bid, the winner has to
pay a price equal to his own bid) or by making his maximum-willingness-to-pay known to the
proxy-bidding agent (who will bid automatically by raising the current high price by the
minimum-increment until his bidder appears as the high-bidder – within the limit of the
maximum-willingness to pay). It is not rational for a bidder to use single bids58 (see Chapter
3.3.2 below). The proxy-bidding mechanism asks bidders to enter the maximum-willingness-
to-pay into the auction’s website. The auction house now has the possibility to make a seller’s
dream come true, posting a fake bid just below the highest reservation price and thereby
raising the price to just a minimum-increment below the high-bidders reservation price. This
would enable the auctioneer to extract the total surplus from the high-bidder, thereby leading
58 Single bids can be rational if rivals are intimidated by jump bids (jump bids are bids that are higher than the
current high-bid plus a minimum-increment. Easley and Tenorio (1999) study jump bidding on the Internet,
Avery (2002) models strategic jump bidding in English auctions.
91
to the first-best outcome. Legally this is not allowed, but technically it is possible. Of course
the auction house has a smaller incentive to engage in such fraudulent activities than would
the seller, because the auction house only receives a certain fraction of the final price as
commission. Nonetheless, the moral hazard incentive is present. If bidders are aware of this
and believe it to be a serious threat (due to the anonymity of the internet), late bidding would
be a logical consequence. Even though I believe the moral hazard explanation to be plausible
and realistic, in the following sections and models we assume that the moral hazard incentive
is of no relevance and search for other explanations.
2.1.2 Interdependent Values
When all bidders have private values and believe their rivals to have private values too, late
bidding is not reconcilable with standard auction theory. There is no reason for a bidder to
hold back his valuation until the very last minutes when none of his rivals is influenced by his
valuation and he is also not influenced by the valuations of his rivals.
In the following section we therefore concentrate on models of interdependent valuations. At
least one bidder has a valuation that depends on private information of another bidder.
2.1.3 General Bidding Model
In the following section I will show bidding behaviour for various models of interdependent
valuations. In section 2.1.3, I use two variants of the model from Part Two; the symmetric
case, where all bidders have the same λ value and the expert-amateur model, where there is
one expert with a λ equal to zero competing against amateurs with a λ greater than zero. In
section 2.1.4 late bidding is tested for a prestige-value model, for the symmetrical and the
expert-amateur (i.e. asymmetrical) case. In section 2.1.6 late bidding is tested for the
Milgrom-Weber general symmetric model of interdependent valuations.
Revenue in the hard close auction is compared to that of the automatically extended auction.
The revenue-comparison is conducted for the case when the representative bidder (for whom
the bidding strategy is calculated) is the winning high bidder.
92
The automatically extended auction is modelled as a Japanese auction (as done by Bajari and
Hortascu, 2000)59. The Japanese auction is modelled like in Part Two, where the valuation
(reservation price) of bidder i is defined by: iii sCp )1( λλ −+= , where ∑≠=
=n
jji
ij
sn 1
1C . Private
signals are drawn from the uniform distribution U∼[0,1]. It is further assumed that bidding
is not costly and that there is no fraudulent misconduct on part of the auction house.
is
2.1.3.1 Model of the Hard Close Auction
One further reason why it is impossible to explain late bidding with standard auction theory is
because the theoretical model used for modelling an English auction - the auction format
predominantly used by Internet auctions is the Japanese (ascending clock) auction. However,
the Japanese auction60 offers no possibility for late bidding because bidders have to log-on to
the auction at the beginning of the auction, have to publicise their exit price and cannot re-
enter the auction once they have logged-off. The Japanese auction does not represent the rules
of the hard-closing Internet auction well, because bidders in hard-closing Internet auctions
have the possibility to place bids until the very last moments of the auction. In the following
analysis I use a model for the hard-closing auction format that represents the Internet auction
more accurately.
The model representing the hard close auction consists of a Japanese auction followed by a
second-price sealed-bid auction61. Bidders are obliged to participate in the Japanese auction,
but can bid zero. Bids in the Japanese auction are binding. The final price of the Japanese
auction is revealed before the beginning of the second-price sealed-bid auction. The second-
price sealed-bid auction represents the “sniping”-phase of the auction; i.e. the very last
minutes of the auction, when there is no more time to react to bids made.
Late bidding Proposition: The high bidder bids zero in the first round (in the Japanese
auction). 59 The automatically extended auction is modelled as a Japanese auction. In reality bidders in automatically
extended auctions have the possibility to wait until the very last moments of the auction to post their bids, but
due to the extension rule there can be no secret last minute bids. Rivals have the possibility to react to bids. 60 Described and modelled in Part Two. 61 Bajari and Hortascu (2000) use this set-up to model an Ebay auction.
93
When planning his bidding-strategy, a bidder is interested in the case where he is the winner.
That is a bidder plans his strategy for the case when he has the highest private signal. In case a
bidder is not the winner, his payoff is assumed to be zero.
The decision is looked at from the point of view of a bidder who believes that his rival - the
price-determining bidder, i.e. the bidder with the second-highest valuation - has an
interdependent valuation. As explained above, when a bidder believes or knows that the price-
determining bidder has an independent valuation he has no incentive to bid late.
The proposition is proved by showing that it is a dominant strategy for this bidder to bid zero
in period one (the Japanese auction) and to wait until period two (the sniping phase) to bid his
valuation. This is true for the case when he himself has an interdependent valuation (the
symmetric case) and when he has certainty about his valuation (he is an expert) but his rival
has an interdependent valuation (his rival is an amateur).
When the representative bidder has a private valuation and the price-determining rival also
has a private valuation – even though the representative high-bidder falsely assumes his rival
to have an interdependent valuation – sniping will be of no advantage. The only time a private
valuation high-bidder is worse off by sniping than by early bidding is when his private signal
is tied with that of his rival (who also has a private valuation). In this case the bidder who
submits his bid earlier wins, but this event has a probability equal to zero.
2.1.3.2 Symmetric Model: All bidders have the same λ
n... number of bidders at the beginning of the auction
k… number of bidders who have already quit the auction
λ... measure of how important the common value is to a bidder, 0 ≤ λ ≤ 1, every bidder has
the same λ
si... private signal of bidder i, the si are uniformly distributed ∼ [0,1]
ek... kth observed exit price
pi … reservation price (valuation) of bidder i
C... common value
94
The auction is carried out in two stages: first a Japanese auction and then a second-price
sealed-bid auction. The Japanese auction ends when the penultimate bidder quits the auction,
the price at which he does this is the final price. Bids in the Japanese auction are binding.
Bidder i’s valuation: ii sCp )1( λλ −+= . The common value is further defined as the mean
private signal of all other bidders: ∑≠
=
n
ji
ij
sn 1
1=i
is
C . As explained in Part 2 Chapter 5, bidder i
only knows his own private signal , but does not know the private signals of all the other
bidders. Bidders estimate the common value in each round.
It is rational for a bidder to make a final bid equal to his reservation price62. However, it might
not be rational for a bidder to reveal his valuation to his rival bidders, knowing that they take
his valuation into account in the common value. By bidding zero in the Japanese auction and
waiting to bid his valuation in the second-price auction, he can keep his valuation secret from
his rivals.
Bidders must decide when to place their bid. They are allowed to bid in both stages of the
auction in the Japanese auction (called truthful) or in the second-price auction (called
sniping). A bidder bases his bidding strategy on the case when he receives the highest signal.
A bidder receives payoff zero when he does not win the auction. When a bidder wins, he
receives a payoff equal to their final-bid minus the second-highest bid.
Below expected payoffs are calculated when bids are posted truthfully (posted in the Japanese
auction) and when bidders snipe (post their bids in the second-price auction). The other
bidders are assumed to bid truthfully, when either A or B bids truthfully.
62 See Part One Chapter 2.1.3 for explanations.
95
B
Truthful Snipe
+−
111nn
λ
+−
111nn
λ
+
−)1(2
321nn
λ
+−
)1(221nnλ
+−
)1(221nnλ
+
−)1(2
321nn
λ
+−
)1(221nnλ
+−
)1(221nnλ
Truthful , ,
A
Snipe , ,
Table 19: Expected Payoffs in case of symmetrical valuations
The expected payoff of a bidder is determined by the probability of winning the auction
multiplied by the expected payoff in case of winning the auction.
Bidder A’s expected payoff is higher from sniping than from truthful bidding. The same is
true for B. It is therefore a dominant strategy for all bidders to snipe. The equilibrium is
(snipe, snipe), which gives the winner the highest possible payoff and correspondingly leads
to the lowest seller revenue.
The probability of winning the auction is equal to the probability of being the high-signal
holder in the symmetrical case. The high-signal holder always wins the auction. Bidding
truthfully or sniping influences the payoff he receives as a winner, but not the chance of
winning. This is because a bidder’s valuation is the weighted average of his signal and the
common value. The private signal of a high-signal holder is (by definition) higher than his
common value. The probability of being the high-signal holder is , which is equal to the
probability of winning. The payoffs when a bidder wins are given in the calculations below.
They are then multiplied by the probability of being the winner, which is , and this will
correspond to the numbers found in Table 19 above.
a.) Calculations when all bidders are truthful
All bidders bid their honest valuation in the Japanese auction. Bids are made as described
above in Part 2 Chapter 5. There is no bidding in the subsequent second-price sealed-bid
auction.
n1
n1
96
Winner Payoff:
The winner’s payoff is composed of his valuation minus the second-highest bid. Bidder i has
a valuation of the form ii sCp *)1(* λλ −+= , therefore the winner’s payoff is calculated as
follows:
1)1()1(* −−−−−+= nn sCsC λλλλ
1)1()1( −−−−= nn ss λλ
Expected Payoff:
E[P]=
+−
−+
−11
1)1(
nn
nnλ
=
+−
11)1(
nλ
b.) Calculations when A snipes and B bids truthfully:
When A does not participate in the Japanese auction, all (n-1) other bidders believe that there
are only (n-1) bidders participating and therefore calculate the common value incorrectly, they
underestimate the true common value. The common value is estimated by estimating the
upper bound of the distribution of the signals s. This is calculated by using the signal
realisations revealed through the exit prices and the number of still active bidders. See Part 2
Chapter 5 for more details on the common value estimation procedure.
When the highest signal (the n-th signal) is missing, the (n-1) mistaken bidders calculate the
, as follows: 3−nC
There are n-1 bidders participating in the Japanese auction. After the (n-3)rd bidder quits the
auction, there are still two bidders remaining. They calculate the common value using the (n-
3) lower signals and then estimating the value of the distribution of the two missing signals.
k…number of bidders who have quit the auction so far
k = n-3
97
3−nC (k)
+−−++++
−= − 2
ˆ)2(...
21
321abknsss
n n
+++++
−= −
− 2][]ˆ[
...2
1 1321
nn
sEbEsss
n
E[C (k)] 3−n
+
−+−+
+−
+++
++− )1(2
1313...
12
11
21
nnn
nn
nnn=
+
−+−+
+−+−+
−++
−=
)1(213
121
)1(2)1(
21
nnn
nnnn
nnn
n
+−
++−
−++
−=
)1(242
)1(266
)1(2)1(
21
nn
nn
nnn
n
+
+−+−
=)1(2
24)1(2
1n
nnnn
)1(21
)1)(2(2)1)(2(
)1(223
21 2
+−
=
+−−−
=
++−
−=
nn
nnnn
nnn
n
A does not join the Japanese auction, but observes its outcome and can therefore observe the
(n-2) lowest signals, from which he can estimate the distribution of the remaining two signals.
He can calculate the common value using the procedure described in Part 2 Chapter 5.6.
2−nC
+−++++
−= − 2
ˆ)(...
11
221abknsss
n n
E[C ] 2−n
+
++−
++−
+++
++− )1(2)1(2
212...
12
11
11
nn
nn
nn
nnn=
=
+−
++−+
−++
− )1(222
11
)1(2)1(
11
nn
nnn
nnn
n
=
+−
++−
−++
− )1(222
)1(224
)1(211 2
nn
nn
nnn
n
=
+−
− )1(211 2
nnn
n
98
=
+−
− )1(2)1(
11
nnn
n
= )1(2 +n
n
When A wins, his payoff is determined by:
Payoff 132 )1()1( −−− −−−−+= nnnn sCsC λλλλ
A’s expected payoff, when he is the winner:
E[P] ][)1(][][)1(][ 132 −−− −−−−+= nnnn sECEsECE λλλλ
= 11)1(
)1(21
1)1(
)1(2 +−
−−+−
−+
−++ n
nnn
nn
nn λλλλ
= 1
1)1()1(2
1+
−++ nn
λλ
= 1
1)1(2
1+
++
−nn
λ
=)1(2
11
1+
−+ nn
λ
When B wins, his payoff is determined by:
Payoff 123 )1()1( −−− −−−−+= nnnn sCsC λλλλ
B’s expected payoff, when he is the winner:
E[P] ][)1(][][)1(][ 123 −−− −−−−+= nnnn sECEsECE λλλλ
= 11)1(
)1(21)1(
)1(21
+−
−−+
−+
−++−
nn
nn
nn
nn λλλλ
= 1
1)1()1(2
1+
−++
−nn
λλ
= 1
1)1(2
3+
++
−nn
λ
=)1(2
31
1+
−+ nn
λ
99
c.) Calculations when A is truthful and B snipes:
Like b.), but payoffs of A and B are reversed.
d.) Calculations when all bidders snipe:
When no bidders participate in the Japanese auction, the final price is zero. In the subsequent
second-price sealed-bid auction naïve bidders could believe that no other bidders are
participating and could take therefore take the common value to be zero, but here bidders are
forward looking – knowing that sniping is possible – and use the maximum-likelihood
method to estimate the common value.
The winner’s payoff is thus calculated as follows:
112 )1(ˆ)1( −−− −−−−+= nnnn sCsC λλλλ
whereby )1(22 +
=− nn
nC (see above)
The expected winner’s payoff:
E[P] )2
1])([][( 1λ
−−= −nn sEsE
11)1(
)1(21
1)1(
)1(2 +−
−−+−
−+
−++
=nn
nn
nn
nn λλλλ
1
1)1()1(2
1+
−++
=nn
λλ
A’s final bid is: AA ss )1(
2λλ −+
B’s final bid is: BB ss )1(2
λλ −+
A wins when AA ss )1(
2λλ −+ > B
B ss )1(2
λλ −+ , i.e. 2A
Ass λ− >
2B
Bss λ− .
B wins when 2A
Ass λ− <
2B
Bss λ− .
The probability of winning is ½ for each bidder.
2.1.3.3 Expert-Amateur
Bidder i’s valuation is determined by ii sCp )1( λλ −+= . There is one bidder with a 0=λ .
He has an independent private valuation and is known as the expert. All other bidders have a 100
0>λ , they are amateurs. A bidder’s λ is known by all bidders. This case could for example
depict an Internet auction, where there is one person who has an entirely private valuation.
When planning his bidding strategy, he forms a belief on the kind of bidders he will encounter
in the auction. He assumes for example that the other bidders are merely Internet surfers who
develop an ad hoc interest in the good. They have a private estimate of the item’s value, but
include the mean valuation of the others in their own valuation. An expert expecting such
rivals has to consider carefully at what point in time he should reveal his true valuation.
ssp =
ttp )
stp )
−1
−1
Table 20: Payoffs in the expert-amateur case. Payoffs shown are expected payoffs.
Amateur
Truthful Snipe
Trut
hful
+−
++ )1(2
21
1n
nn
ptt λ ,
++
−)1(2
1)1(nn
nptt λ
+−
++ )1(2
11
1nn
npts λ ,
+
−+
−)1(21
1)1(nn
npts λ
Exp
ert
Snip
e
+−
++ )1(2
11
1nn
npst λ ,
++
−)1(2
1)1(nn
npst λ
+−
++ )1(2
11
1nn
npss λ ,
+
−+
−)1(21
1)1(nn
npss λ
tttsst ppp >=
= Prob )1()1(2 ME s
nns λλ −++
>(
= Prob )1()1(2
1ME s
nns λλ −++
( −>
As can be seen from the table above, the expert’s payoff is higher when he chooses to snipe
than when he bids truthfully, given that the amateur bids truthfully. The payoff of the expert is
equally high whether he chooses to snipe or to bid truthfully, given that the amateur snipes.
The expert thus has a weakly dominant strategy, to snipe. It is a weakly dominant strategy for
the amateur to bid truthfully, because his payoff is slightly higher when all bidders bid
truthfully. The equilibrium reached will therefore be (snipe, truthful), which yields the same
payoff as the case (snipe, snipe).
101
The amateur’s probability of winning is dependent on n and λ . This is because the amateur
includes the other bidder’s private signals in his valuation and by doing so, lowers his
valuation in comparison to his private signal. The probability of being the high-signal holder
is n1 , but the probability of winning is now higher for the expert than for the amateur
(because the amateur’s 0>λ is always higher than that of the expert, which is zero). The
probability of winning is at least as high for a bidder when he snipes, as when he bids
truthfully.
a.) Calculations when all bidders are truthful:
In this case all bidders participate in the Japanese auction and there is no more bidding in the
subsequent second-price sealed-bid auction. The payoffs are as follows,
Expert’s payoff, when the expert wins: 12 )1( −− −−− nnn sCs λλ
Amateur’s payoff when the amateur wins: 12 )1( −− −−+ nnn ssC λλ
=−2nC)1(2 +n
n
Expert’s payoff when expert wins:
1)2( )1( −− −−−= nnnE sCsP λλ
11)1(
)1(21 +−
−−+
−+
=nn
nn
nn λλ
)1(21
11
1+
−+−
++
=nn
nn
nλλ
)1(2
)1(21
1+−−
++
=n
nnn
λ
)1(2
21
1+−
++
=n
nn
λ
Amateur’s payoff when amateur wins:
1)2( )1( −− −−+= nnnM ssCP λλ
1)1(21
1+
−+
++
=n
nnn
nλλ
102
)1(2
21
1+
−+
+=
nnn
nλ
)1(21
1+
−+
=nn
nλ
The final bid of the expert is . The expected final bid of the amateur is Es
Msnn )1(
)1(2λλ −+
+.
The expert wins if ME snns )1(
)1(2λλ −+
+> .
The amateur wins if ME snns )1(
)1(2λλ −+
+< .
Es is distributed U ∼ [ ]1,0 .
Msnn )1(
)1(2λλ −+
+is distributed, U ∼
−+
++)1(
)1(2,
)1(2λλλ
nn
nn .
The probability of winning is dependent on and on n λ . The expert’s probability of winning
is larger when λ is larger and n is smaller. The reverse is true for the amateur.
b.) The expert snipes, the others bid truthfully:
When only one bidder participates in the Japanese auction, the Japanese auction ends at a
price equal to zero, because the rule says that the auction ends when the penultimate bidder
quits the auction, which is zero in this case. In the subsequent second-price sealed-bid auction
all bidders post a bid equal to their reservation price, ii sCp )1( λλ −+= . As no bids were
posted in the Japanese auction, bidders could assume that the common value thus is zero,
however as forward-looking bidders they assume that it is still possible that other bidders are
participating and estimate the common value as described in 2.1.3.2 c.).
The final bid of the expert is . The final bid of the amateur is Es Msnn )1(
)1(21 λλ −+− .
When the expert wins, he receives the following payoff:
13 )1( −− −−−= nnnE sCsP λλ
][)1(][][ 13 −− −−−= nnn sECEsE λλ
103
= 11)1(
)1(21
1 +−
−−+−
−+ n
nnn
nn λλ
= )1(2
111
11
−−
−+−
++ n
nnn
nλλ
=)1(2
11
1+−
++ n
nn
λ
When the amateur wins, he receives the following payoff:
12 )1( −− −−+= nnnM ssCP λλ , whereby =− ][ 2nCE)1(2 +n
n
21 −− +−−= nnnn Csss λλ
The amateur’s gain from winning is Mn sC )1(2 λλ −+− and not Mn sC )1(3 λλ −+− , because ex
post all signals are revealed and he can calculate the common value with full information.
In case the amateur wins, he expects the following payoff:
][][][][][ 21 −− +−−= nnnnM CEsEsEsEPE λλ
)1(211
11 +
++
−+−
−+
=nn
nn
nn
nn λλ
)1(2)1(2
21
1+
++
−+
=nn
nn
nλλ
)1(21
1+
−+
=nn
nλ
The final bid of the expert is . The final-bid of the amateur is Es Msnn )1(
)1(21 λλ −++− .
The expert wins, if ME snns )1(
)1(21 λλ −++−
> .
The amateur wins, if ME snns )1(
)1(21 λλ −++−
< .
Es is distributed U ∼ [ ]1,0 .
104
Msn
n )1()1(2
1 λλ −++− is distributed, U ∼
−+
+−
+− )1(
)1(21,
)1(21 λλλ
nn
nn .
c.) The expert bids truthfully, the amateurs snipe:
In this case the Japanese auction takes place with (n-1)-bidders. From the Japanese auction,
they calculate the final common value when the (n-3)rd bidder quits the auction. Calculation of
the common value is presented above in Chapter 2.1.3.2.
The final bid of the expert is . The final bid of the amateur is Es MM ss )1(2
λλ − .
When the expert wins, his payoff is calculated as follows:
EP 11 )1(ˆ−− −−−= nnn sCs λλ
11 )1(
2 −− −−−= n
nn s
ss λλ
)2
1(1λ
−−= −nn ss
The expert’s expected payoff in case of winning:
][ EPE )2
1]([][ 1λ
−−= −nn sEsE
)2
1(11
1λ
−+−
−+
=nn
nn
)1(2)1(
11
+−
++
=nn
nλ
)1(2
)1(2+−+
=n
nλ
When the amateur wins, his expected payoff is calculated as follows:
MP = 12 )1( −− −−+ nnn ssC λλ
=−2nC)1(2 +n
n
The amateur’s payoff in case of winning: 105
][ MPE ][][)1(][ 12 −− −−+= nnn sEsECE λλ
= 11
1)1(
)1(2 +−
−+
−++ n
nn
nnn λλ
= )1(211
1+
++
−+ n
nn
nn
λλ
= )1(2
21
1+
−+
+ nnn
nλ
= )1(21
1+
−+ n
nn
λ
The final bid of the expert is . The final-bid of the amateur is Es MM ss )1(2
λλ −+ .
The expert wins, if MM
E sss )1(2
λλ −+> .
The amateur wins, if MM
E sss )1(2
λλ −+< , i.e. when 2M
MEsss λ−< .
Es is distributed U ∼ [ ]1,0 .
2M
MEsss λ−< is distributed U ∼ [ ,0 ]
21 λ− .
If 2
1 λ−>Es , then the expert always wins. This event occurs with probability
2λ .
If 2
1 λ−<Es , then the expert wins half the time. This event occurs with probability
−
21
21 λ .
The probability that the expert wins is thus, 42
12
λλ−+ =
421 λ+ . The probability that the
amateur wins is 42
1 λ− .
d.) All bidders snipe:
In this case no bids are posted in the Japanese auction, which ends with a final price of zero.
Bidders could assume from the outcome of the Japanese auction that there are no other
bidders participating, however knowing of the possibility that sniping is possible, they take
106
this into account and estimate the common value using the maximum likelihood method (see
above under 2.1.3.2.c.).
The expert’s payoff when he is the winner:
EP 11 )1(ˆ−− −−−= nnn sCs λλ
)2
1(1λ
−−= −nn ss
The expert’s expected payoff when winning:
][ EPE )2
1]([][ 1λ
−−= −nn sEsE
)1(2
)1(2+−+
=n
nλ
The amateur’s payoff, when he is the winner:
MP 12 )1( −− −−+= nnn ssC λλ
The expected amateur’s payoff when he wins:
][ MPE ][][)1(2
][1
2−
− −−+= nnn sEsE
CEλλ
11
1)1(
)1(2 +−
−+
−++
=nn
nn
nn λλ
1
11)1(2 ++
+−
+=
nnn
nn λλ
)1(21
1+
−+
=nn
nλ
The final bid of the expert is . The final-bid of the amateur is Es MM ss )1(2
λλ −+ .
The expert wins, if MM
E sss )1(2
λλ −+> .
The amateur wins, if MM
E sss )1(2
λλ −+< , i.e. if 2M
MEsss λ−< .
107
Es is distributed U ∼ [ ]1,0 .
2M
MEsss λ−< is distributed, U ∼ [ ,0 ]
21 λ− .
If 2
1 λ−>Es , then the expert always wins. This event occurs with probability
2λ .
If 2
1 λ−<Es , then the expert wins half the time. This event occurs with probability
−
21
21 λ .
The probability that the expert wins is thus 42
1 λ+ . The probability that the amateur wins is
421 λ− .
2.1.3.4 Resulting Bidding Behaviour
Late bidding Result: In the model above I show that a bidder will receive higher payoff by
bidding zero in period one (during the Japanese auction) and waiting until period two (the
sniping period) to bid his true valuation, than by bidding his true valuation in period one.
This is true in the expert-amateur setting; i.e. when a bidder has an independent private
valuation and expects his rivals to have some uncertainty with regard to their valuation
)0( >λ . In the symmetric setting, i.e. when all bidders have a valuation composed of a
common and a private value element with the same λ , it is better for a bidder to snipe, if his
rivals are not sniping; in case the rivals are sniping, the payoff is equal whether a bidder
snipes or not. Because all bidders devise their bidding strategy for the case of being the high
signal holder, it is expected that all bidders will engage in sniping. The expected seller
revenue is equal to:
][)1(2
][][ 1
1)( −
− −+= nn
snipingHardClose sEsE
RE λλ
11)1(
)1(21
+−
−++−
=nn
nn λλ
11
)1(21
11
+−
−+−
++−
=nn
nn
nn λλ
108
)1(2
22111
++−−
++−
=n
nnnn λ
)1(21
11
+−
−+−
=nn
nn λ
2.1.3.5 Revenue Comparison: Hard Close and Automatically Extended Auction
In the automatically extended auction, the revenue would be equal to the case when all
bidders bid truthfully:
=][ AutoExtRE ][)1(][ 1−−+ nsECE λλ
11)1(
)1(22
21
+−
−+
+
−+=
nn
nnn λλ
The revenue in the automatically extended auction is equal to the revenue in the hard close
auction when all bidders bid truthfully. It is shown that at least bidders in the hard close
auction will engage in sniping, which lowers seller revenue. In turn the hard close auction
leads to lower seller revenue than the automatically extended auction. The Ebay auction
would thus be expected to yield lower seller revenue than the Amazon auction.
2.1.4 Prestige Value Model
One type of interdependent valuation that can be found is that of bidders having a valuation
with additive utility. Bidders of this type receive utility from their own private valuation and
utility from the valuation a participating expert bidder attaches to the good.
Similar interdependent valuations are used in Krishna (2002, p.126), Izmalkov (2001), and
Perry & Reny (2001). An interdependent valuation of this kind might also arise when it is
painful for a bidder to lose, that is, when his payoff from losing an auction is negative. It is
equivalent to say that a bidder of this kind attaches extra value to winning and this value is
thus dependent on the other bidder’s valuation.
A valuation of this kind may be realistic for very rich buyers when bidding for unique goods
such as Impressionist art. When a Van Gogh painting is offered at an auction, bidders will
most probably have an idea of their maximum willingness to pay, but are likely to be flexible
within a certain range – as long their budget constraint is not surpassed. Van Gogh paintings
are rarely offered on the market, so that a true fan (someone who values the possession of the
109
painting very highly) with a high budget will adapt his valuation to the “market”, i.e. his
rivals’ bids.
A valuation of this kind might also be realistic for goods where the budget constraint is not a
strong influence, for example in the case of cheap but unique paraphernalia offered on the
Internet.
2.1.4.1 Symmetrical Case
Two bidders: Bidder A and bidder B.
Two periods. Bidders can bid in period one and/or in period two. Bids made in period one are
publicised before period two. The bidder who posts the highest bid wins. The final price is
determined by the high-bid of the non-winning bidder.
Each bidder draws a private signal from U ∼[ before the auction begins. The bidders do
not know the signal drawn by the other bidder, they only know the signals are uniformly
distributed on [ .
s ]1,0
],0 b
The valuations of the two bidders are as follows:
BAA ssV α+=
ABB ssV α+=
s ...private signal
V ...valuation
itV …valuation of bidder in period t
b ...bid
210 <<α
B
Early Snipe
Early )1)((21 α−− BA ss , )1)((
21 α−− AB ss )1)((
83 α−− BA ss , )1)((
85 α−− AB ss
A Snipe )(
85
BBA sss −+α , )(83
AAB sss −+α )(21
BBA sss −+α , )(21
AAB sss −+α
Table 21: Payoff Table for symmetrical bidders
110
The expected payoff is determined by the probability of being the high bidder multiplied by
the payoff that this bidder receives when he is the winner.
A has a higher expected payoff from sniping than from truthful bidding, given that B bids
early. This is because by bidding in period two, A avoids the prestige-effect of B that drives
the price up. A has a dominant strategy to snipe. The same is true for B, respectively. The
equilibrium will therefore be (snipe, snipe). This makes it impossible for a bidder to infer
anything about the other bidder’s valuation from his bid timing, because in any case a bidder
will bid late. Calculations are given below.
The probability of winning is equal to the probability of being the high-signal holder in the
cases (snipe, snipe) and (truthful, truthful). This is because both bidders have the same
informational structure, when they both choose to bid at the same time. In these two cases the
probability of winning is ½ for each bidder and thus expected payoffs are equal to the payoffs
given in the respective boxes above multiplied by ½.
When one bidder decides to snipe and the other bids truthfully, the probability of winning is
higher for the bidder who snipes than for the truthful bidder. The sniper wins when his signal
is greater than his truthful rival’s signal multiplied by )1( α− . The sniper’s probability of
winning is greater than that of the truthful rival, because 0>α . This will increase a bidder’s
incentive to engage in sniping, thereby strengthening the result given above that the
equilibrium outcome will be (snipe, snipe).
a.) Both bidders bid early:
Valuations in the first period: V AA s=1
V BB s=1
and both bid accordingly.
Second Period:
Now, A forms the belief that B’s valuation V BB s=
B forms the belief that A’s valuation V AA s=
111
A updates his valuation to V BAA ss α+=2
B updates his valuation to V ABB ss α+=2
and both bid accordingly.
A’s payoff, when he wins )1)(( α−−= BA ss
B’s payoff when he wins )1)(( α−−= AB ss
Each bidder’s probability of winning is equal to ½.
b.) A snipes, B bids early
First Period:
BB sV =1 and B bids accordingly
AA sV =1 , but A does not post a bid
Second Period:
BAAA bsVb α+== 22
B does not post a bid in the second period.
A’s payoff, when he wins: BBA sss −+α
A wins if BBA sss −>α
B’s payoff, when he wins: BAAB ssss αα −−+ = )1)(( α−− AB ss
B wins if BBA sss −<α
A’s probability of winning is higher than B’s probability of winning.
A wins if BBA sss α−> , i.e. if BA ss )1( α−> .
UsA : ∼[0,1]
Bs)1( α− :U ∼[ )]1(,0 α− , because ∼[0,1] UsB :
If )1( α−>As , A always wins. Prob[ )]1( α−>As =1 )1( α−− = α .
If )1( α−<As , A wins half of the time. Prob[ )]1( α−<As =2
1 α− .
Prob[ ])1( BA ss α−> =2
1 αα −+ =
21 α+
112
The probability that A wins is2
1 α+ and the probability that B wins is 2
11 α+− .
Expected probability that A will win:
E[Prob 625.085
225.1
225.01
2][1]])1( ===
+=
+
=−>αα EEss BA[
Expected probability that B will win:
E[Prob 375.083
851
225.11
225.011
2][11]])1( ==−=−=
+−=
+−=−<
αα EEss BA[
c.) A bids early, B snipes
First Period: AA sb =1
B does not post a bid.
Second Period:
B updates his valuation to V ABABB bsss αα +=+=2 .
A does not post another bid.
A’s payoff when he wins: )1)(( α−− BA ss . A wins if ABA sss α+> , i.e. if BAA sss >−α .
B’s payoff when he wins: ( )1)( α−− BB ss . B wins if AAB sss >+α , i.e. if AAB sss α−> .
B’s probability of winning is 2
1 α+ , his expected probability of winning is 85 .
A’s probability of winning is 2
11 α+− , his expected probability of winning is
83 .
For calculations see b.) above.
d.) Both bidders snipe
1st Period: no bids
2nd Period: and AA sb =2 BB sb =2
A’s Payoff, when he wins, i.e. if : BA ss > BBA sss −+α
B’s Payoff, when he wins, i.e. if : BA ss < AAB sss −+α
A wins with probability ½, B wins with probability ½.
113
2.1.4.2 Expert-Amateur Case
In the section below, bidding behaviour in the prestige value model is looked at for the case
where one bidder has certainty about his valuation, he is known as the expert; and the other
bidder is uncertain about his valuation, he is an amateur. Both bidders have a valuation of the
type presented above: V jii ss α+= , but the expert has an 0=α , while the amateur has an
0>α . This case is realistic for Internet auctions, where bidders are often very different and
can be thought to have asymmetrical valuations. One bidder, sure of his valuation, competes
with a bidder who is uncertain about his valuation and for whom it is important how highly
the expert values the good.
Two bidders: an expert E and an amateur M
Two periods. Bidders can bid in period one and/or in period two. Bids made in period one are
publicised before period two.
The valuations of the two bidders are as follows:
EMM ssV α+=
EE sV =
b ...bid
s ...private signal
V ... valuation of bidder i in period t
210 <≤α
In order to find out whether it is more profitable for the expert to bid truthfully or to engage in
sniping, his payoffs are calculated and presented in the table below. The corresponding
calculations can be found from a.) to d.).
Amateur
Truthful Sniping
Truthful )(83
EME sss α−− , )(85
EEM sss −+α )(83
EME sss α−− , )(85
EEM sss −+α
Expert Sniping )(
21
ME ss − , )(21
EEM sss −+α )(21
ME ss − , )(21
EEM sss −+α
Table 22: Expected payoff table: Prestige value model for the expert-amateur case.
114
The expert benefits from sniping. His expected payoff is higher. By bidding in period two, the
expert can recreate a second-price sealed-bid auction and avoids the prestige-effect of the
amateur that drives the price up. It is a dominant strategy for the expert to engage in sniping.
The same is true for the amateur. This makes it impossible for a bidder to make an inference
on the rival’s valuation, because in any case the rival will snipe.
The probability of winning is higher when a bidder engages in sniping than when he bids
truthfully. Calculations can be found below from a.) to d.).
Calculations:
a.) Calculations when both bidders bid early
First Period: EEE sVb == 11
MEM sVb == 11
Second Period:
Amateur updates valuation to V EMM ss α+=2 and bids accordingly.
Final Price:
If the expert wins the final price is EM ssP α+= .
If the amateur wins the final price is EsP = .
Expert’s payoff when he wins: EME sss α−− .
Amateur’s payoff when he wins: EMEEM sssss )1( αα −−=−+ .
If EEM sss >+α , i.e. EEM sss α−> , then the amateur wins.
If EME sss α+> , i.e. EEM sss α−< , then the expert wins.
The expert’s probability of winning is 2
11 α+− , his expected probability of winning is
83 .
The amateur’s probability of winning is 2
1 α+ , his expected probability of winning is 85 .
For calculations see 2.1.4.1 b.) above.
115
b.) Calculations when the amateur bids early and the expert bids late
First Period:
1Mb MM sV ==
The expert does not post a bid.
Second Period: EEE sVb ==2
Final price if expert wins: Ms
Final price if amateur wins: Es
Expert’s payoff if he wins: . Expert wins if . ME ss − ME ss >
Amateur’s payoff if he wins: EEM sss −+α . Amateur wins if ME ss < .
Both bidders have a probability of winning of ½.
c.) Calculations when the expert bids early and the amateur snipes
First Period:
EEE sVb ==1
Amateur does not post a bid.
Second Period:
Amateur updates his valuation to EMMM bsVb α+==2 .
Expert does not post a bid in the second period.
Final Price: EM bs α+
Expert’s Payoff: EME sss α−− , E wins if EEM sss α−< .
Amateur’s Payoff: EEM sss −+α , wins if ME ss >− )1( α .
M wins if )1( α−> EM ss . E wins if )1( α−< EM ss .
The expert’s probability of winning is 2
11 α+− , his expected probability of winning is
83 .
The amateur’s probability of winning is 2
1 α+ , his expected probability of winning is 85 .
For calculations see 2.1.4.1 b.) above.
116
d.) Calculations when both bidders snipe
First Period: no bids
Second Period:
EEE sVb ==2 and b MMM sV ==2
Final Price: Ms
Expert’s Payoff when he wins: . He wins if . ME ss − ME ss >
Amateur’s payoff when he wins: EEM sss −+α . He wins if . EM ss >
Each bidder’s probability of winning is ½.
2.1.4.3 Result of Prestige Value Auctions
Irrespective of whether bidders have symmetrical or asymmetrical valuations, all bidders will
engage in sniping and thus the equilibrium final price in the prestige-value auction will
always be equal to . The equilibrium seller’s payoff is thus equal to that of a second-
price sealed-bid auction.
ndHighests2
2.1.4.4 Payoffs in the Automatically Extended Auction
a.) Symmetrical Valuations With α > 0:
The final price and thus the seller revenue can be infinitely high in an Internet auction ending
by automatic extension. In an Internet auction of the Amazon type, there are not only two
periods in which to post bids, but instead the auction continues as long as bids are posted.
When bidders have a valuation of the type V jii ss α+= , bidders will want to revise their
valuation upwards every period. This process can continue infinitely.
Imagine for example the following scenario in the Japanese auction: In the beginning of the
auction a bidder i plans to stay in the auction for at least until the counter reaches his private
signal. But when the current price is equal to −is minimum increment, bidder i can infer that
the high bidder’s valuation must be greater than or equal to (because the current price is
calculated as the second highest bid plus a minimum increment). Bidder i thus updates his
valuation to V
is
jii ss α+=
is
, whereby he can deduce that , therefore updates his valuation
toV
ij ss ≥
ii s α+= . The high bidder also updates his valuation. The auction continues infinitely,
117
thereby raising the seller revenue to extreme levels. A self-reinforcement effect of this kind
creates a clear seller preference for auctions with automatic extension.
b.) Asymmetrical valuations
In this case bidders have valuations of the kind V jii ss α+= , but not all bidders have the
same α value, because at least one bidder has an 0=α . Here there are two bidders, the
amateur with 0>α and the expert E with 0=α . In the beginning of the auction, both bidders
plan to stay active at least until the counter reaches their private signal. When the current price
is equal to min. increment, the amateur knows that the expert’s signal must be greater
than or equal to his own, . The amateur then updates his valuation to: V
−Ms
ME ss ≥ EMM ss α+= .
The auction ends when the first bidder exits the auction. This is the case when the current
price is equal to or when the minimum increment is smaller than V . The auction ends
when the first of the two ending conditions is reached. Interdependent valuations of the kind
presented above create a strong incentive for experts to engage in sniping, because the
amateur - when informed about the expert’s valuation - can revise his valuation to values
higher than that of the expert.
Es M
2.1.5 Payoff Comparison: Hard Close and Automatically Extended Auction
In the model 2.1.3, where bidders have valuations of the type ii sCp )1( λλ −+= , the
equilibrium outcome is that all bidders engage in sniping. The seller revenue from the hard
close auction is higher than that from the automatically extended auction. This is true for both
the symmetrical and the asymmetrical case.
In the prestige value model the seller revenue from the equilibrium outcome in the hard close
auction is lower than that from the automatically extended auction. This is true irrespective of
whether bidders have symmetrical or asymmetrical valuations (shown above).
2.1.6 Milgrom-Weber Model
2.1.6.1 Bidding in the hard close auction
The hard close and the automatically extended auction are both sub-variants of the English
auction. If we wish to compare the two formats in the general symmetrical model of
interdependent valuations, as proposed by Milgrom-Weber, we can use their model of the
118
Japanese auction63 to model the automatically extended auction (see above for the reasoning).
The hard close auction is characterised by the fact, that the last and thus highest bids can be
submitted secretly, unobservable to the rest of the bidders. The hard close auction can be
modelled as the Japanese auction with one modification: the bidder with the highest private
signal has the possibility to participate secretly in the Japanese auction. In this case the
Japanese auction is of the usual kind, except for the fact that all bidders (other than the high-
bidder) erroneously believe that there are only (n-1)-bidders participating in the auction.
When calculating their valuation based upon the exit prices observed, they are not able to take
the highest signal into account, as it is unknown to them. All bidders except the high-bidder
therefore believe to have a lower valuation than they actually have and will exit too soon. The
high-signal bidder will win the auction, but at a final price lower than in the case when his
participation is observable.
Modifying the Japanese auction, so that the high-signal bidder can participate secretly, leads
to lower seller revenue, because all other bidders calculate their valuation under the false
belief that there are only (n-1) bidders participating, who have the lower (n-1) signals drawn
from the distribution.
If the high-signal owner has the possibility to participate secretly, he will choose to do so,
because it lowers the final price, thereby increasing his payoff. It is the interdependency of
signals that creates an incentive for the high-bidder to hold back his signal. The hard-closing
auction of the Ebay type allows bidders to hold back their signal. When a bidder decides
whether to participate openly or secretly, he will base his decision on the assumption that he is
the high-signal owner. In all other cases he does not have the possibility to attain a positive
payoff anyway. That is why the decision is looked at from the perspective of the high-signal
owner.
Because all bidders base their bidding-decision on the assumption that they are the high-signal
owner, that would result in all bidders participating secretly. In this case the outcome would
resemble that of the second-price sealed-bid auction in the Milgrom-Weber model. The proof
for why the expected final price is lower when the highest signal is not revealed is similar to
the proof presented in Milgrom-Weber (1982) for why the Japanese auction leads to higher
63 For an explanation of the Milgrom-Weber Model see Part One Chapter 3.3.2.
119
seller revenue than the second-price auction (and to the proof of why it is more profitable for
the seller to reveal public information).
A bidder’s valuation is a function of all bidders’ quality signals . .
The function u is increasing in all its arguments and symmetric in { .
nXXX ,...,, 21
ijjX ≠}
)(: ii XuV =
Bidder 1 observes the quality signal . 1X
121 ,...,, −NYYY are defined to be the signals in descending order of respectively. NXXX ,...,, 32
Therefore bidder 1 has a valuation )...,,( 1111 −= NYYXuV .
Affiliation of the signals ( implies that the conditional expected values of V , ),1 YX 1
=:),,( 211 yyxw [ 1VE ,1 11 yY1 xX = ]22, yY ==
and v =:),( 21 yy [ 1VE ,11 yY = ]22 yY =
are strictly monotonically increasing in all their arguments. v is the conditional
valuation of bidder 1 after having observed the signals Y
),( 21 yy
1y1 = and Y 2y2 = . is the
conditional valuation of bidder 1 after having observed the signal realizations ,
, Y .
),,( 211 yyxw
11 xX =
11 yY = 22 y=
Expected price in the normal Japanese auction compared to the secret Japanese auction:
][ JapPE = ]}{),,([ 11111 YXXYYwE >
]}{),([ 2122 YYYYvE >≥
= . ][ SecretPE
When one bidder can keep his valuation secret and this is the bidder with the highest signal,
all other bidders form a mistakenly lower expected value of the good. The second-highest
bidder determines the price and he will bid too low in the secret auction (hard close auction).
In the automatically extended auction, where no bidder can keep his signal secret, the second-
highest bidder will be able to take account of the highest signal when forming his valuation. It
follows from the assumption that u is increasing in all its arguments, that the expected price is
lower in the hard closing auction than in the automatically extended auction.
120
2.1.7 General Prediction for Interdependent Valuations
In the general model, in the prestige value model and in the Milgrom-Weber model bidders in
the hard close auction bid zero in the first period (the Japanese auction) and bid their true
valuation in the sniping period. A bidder holds back his valuation until the sniping period
when he knows (or believes) that his rivals do not have independent private valuations, but
are instead uncertain about their valuation. The advantage of late bidding is that the expected
price is lowered and the expected payoff increased, without changing the probability of
winning the auction.
In the general bidding model, the prestige-value model, and the Milgrom-Weber Model, the
expected auction revenue of the hard close auction is higher than that in the automatically
extended auction.
2.1.8 Late bidding with Respect to the Ending-Rule
Late bidding (sniping) is expected in the hard close auction when bidders expect the second-
highest bidder to have a valuation of λ greater than zero (shown above in Part Five Chapters
2.1.3, 2.1.4 and 2.1.6).
No late bidding is expected in the automatically extended auction.
2.1.9 Late Bidding According to Good Type
No late bidding is expected in automatically extended auctions. Predictions for late bidding in
hard close auctions for computers, cars and paintings:
a.) Computers: Bidders are expected to have a λ equal to zero and believe others to have a
λ equal to zero, too. Valuations are expected to lie within a very narrow distribution. No late
bidding is expected.
b.) Cars: Some bidders are expected to have interdependent valuations, due to uncertainty
about their valuation. Late Bidding is expected.
c.) Art: Many bidders are expected to have interdependent valuations due to fashion and
prestige reasons. Late Bidding is expected.
121
2.2 Seller’s Choice of Ending-Rule
Clearly, sellers prefer a hard close to an automatically extended ending for selling computers.
In case of cars, there are more successful auctions ending by automatic extension than hard
close and the same is true for art auctions.
The seller’s preference for auctions ending by automatic extension can be explained with the
results from Section 2.1 on late bidding. In this section it is shown that an auction ending by
hard close yields lower seller revenue than an automatically extended auction. Based on this
result, sellers are expected to prefer automatically extended auctions.
Sellers prefer the hard close ending in case of computers, because computers are readily
available at a known market price. Computer auctions do not conform to the model presented
in Section 2.1, because participating bidders have no uncertainty about the true value. The
reason why sellers prefer the hard close ending for computers is because buyers aiming for the
lowest possible price will prefer shorter and time-constrained auctions that accumulate as few
rival bidders as possible.
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3. Empirical Investigation
As introduced in Part Five Chapter Two, economists have begun to study Internet auctions.
Bajari and Hortacsu (2000) use Ebay coin auctions to estimate the winner’s curse and the
expected profit necessary to motivate entry. Katkar and Lucking-Reiley (2001) study Ebay
Pokemon auctions to find a seller’s motivation for using reserve prices.
Roth and Ockenfels (2000) were the first to publish empirical research on the late bidding
phenomenon. They study the last twelve hours of Ebay and Amazon auctions and find that
many bids are submitted toward the end of these twelve hours. Furthermore, they find time-
invariance when plotting the cumulative number of bids submitted against time for the last
twelve hours, six hours, three hours, last hour, and last ten minutes.
In the following section I empirically investigate my theoretical late bidding predictions
(presented in Part Five Chapter Two). Late bidding is expected in the hard close auction but
not in the automatically extended auction. Especially in the case of asymmetrical valuations
(the Expert-Amateur model), the expert is expected to enter the hard-closing auction in the
very last moments of the auction. In all three models the expected seller’s revenue is higher in
the automatically extended auction than in the hard close auction. Therefore sellers are
expected to prefer the automatically extended auction.
Yahoo auctions are chosen for investigating the theoretical predictions, because Yahoo offers
the seller the possibility to choose between the hard close and the automatically extended
auction. Using Yahoo data, I can compare bidding behaviour for two different ending rules
and determine which ending rule the seller chooses more often. Data is taken from three
categories of goods in order to further differentiate the results according to the type of good.
As noted in Part Five Chapter 2.1.9, different types of goods imply different bidder
valuations. Computer buyers are expected to have little uncertainty about their valuation;
0=λ , so late bidding is not expected. Car buyers are expected to have some uncertainty
about their valuation – asymmetrical valuations (Expert-Amateur case) are expected resulting
in late bidding by at least some bidders (the experts). Buyers of Art sold on Yahoo auctions
are expected to have some uncertainty about their valuation due to reputational effects,
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therefore the behaviour is expected according to the prestige value model – possibly with
asymmetrical valuations. Late bidding is expected.
A sample of three hundred Yahoo auctions – completed between November 2000 and January
2001 – downloaded from the Yahoo website, were included in the data-analysis. Only
auctions with at least two bidders were included. Hundred auctions were taken from each of
three categories: computers, cars and art. Computer auctions were taken from the sub-
categories: laptops and personal computers; cars from the sub-category general, and art from
the sub-categories paintings and drawings.
The following central hypotheses were tested empirically:
Hypothesis 1: Internet auctions are characterised by late bidding, where late bidding means
that there is a strong over-proportional price increase at the end of the auction.
Hypothesis 2: The over-proportional price increase toward the auction end is stronger for
auctions with a hard close than for those with an automatically extended ending.
Hypothesis 3: There is more late bidding when there is uncertainty about the value of the
goods, therefore more late bidding in art than in computer-auctions.
Hypothesis 4: The winning bidder enters the auction shortly before the auction end.
Hypothesis 5: The winning bid is posted as a proxy bid.
Hypothesis 6: Sellers prefer automatically extended auctions.
3.1 Late Bidding
3.1.1 Existence of Late bidding
Hypothesis 1: Internet auctions are characterised by late bidding, where late bidding means
that there is a strong over-proportional price increase at the auction end.
3.1.1.1 Complete Auction Duration
Hypothesis 1 is tested by plotting the bidding path, the current price against time. Hundred
auctions were surveyed for each of the three product groups. The seller is allowed to choose
the auction duration; therefore auctions are of different durations. The starting and final price
was also different for almost every auction surveyed. The different time lengths and different
minimum and end prices were made comparable by setting the final price equal to one and the
minimum price chosen by the seller equal to zero. On the x-axis the starting time was
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normalised to zero and the end time set equal to one. The auction time duration was then split
into hundred intervals from zero to one, i.e. intervals from 0, 0.01, 0.02, 0.03…to 1. The y-
axis in the graphs below represent the increase in the normalised price from one period to the
next, whereby the minimum price in period 0.0 was taken as 0 and the final price in period 1.0
was taken as one.
Observation 1: Internet auctions are characterised by late bidding; late bidding meaning that
there is a strong over-proportional price increase at the auction end. This is true for all three
categories of goods. The bidding paths for computers, art and cars are shown below. The price
increase is strongest in the last hundredth of the auction duration for all three categories of
goods.
Cars, Art, and Computers
0 0,05 0,1
0,15 0,2
0,25
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1Time
Pric
e In
crea
se Cars
Art
Computer
Figure 3
Computers
0 0,05 0,1
0,15 0,2
0,25
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Time
Pric
e In
crea
se
Figure 4
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Cars
0 0,05 0,1
0,15 0,2
0,25
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Time
Pric
e In
crea
se
Figure 5
Art
0 0,05 0,1
0,15 0,2
0,25
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Time
Pric
e In
crea
se
Figure 6
3.1.1.2 Last Twelve Hours
Auction durations vary widely from only one day to twelve days, it is thus interesting to find
out how high the price is twelve hours before the auction end and to magnify the time period
right before the auction end.
In the graphs below, the final price is normalised to one and the minimum price chosen by the
seller set to zero. This method makes auctions with different starting and final prices
comparable. The bidding curves shown in the graphs below represent the mean bidding curve.
Out of the hundred auctions in each category of goods, fifty were hard close and fifty
automatically extended auctions.
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Last Twelve Hours
00,10,20,30,40,50,60,70,80,9
1
0:00 2:24 4:48 7:12 9:36 12:00
Time
Pric
e
Cars
Computers
Art
Figure 7
Observing only the last twelve hours of the average auction path, we find that the price
increases very strongly shortly before the auction ends. This is especially true for art auctions:
twelve hours before the auction end, the current-price has only reached 52% of its final price
on average.
Last Hour
0,50
0,60
0,70
0,80
0,90
1,00
11:00 11:12 11:24 11:36 11:48 12:00
Time
Pric
e
Cars
Computers
Art
Figure 8
One hour before the auction end, the current price has only reached 73% of the final price in
the average art auction.
3.1.2 Late bidding: Dependency on Ending-Rule
Hypothesis 2: The over-proportional price increase toward the auction end is stronger for
auctions with a hard close than for those with an automatically extended ending.
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3.1.2.1 Complete Bidding-Path
In the graphs below, 0 indicates hard-closing auctions, 1 indicates automatically extended
auctions.
a.) Art
Art
00,10,20,30,40,50,60,70,80,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Time
Pric
e Art 0
Art 1
Figure 9
The mean bidding curve of hard closing and automatically extended art auctions are rather
similar. In both cases the bidding curve has the steepest rise at the end of the auction duration.
Sniping seems to be prevalent in all kinds of art auctions.
b.) Cars
Cars
00,10,20,30,40,50,60,70,80,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Time
Pric
e Cars 0
Cars 1
Figure 10
The mean bidding curve of car auctions with a hard close and car auctions with automatic
extension differ significantly. In auctions with automatic extension the current high bid rises
quickly (after about a tenth of the auction duration) to 40% of its final price. In auctions with
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a hard close, the steepest price rise takes place close to the end of the auction. Sniping occurs
with significantly higher intensity in hard closing car auctions than those ending by automatic
extension.
c.) Computers
Computers
00,10,20,30,40,50,60,70,80,9
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Time
Pric
e Computer 0
Computer 1
Figure 11
The largest increase in the current high bid occurs towards the very end of the auction. That
is, there is significant late bidding on Yahoo auctions. The amount of late bidding is slightly
higher for automatically extended auctions.
Observation 2: Late bidding is stronger for hard closing than automatically extended car
auctions. The opposite is true for art auctions.
When observing the complete bidding path, we find that the prediction that hard close
auctions lead to more late bidding than automatically extended auctions does not prove to be
correct for computer and art auctions. Hard closing auctions clearly lead to late bidding (the
bidding path lies below the 45° line until the very end of the auction duration), but so do
automatically extended auctions! The theoretical prediction corresponds to the empirical
findings regarding car auctions: there is significantly more late bidding in hard closing than
automatically extended auctions. Computer auctions show a little more late bidding for
automatically extended auctions.
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Explanations for the empirical finding that bidders do not treat automatically extended
auctions as expected, could lie in bidder irrationality (inexperience) or in the usage of a
“wrong” model. Before this question is analysed further the hypothesis is tested again, but this
time only for the bidding path in the last twelve hours of the auction duration. As stated
above, auction durations vary widely, it is therefore interesting to magnify the time period
right before the end. Roth and Ockenfels (2000) studied the cumulative number of bids
submitted in the last twelve hours of hundreds of Ebay and Amazon auctions. They find that
significantly more late bids are submitted in Ebay than in Amazon auctions. Here in contrast
we are not looking at the cumulative number of bids submitted, but at the price path plotted
against time.
3.1.2.2 Last Twelve Hours
Last 12 Hours
0,30,40,50,60,70,80,9
1
0:00 3:00 6:00 9:00 12:00
Time
Pric
e
Cars 0
Cars 1
Computers 0
Computers 1
Art 0
Art 1
Figure 12
The bidding path in the last twelve hours shows the same results as the complete auction path:
the theoretical prediction that there is more late bidding in hard-closing than in automatically
extended auctions is true for cars, but the opposite is true in the case of art. Computer auctions
show little difference in late bidding with respect to the ending rule.
3.1.2.3 Reasons for Late Bidding in Automatically Extended Auctions
When bidders do not expect their rivals to come back in the last few minutes of the auction,
that is when they expect their rivals to treat the automatically extended auction just like a hard
close auction, then there is no need to bid early and then there is a late bidding incentive just
like in a hard close auction. This could be realistic in an expert-amateur scenario where the
amateurs do not foresee that the experts might post late bids that would in turn raise their own
valuation. Naïve amateurs could create such a result. Another explanation could be a high cost
for amateurs to return to the auction at the tentative auction end in order to check whether
they were overbid. If this cost is very high and experts are aware of this, they will engage in
late bidding just like in a hard close auction. Late bidding in the automatically extended 130
auction is only rationally justifiable when at least some of the bidders are either naïve or have
very high costs of late bidding that deter them from doing so.
Empirical findings support the proposition that many bidders do not update their valuation in
the very last minutes of automatically extended Yahoo auctions. For example, only two out of
the fifty automatically extended computer auctions showed bidding in the last five minutes of
the auction (only when there are bids in the last five minutes of the auction, does the
extension time come into play).
3.1.3 Late bidding: According To Type of Good
Hypothesis 3: There is more late bidding when there is uncertainty about the value of the
goods, therefore more late bidding in art than in computer auctions.
Bidders can be assumed to have very little uncertainty about the value of a computer, some
uncertainty with respect to the value (quality) of a car and a lot of uncertainty about the
common value (market value and prestige value) of art.
Observation 3: There is far more late bidding in art than in car or computer auctions.
00,10,20,30,40,50,60,70,80,9
1
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0Time
Pric
e
ArtComputers
Cars
Figure 13
131
Automatically Extended Auctions
0
0,2
0,4
0,6
0,8
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Time
Pric
e
Art 1
Computer 1
Cars 1
Figure 14
Figure 15
Hard Close Auctions
0
0,2
0,4
0,6
0,8
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Time
Pric
e
Art 0
Computer 0
Cars 0
3.1.3.1 Art: Strongest Late bidding
The goods sold in the category art are paintings by unknown painters, often imitations of
famous paintings. Their value cannot be found in a catalogue, so that there is some
uncertainty about the “market value”. When paintings are not by someone “famous”, it might
be important to a buyer whether other people also think that the artwork is beautiful. A bidder
might allocate a personal valuation to a painting, but is ready to pay more if others are also
interested in the object.
In the case of asymmetrical valuations, whereby amateurs have a prestige value, experts
(having a completely private valuation) will be especially careful to disclose their valuation to
amateurs, because not only do they risk having to pay a higher price, as in the case of the
general bidding model (presented in Part Five Chapter 2.1.3) but also do they risk losing the
object to the prestige value bidder. The probability of winning is dramatically lowered when
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an expert discloses his valuation to an amateur with a prestige value (as presented in Part Five
Chapter 2.1.4).
3.1.3.2 Computers: Late bidding Similar For Both Ending Rules
Observation 4: Both hard-closing and automatically extended computer auctions are
characterised by late bidding.
Computer prices are well known. Computers in the survey are generally “second-hand” even
though they are often described as being “brand-new”.
Computers are a readily available good for which prices are well known. Buyers thus have
little uncertainty about the value of a computer. There is some uncertainty regarding the
quality of the computer, but a bidder cannot assume a competing bidder has better information
about a particular computer than he does himself. Assuming that bidders have little or no
uncertainty with respect to their valuation of the computer and assume the same for their
rivals, bidders post their bids early under both ending rules: there is no strategic incentive to
delay bidding when a bidder has a private valuation and assumes that his rivals also have
private valuations.
One reason why bidders might nonetheless engage in late bidding is that computers are not a
unique good, but are available in unlimited quantity. Bidders know the market value and can
base their own valuation on the market value. The Internet offers the possibility to find a
computer at a low price, particularly if – by chance – one happens to be the only one to notice
the “cheap computer”. The lack of scarcity of computers eliminates the need for bidding
early: a bidder can follow a number of computer auctions simultaneously, choosing the
auction with the lowest current price and the least remaining time. Delaying the bid can be
seen as a strategic decision in terms of the option value of finding an even cheaper computer.
Computers are readily available; therefore a buyer will not be prone to revising his bid
upwards after observing last-minute bid flurries.
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3.1.3.3 Late bidding In Car Auctions: Strongly Dependent On Ending-Rule
The goods sold in the category cars are used vehicles, some of which are old-timers. The
value of used cars can be found in catalogues and can be compared with prices charged by
certified dealers. The value of a car does however depend strongly on its condition, which is
unobservable by bidders on the Internet. Whereas real dealers might provide certification for
the car’s quality, the cars sold on the Internet do not come with a certification. Some bidders
will be better informed about the value of a used car than others.
Cars are the only type of good where automatically extended auctions are characterised by a
lot less late bidding than hard close auctions. Reasons for the unexpected late bidding in
automatically extended computer and art auctions are discussed in Chapter 3.1.2.3. Possible
reasons are high costs of returning to an auction in its last moments and naiveté of the bidders
with uncertain valuations (i.e. the amateurs). Cars conform better to the theoretical
predictions, because cars differ from (used) computers and art (by mostly unknown artists) in
that they are far more expensive – so that the costs of returning to the auction in the last few
moments to make possible updates can be assumed to be negligible. The high price of a car
makes bidders believe that competing bidders will follow the auction very closely and bids
might be posted at any time. Due to the high price of cars, the minimum increment is higher
in absolute terms. If the bidder with the highest private signal engages in late bidding but
lower valuing bidders bid early, it is possible that a rival bidder submits a bid just below the
high bidder’s valuation, so that the current bid plus the minimum increment would now
surpass the high-bidder’s valuation. In the case of high minimum increments (as is the case
with expensive items like cars) late bidding lowers the probability of winning, when the rivals
do not engage in late bidding.
In an automatically extended auction, a bidder with foresight knows that he cannot hide his
valuation from his rivals, because the “extension time” gives rivals – with interdependent
valuations - time to react to them. Due to the lower probability of winning an auction in case
of high minimum increments, bidders are not expected to engage in late bidding in
automatically extended car auctions.
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3.1.4 Operational Investigation of Late Bidding
In the section above the dynamic price-formation in Yahoo auctions was presented; price was
plotted against time. It turns out that the price increases very strongly and over-proportionally
towards the very end of the auction. In order to provide an in-depth treatment of the empirical
study of late bidding, the price-formation is also considered operationally.
In the following section late bidding is investigated operationally: the current price is plotted
against the last twenty bids in the auction course. As before, the price is normalised from zero
(the starting price) to one (the final price). The average bidding curves – using the same data
as above - are shown, separated according to good type and ending rule. The curve cuts the y-
axis at the fraction of the price that is reached at the twentieth bid before the auction ends.
Operational Investigation of the last twenty bids made:
Hard Close - Operational
00,10,20,30,40,50,60,70,80,9
1
1 3 5 7 9 11 13 15 17 19
Last Twenty Bids
Pric
e Art 0Cars 0Computers 0
Figure 16
Auto. Extended Auctions - Operational
00,10,20,30,40,50,60,70,80,9
1
1 3 5 7 9 11 13 15 17 19
Last Twenty Bids
Pric
e Art 1Cars 1Computers 1
Figure 17
135
00,10,20,30,40,50,60,70,80,9
1
1 3 5 7 9 11 13 15 17 19
Last Twenty Bids
Pric
e
Computers 0
Computers 1
Cars 0
Cars 1
Art 0
Art 1
Figure 18
As observed in the dynamic bidding paths (in Chapters 3.1.1, 3.1.2 and 3.1.3), the operational
bidding path also shows that the price increases most strongly, over-proportionally towards
the end. Furthermore, conforming to our previous results, art auctions show stronger late
bidding than do cars and computers. The operational bidding paths confirm hypothesis one
and three. Hypothesis two is not confirmed for art and computers, as in the case of the
dynamic bidding path.
In their work on Internet auctions, Roth and Ockenfels (2000) find an interesting result. They
find that Ebay and Amazon auctions are time-invariant, that is the cumulative number of
submitted bids plotted against time follows a similar pattern during the last twelve hours, last
six hours, last three hours, last hour, last thirty minutes and last ten minutes before the auction
end. I tested whether Yahoo auctions are also time-invariant.
3.2 Time-Invariance
The graphs below show the cumulative number of bids submitted (the number of bids
submitted until a given point in time as a part of the total amount of bids submitted during the
whole auction duration). The high bid of every auction in the sample was standardised to one
for the relevant time frame. The time was then split into deciles. In the case of the
136
automatically extended auctions, the end was taken as the tentative end, so that automatic
extensions – as long as there were no further bids during the extension - did not change the
auction end. That explains why the curve for the last ten minutes is not flat in the last five
minutes, even though the ending-rule specifies that the last five minutes have to be bid-free.
Observation 5: Hard-closing auctions show time-invariance with respect to the bidding path.
Automatically extended auctions produce only a weak form of time-invariant bidding paths.
Computers Hard Close
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0 1 2 3 4 5 6 7 8 9 10
Time (in deciles)
Bid
s Su
bmitt
ed
12h6h3h1h
30min10min
Figure 19
Figure 20
Computers Auto Extension
00,10,20,30,40,50,60,70,80,9
1
0 1 2 3 4 5 6 7 8 9 10
Time (in deciles)
Bids
Sub
mitt
ed 12h6h3h1h30min10min
Cars Hard Close
00,10,20,30,40,50,60,70,80,9
1
0 1 2 3 4 5 6 7 8 9 10Time (in deciles)
Bids
Sub
mitt
ed 12h6h3h1h30 min10 min
Figure 21
137
Cars Auto Extension
00,10,20,30,40,50,60,70,80,9
1
0 1 2 3 4 5 6 7 8 9 10Time (in deciles)
Bids
Sub
mitt
ed 12h
6h
3h
1h
30 min
10 min
Figure 22
Art Hard Close
00,10,20,30,40,50,60,70,80,9
1
0 1 2 3 4 5 6 7 8 9 10
Time (in deciles)
Bids
Sub
mitt
edt
12h
6h
3h
1h
30 min
10 min
Figure 23
Art Auto Extension
00,10,20,30,40,50,60,70,80,9
1
0 1 2 3 4 5 6 7 8 9 10Time in deciles
Bids
Sub
mitt
ed
12h
6h
3h
1h
30 min
10 min
Figure 24
The investigation with respect to time-invariance shows that hard close computer and art
auctions show time-invariance and so do hard close car auctions, but slightly weaker. None of
the automatically extended auctions show time-invariance. Three hundred auctions were
surveyed in total, thus fifty in each of the six categories - split according to good type and
ending rule. It would be interesting to check whether taking a larger sample would also lead to
time-invariance for automatically extended auctions or whether the discrepancy remains.
138
One interesting result with respect to bidding behaviour in the last few minutes is that there
seems to be less very late bidding in hard close than in automatically extended auctions. The
cumulative number of bids submitted in the last ten minutes rises earlier for hard close
auctions than for automatically extended auctions.
3.3 Winner’s Bidding Behaviour
Hypothesis 4: The winning bidder enters the auction shortly before the auction end.
3.3.1 Entry Time of Winner
In the theoretical models of the hard close auction presented above, it was predicted that every
bidder who knows (or believes) that his rival (the bidder with the second-highest valuation)
has an interdependent valuation, i.e. a 0>λ (in the general model) or an 0>α greater than
zero (in the prestige-value model), will postpone bidding his valuation to the sniping period.
If all bidders believe their rivals to have a λ (or α ) greater than zero – whether the general
bidding model or the prestige value model - this will lead to the most extreme form of
sniping: every bidder will post his bid in the last possible moment. This will lead to lower
seller-revenue in the hard close than in the automatically extended auction.
In the following section we investigate when the winning bidder enters the auction. Yahoo
auctions publicise the complete bidding course for closed auctions, i.e. auctions that have
already ended. This feature makes observable when the winning bidder first posts a bid. The
entry time is given as a fraction of the total time, where the auction end is normalised to one
and the start to zero. The average entry time is calculated with the same data as used in the
previous chapters – classified according to the good and the ending rule.
Entry Time of Winner:
Entry Time of Winner
Hard Close 0,99 Computers
Automatic Extension 0,81
0,93 Art
Automatic Extension 0,93
Hard Close 0,97 Cars
Automatic Extension 0,92
Hard Close
Table 23
139
Observation 6: The winning bidder enters the auction at a very late point in time. The winner
enters later in hard closing than in automatically extended auctions.
It can clearly be seen that the winner enters the auction at a very late point in time in all kinds
of auctions. In computer auctions the winner enters in the last hundredth of the auction, in art
auctions after 93% of the auction duration has gone by and in car auctions after 97% of the
auction has gone by on average. The winner enters later in hard close than in automatically
extended auctions. This result conforms to the theoretical predictions.
From this result we can deduce that the winner is a bidder who believes (or fears) that his
rivals have interdependent values.
3.3.2 Winning Bid: Single Bid or Proxy Bid?
As introduced in the theoretical investigation in Chapter Two, bidders have the possibility of
making two types of bids: single bids and proxy bids. Single bids correspond to proxy bids
when they are just the minimum increment above the current price. Updating a single bid
takes far more time than letting the bidding agent bid automatically, a bidder might thereby
also lose out on being the high-bidder (because when two bids are made that are equally high,
the first one is the winner). Why use a single bid when a proxy bid is much more time-
efficient and because of being faster has a higher probability of winning? Furthermore proxy
bids allow a bidder to pay the lowest possible price in case of winning – namely a price equal
to the second highest bidder’s maximum willingness to pay plus the minimum increment. The
same is not true for single bids, where the winner might end up paying far more than the
second-highest bid.
A bidder posting single bids has to form his bid according to his first-price auction strategy.
The first-price auction strategy is dependent on a bidder’s risk-attitude. There is a trade-off
between a lower probability of winning and a higher expected payoff, making the bidding
strategy more complicated and making an inefficient outcome more probable. As suggested
by Avery (2002), jump-bids (single bids higher than the current-price by the minimum
increment) can have one advantage, under certain assumptions they can signal toughness and
can scare competing bidders. Altogether there are more reasons speaking in favour of proxy
bids. Therefore we arrive at the following hypothesis:
140
Hypothesis 5: The winning bid is posted as a proxy bid.
Yahoo publicises the type of bid used in the bidding course of closed auctions on its website.
The following information was taken from the Yahoo website as part of the same data set of
three hundred auctions as used above.
Winner's bid
Proxy Bid Single Bid
Art 95% 5%
Cars 58% 42%
Computers 99% 1%
Table 24
In accordance with our theoretical prediction it follows that the winning bid in art and
computer auctions is almost always a proxy bid. Winning proxy bids are more frequent for
cars, but there is also a large number of winning single bids. The use of single bids can be
explained as a mistake (naivety) or as a strategic decision to intimidate rivals. Most plausibly
single bids are used when sellers set a reserve price and a bidder decides to bid the minimum
price at which the seller is willing to sell the good to him. The buy-price serves as insurance
for the seller, so that he is not obliged to sell his car at too low a price. The high number of
winning single bids for cars can be seen as an indication that the market for cars is rather thin,
with often only one buyer willing to pay a price acceptable to the seller64.
3.3.3 Winning Proxy-Bids In Comparison To Second-Highest Bid
As postulated in Hypothesis 4, bidders are expected to bid using the proxy bidding
mechanism. However, the empirical investigation shows that winning bids are sometimes
single-bids. A single bid does not guarantee that the final price is only a minimum-increment
above the second-highest (single or proxy) bid. By looking at the second highest bid
(measured as a fraction of the final price), we can find out whether winning single-bids
strongly outbid the next highest bid.
64 Once a bid is posted that is higher than the seller’s reserve price, the seller’s reserve-price is erased from the
website, so that this explanation cannot be verified with the data available on the Yahoo website for closed
auctions.
141
Second highest bid as fraction of final price
Winning bid is single bid Winning bid is proxy bid
Art 0,77 0,95
Cars 0,69 0,98
Computers 0,86 0,98
Table 25
The final price ends up being much higher than the second-highest bid when the winner posts
a single bid instead of a proxy bid.
3.4 Seller’s Choice of Ending-Rule
The theoretical predictions of Chapter 2 suggest that when the second-highest bidder has an
interdependent valuation (as is postulated is often the case in Internet auctions), the final price
and seller revenue will be lower in the hard closing than in the automatically extended
auction. The revenue is lower in the hard-closing auction due to the strategic incentive for late
bidding.
Hypothesis 6: Sellers prefer automatically extended auctions.
3.4.1 The Preferred Ending-Rule
The Yahoo website publicises the type of ending rule used in every auction. The data sample
leads to the following results:
Percentage of auctions ending by:
Hard Close Auto Extension
Art 18% 82%
27% 73%
Computers 91% 9%
Cars
Table 26
From the chart above we can see that sellers chose the hard close ending far more often than
the automatic extension ending for art and cars. In the case of computers, sellers chose the
hard close ending more often. 142
The theoretical prediction and empirical evidence that sellers prefer automatically extended
auctions for art is also confirmed by Mr. Wedenig from the Austrian auction house
Dorotheum, who complains that Ebay – with whom the Dorotheum cooperates concerning its
Internet auctions - does not let Dorotheum extend auctions, even though this would improve
profitability according to Mr.Wedenig. He believes that Ebay does not allow a change in its
rules in order to have consistency throughout all product categories65.
Ms. Suttner from the London Department of Sotheby’s provides further support for the
empirical findings and says: “Extensions for art auctions can lead to strong price increases”66.
Sotheby’s.com uses surprise-extensions when they see that there is strong bidding activity
towards the end of the auction.
The result for art and cars corresponds to the theoretical prediction, whereas the result for
computers does not. As explained in the chapter above, computers do not prove to be a good
with interdependent valuations, but instead a good with a well-known common value. Despite
the similarity of names that does not mean that the common value model applies here. On the
contrary, bidders have no (or hardly any) uncertainty about their valuation, i.e. have a λ equal
to zero. In this case there is no strategic late bidding incentive as predicted for the models of
interdependent valuations. As discussed above in Chapter 2.1, there might still be an incentive
for bidders in computer auctions to engage in late bidding: bid delay has an option value, the
option value of being able to find a cheaper computer sold in one of the many other
(simultaneous or slightly later) computer auctions. This incentive does not, however, differ
with respect to the two ending rules.
The arguments above would explain why a seller would be indifferent between the two
ending rules for computers, but why would he prefer the hard close ending? If the main
objective of a buyer in a computer auction is to bargain hunt, that is to find a low price for a
good he knows his valuation for with certainty, then he will not be willing to update his
valuation in the extension period. He does not need to bid up to his true reservation price,
because there are so many other computers offered on the Internet. In fact the possibility that
an item can get snatched away by a rival in the extension period makes an automatically
65 Interview with Mag. Wedenig, Dorotheum Internet Auctions GmbH, quoted in Wagner (2002) p.62. 66 Interview with Mag. Suttner, Sothebys.com, London Department, quoted in Wagner (2002) p.37.
143
extended auction less attractive for a buyer and he will concentrate on those auctions were
there is a higher chance of a bargain. The hard close auction is preferred by bargain hunters
and the seller being otherwise indifferent thus prefers the hard close ending to attract buyers.
3.5 Successful Matchings
3.5.1 Average Number of Bids
The last question investigated in the empirical study of Internet auctions is: how successful
are Internet auctions? Out of the number of items offered, how many lead to a successful
matching, that is a price equal or above the seller’s minimum or reserve price?
Yahoo does not list auctions on its closed-auctions website, that received zero bids. In
contrast the closed auction website of Ebay lists all auctions conducted including those that
ended without having received a single bid. The following survey was therefore conducted
using three hundred Ebay auctions, hundred auctions from each of the three categories. The
survey was conducted on March 7th, 2001.
The mean, median and modal number of bids is given for each of the three categories, as well
as the mean, median and modal price67 attained.
Observation 7: Many goods do not receive a single bid. Most auctions are characterised by
low bidding activity.
Pocket-watches Laptops Paintings
Bids Price
(in $)
Bids Price
(in $)
Bids Price
(in $)
Mean 2 20 8,9 574,4 1,9 75
Mode 0 --- 1 --- 0 ---
Median 1 7 4 126,3 0 27
Table 27
A large proportion of Internet-auctions are unsuccessful, in that there is not even one bidder
who is willing to make an acceptable bid, i.e. pay a price above the seller’s minimum-price.
The fact that no bid is made can be caused by lack of demand at the seller’s unrealistically 67 Only successful auctions are included in the calculation of the average price.
144
high starting or reservation price. Or it could be caused by lack of demand for the product that
is quite independent of the price. As we can see in this survey zero was the number of bids
occurring most often (the mode) in the hundred auctions surveyed for the categories pocket-
watches and paintings.
3.5.2 Buy-Price
The “buy-price” as it is called in Yahoo terminology corresponds to the seller’s reserve price
in economic terms. Internet sellers often use secret buy-prices. When a good has a secret buy-
price that has not yet been met, this information is displayed on the web site, but without
telling the bidders how high the buy-price is. A seller reserve price has the advantage of
eliminating the risk that a seller has to sell a good (which has a specific value above zero to
himself) for a value below his own valuation. A reserve price however reduces the number of
participants, because it scares off some low valuation bidders.
In this section the final prices are compared to the seller reserve prices, investigating how
many auctions have final prices below the seller’s reserve price and how many just match the
seller’s reserve price. The Yahoo-website for closed auctions lists all auctions with reserve
prices, where the final price remained below or equal to the seller’s reserve price. Auctions
with reserve prices that were met cannot be identified as such, as the information that a
reserve price was used is eliminated once the reserve price has been met. According to the
auction rules, a seller is only required to sell his good when the final price surpasses his
reserve price by at least the minimum increment.
A Yahoo data set of three hundred auctions, hundred auctions per category of goods is used.
Seller Reserve Price
Not reached Just reached
Cars 17% 25%
Art 7% 9%
Computers 15% 11%
Table 28
145
Observation 8: Many auctions end with the price just matching the seller’s reserve price.
Out of the auctions that receive a positive number of bids, a notable amount ends without the
seller’s reserve price having being reached. Comparing the different categories of goods, cars
have the highest percentage of auctions where the seller’s reserve price is not reached,
followed by computers and the least amount for art.
Hard close car auctions often end with a price just matching the seller’s reserve-price: about
35% of the hard close car auctions and 29% of the hard close computer auctions, but only
11% of the hard close art auctions close at a price just equal to the seller’s reserve price.
This result is surprising in the sense that a seller is not obliged to sell a good to a bidder
making a bid just equal to the seller’s reserve price. The high frequency of auctions ending
with the reserve price just being matched suggests that sellers might be willing to sell their
good for their reserve price.
4. Results of the Empirical Analysis
4.1 The Four Main Hypotheses and the Empirical Evidence
Hypothesis 1: Internet auctions are characterised by late bidding, where late bidding means
that there is a strong over-proportional price increase at the end of the auction.
Empirical Evidence: Empirically confirmed.
Hypothesis 2: The over-proportional price increase toward the auction end is stronger for
auctions with a hard close than for those with an automatically extended ending.
Empirical Evidence: Empirically confirmed for cars, not confirmed for art and computers.
Hypothesis 3: There is more late bidding when there is uncertainty about the value of the
goods, therefore more late bidding in art than in computer auctions.
Empirical Evidence: Empirically confirmed.
146
Hypothesis 4: The winning bidder enters the auction shortly before the auction end.
Empirical Evidence: Empirically confirmed.
Hypothesis 5: The winning bid is posted as a proxy bid.
Empirical Evidence: Empirically confirmed for art and computers, not for cars.
Hypothesis 6: Sellers prefer automatically extended auctions. Empirical Evidence: Empirically confirmed for cars and art, not for computers.
4.2 Further Important Results
Time-invariance is found for hard closing auctions, and in a weak form for automatically
extended auctions.
A large proportion of Internet-auctions are unsuccessful, in that no bidder is willing to pay
a price acceptable to the seller.
Many auctions (especially car auctions) end with a price just matching the seller’s reserve
price.
5. Conclusion In this section I investigate bidding behaviour on the Internet, in particular late bidding. I find
that bidders have an incentive to engage in late bidding in the hard close auction. A bidder
protects himself from low valuing bidders with interdependent values who would raise their
bid if learning about the high bidders value. Late bidding lowers the price and seller-revenue.
When prestige concerns are present in the automatically extended auction, the price can rise
explosively due to self reinforcing effects.
Yahoo auction data confirm the strong presence of late bidding both with respect to the price
path and the winner’s entry time. The data show that late bidding is not limited to the hard
close auction but is also present in the automatically extended auction, especially for goods
available in abundant quantity. The strategic incentive to delay bidding arises because of the
option value of finding an even cheaper model. Furthermore, the seller’s preference for
automatically extended auctions is empirically confirmed.
147
PART SIX: Conclusion
In my thesis I investigate bidding behaviour in auctions both empirically and theoretically,
focusing on Internet auctions and models of interdependent valuations.
Empirical results from an experiment with real consumption goods show that the secret
second-price sealed-bid auction leads to significantly higher seller revenue than the open
Japanese auction. This result coincides with previous field-studies and laboratory results, but
contradicts theoretical predictions. Furthermore, bidders are found to have a tendency to
“stick together with their valuations”. They are able to satisfy this tendency in the open
Japanese auction but not in the secret second-price sealed-bid auction.
In a general model of interdependent values but independent signals, dynamic price-formation
in the Japanese auction is modelled. The updating procedure shows how bidders form their
valuations endogenously, using the information revealed through the exit prices of their rivals.
As a result, the Japanese auction generates lower revenue than the second-price sealed-bid
auction.
In my investigation of Internet auctions I find late bidding concerning both price-formation
and the timing of the winning bid. I show that bidders with interdependent valuations have an
incentive to hold back their bid until immediately before the auction end in the hard close
auction. Bidders hold back their valuation in order not to incite their rivals with
interdependent valuations to revise their valuations upwards. Interdependent valuations can
arise due to common value or reputational effects. The consequent late bidding in the hard
close auction lowers seller revenue, whereas reputational concerns can lead to exorbitant
seller revenue in the automatically extended auction.
Late bidding is empirically confirmed by Yahoo auction data both with respect to the
winner’s entry time and the price path. Empirical results show that late bidding is not limited
to the hard close auction, but is also present in the automatically extended auction. The
theoretically predicted seller’s preference for automatically extended auctions is empirically
confirmed.
148
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