SØK/ECON 535 Imperfect Competition and Strategic Interaction
AUCTIONS
Lecture notes 05.11.02
Introduction
Auctions are commonly used as a selling or buying mechanism:
art, antiques, used cars
oil, metals, coffee, fish, electricity
treasury bills, certificates
procurement, privatisation
concessions, licences
”Money out of thin air”
Allocation of spectrum rights
UMTS auctions
Great Britain: 5 licenses, 37.1 billion Euro, 129.4 Euro/pop/license
The Netherlands: 5 licenses, 2.7 billion Euro, 33.7 Euro/pop/license
Issues
resource, property rights, allocation
auction or ‘beauty contest’?
auction format
incumbents, entrants, market structure, competition
The Market Mechanism in detail
One agent with complete market power
can determine trading rules/institution
auction versus negotiations
Asymmetric information
do not know value/cost of opponent(s)
Formalised trading rules
‘anti corruption mechanism’
Classification of auction types
number of units
simultaneous or sequential trading
sale and/or purchase
price determination
Game theoretic models
strategic interaction
well-defined ‘rules of the game’
normative versus positive theory
maximisation of income, or minimisation of costs
efficiency
Elementary auction theory
Overview
a simple model
auction formats
The Revenue Equivalence Theorem
The base model
One object for sale
A given number of risk-neutral bidders (n)
No reservation (minimum) price
more precisely: reservation price equal to smallest possible valuation
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Valuations
Let vi be the value of the object to (maximum willingness to pay of) bidder i, i =
1,2,...,n
each bidder knows his or her valuation;
common knowledge that vi is drawn from a distribution Fi with support
[v ,v ] .
We assume
symmetry: Fi(v) = F(v), i = 1,2,...,n
example: uniform distribution on [0,1]
Eksperiment no 1 – sealed-bid first-price auction
Valuations = birth day
January 1 = 1, February 3 = 31+3 = 34, May 17 = 31+28+31+30+17 = 137
and so on
Sealed bids (secret bidding)
each bidder notes his or her bid
Maximum bid wins
First-price auction
the winner pays a price equal to his or her bid
Pay off = valuation – price, if winner; 0 otherwise
Sealed-bid first-price auction – optimal strategies
Risk neutral bidders choose the strategy that maximises expected pay off.
Let Gi(b) be the probability that bidder i wins with bid b
Expected pay off for i:
π i (b,v i ) = [v i − b ]Gi (b )
F
Necessary condition for optimal bid:
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F
dπ i
= [v i − b ]Gi′(b ) − Gi (b ) = 0
db
or
b = vi −
Gi (b )
Gi ′ (b )
Note: bid is shaded below valuation.
Sealed-bid first-price auction - equilibrium
Let bi(v) be i’s strategy, i.e. the bid as a function of valuation.
Symmetry: suppose all follow the same strategy, i.e., bi(v) = b(v), all i.
The probability of winning with a bid b is Gi (b ) = G(b ) = [F (v )]
n −1
Optimal strategy: b(v ) = v −
G (b)
b′ (v ) F (v )
=v −
G′ ( b )
n − 1 F ′(v )
Example: uniform distribution: b (v ) =
n
n −1
n
.
n −1
v
n
2
3
5
10
25
50
100
∞
50%
67%
80%
90%
96%
98%
99%
100%
Experiment no 1 cont – sealed-bid second-price auction
Sealed-bid (secret) bidding
Maximum bid wins
Winner pays a price equal to the second-highest bid (i.e., highest loosing bid)
Sealed-bid second-price auction – optimal strategies
Dominant strategy to bid valuation (b = vi)
b > vi: risk paying price higher than valuation
b < vi: risk loosing even though price is less than valuation
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Vickrey auction
price paid independent of own bid
bid affects probability of winning only
Experiment no 1 cont – open descending auction
Price reduced from a high value until only one bidder accepts.
Winner pays a price equal to accepted price.
This auction format is used in, for instance, the Netherlands (flowers), Israel
(fish) and Canada (tobacco)
Open ascending auction – optimal strategies
Gevinst ved å vente på en reduksjon i salgsprisen fra b til b-db (db er ”liten”):
dπ i = Gi (b )db − [v i − b ]Gi′ ( b ) db
N
N
Optimalt å stoppe når dπ i = 0 , hvilket impliserer at optimalt bud er
b = vi −
Gi ( b )
Gi′ ( b )
Nederlandsk auksjon strategisk ekvivalent med lukket førsteprisauksjon
”åpen førsteprisauksjon”
Experiment no 1 cont – open ascending auction
English auction
bidders over-bid each other
alternatively, an auctioneer increases the price until only one bidder
accepts
Japanese variant
price increased gradually
bidders who have withdrawn cannot re-enter
Winner pays final price
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the highest price accepted by the penultimate bidder
English auction – optimal strategies
Dominat strategy to accept all bids below valuation
English auction equivalent to sealed-bid second-price auction
not strategically equivalent because the set of possible strategies is
different (in English auction strategies can be conditioned on competitors’
behaviour)
Bidding strategies – summary
Private values: vi, i = 1,2,...,n, independently (identically) distributed
Sealed-bid first-price auction
equilibrium strategies (uniform distribution): b(v ) =
n −1
v
n
Sealed-bid second-price auctions
dominant strategy to bid valuation: b(v) = v
Open descending (Dutch) auction
strategically equivalent to sealed-bid first-price auction
Open ascending (English) auction
equivalent to sealed-bid second-price auction
Selling prices
Sealed-bid first-price auction
F
1
expected price equal to expected highest bid: EP = ∫ b(v )dF (v ) , where
F 1 = [F ] is distribution of highest valuation.
n
Sealed-bid second-price auction
expected price equal to expected second-highest valuation:
n
n −1
EP A = ∫ vdF 2 (v ) , where F 2 = [F ] + [ n − 1][1 − F ][F ] is the distribution
of the second-highest valuation.
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F
A
N
E
Uniform distribution: EP = EP = EP = EP =
n −1
n +1
n
2
3
5
10
25
50
100
∞
n −1
n +1
33%
50%
67%
82%
92%
96%
98%
100%
The Revenue Equivalence Theorem
We are looking at mechanisms for determining which of n bidders should
receive an object and which prices bidders (incl. losers) should pay.
Assume each of the n potential risk-neutral bidders has a privately known
valuation for the object drawn from the same, strictly increasing and
continuous distribution function F on [v ,v ] .
Then it can be shown that any mechanism that
(i)
allocates the object to the bidder with the highest valuation, and
(ii)
bidders with the lowest possible valuation v obtains zero payoff,
results in the same expected selling price and the same expected surplus for
a bidder with valuation v.
Examples of auctions that results in same selling price:
sealed-bid first-price, sealed-bid second-price, English, Dutch, all pay
Even if these formats result in the same expected selling price they do not
necessarily lead to the highest possible selling price
if the object must be sold, they do provide maximum price;
mechanisms that lead to higher price must involve positive probability for
either (i) bidder with highest valuation does not obtain the object or (ii)
object is not sold even if there are bidders with valuations higher than that
of the seller.
Generalisation of the model to
correlated (affiliated) valuations
risk aversion
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asymmetries
Optimal (revenue-maximising) auctions
Issues
auction format
reservation (minimum) price
competition (entry)
information provision
Auction format
The Revenue Equivalence Theorem
Risk
Strategic complexity
Co-ordinated bidder behaviour (collusion)
Seller risk
Given the highest value among bidders, the selling price is
certain in the Dutch auction (equal to the bid of the bidder with the highest
valuation), but
uncertain in the English auction (equal to the second-highest bid).
Consequently, the selling price in the Dutch auction (second-order)
stochastically dominates that of the English auction.
A risk averse bidder will therefore prefer a Dutch to an English auction format
by a corresponding argument, he or she would prefer the sealed-bid firstprice format to the sealed-bid second-price format.
Bidder risk
In the English and sealed-bid second-price auction buyers’ attitude towards
risk does not affect behaviour
bidding at valuation is a dominant strategy.
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However, in the Dutch and sealed-bid first-price auction, risk aversion leads to
more aggressive bidding
aggressive bidding reduces a bidder’s risk of losing the auction when the
price is below his or her valuation.
Uncertainty about participation (number of participants and their types) may
also lead to more aggressive bidding (and hence higher selling price).
Strategic complexity
The English and sealed-bid second-price auctions are strategically simple
there is a dominant strategy;
although this may be more difficult to see in the sealed-bid second-price
auction.
I the Dutch and sealed-bid first-price auctions optimal strategies depends on
expectations about competitors’ behaviour.
Coordinated action among bidders
By co-operating (colluding) bidders may increase their (total) gain, although
such collusion involves
side payments;
the risk that some bidders ’cheat’.
Open versus sealed-bid formats:
in the English auction bidders may outbid (’punish’) bidders who do not cooperate;
hence there is less to gain from ‘cheating’ and so easer to achieve
collusion.
First-price versus second-price format (sealed-bids):
since, in the second-price format, a higher bid increases the chance of
winning and not the price one has to pay, the incentive to ’cheat’ is greater
’Bidding rings’
bidders co-operate over a series of auctions;
relevant for the choice between sequential and simultaneous formats.
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Choice of auction format – summary
English and sealed-bid second-price auctions
strategically simpler
Dutch and sealed-bid first-price auctions
less price-risk for the seller
more aggressive bidding among risk-averse bidders
less danger of collusion among bidders
Reservation price
The seller determines a minimum price R such that the object is sold only if
there are bids above this price.
Bidding behaviour will typically be affected by reservation prices
the equilibrium strategies in a sealed-bid first-price auction with symmetric
and uniform distribution of valuations become:
v if v ≤ R

n −1
b (v ) =  n − 1
R R 
v
otherwise
+

n  v 
 n
More aggressive bidding the higher the selling price, but also risk of no sale
expected selling price in a sealed-bid first-price auction with symmetric and
uniform distribution of valuations become:
EP F ( R ) =
2n 
n −1

+ R n 1 −
R
n +1
 n +1 
Optimal reservation price
Trade-off between gain from more aggressive bidding against loss from no
sale.
Optimal (i.e. price-maximising) reservation price in a sealed-bid first-price
auction is given by
R
F ′ (v )
1 − F (v )
=1
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Note: independent of number of bidders.
Uniform distribution
optimal reservation price: R* = ½
maximum expected selling price: EP
F*
n −1
1  1
=
+
n + 1 n + 1  2 
n
n
2
3
5
10
25
50
100
∞
R=0
.3333
.5000
.6667
.8182
.9231
.9608
.9802
1
R = R*
.4167
.5313
.6719
.8183
.9231
.9608
.9802
1
Competition
Increased participation
increases probability of high-valuation bidders, and
may lead to more aggressive bidding.
Increase in selling price when participation increases from n to n+1 (no
reservation price, uniform distribution):
EP n +1 − EP n =
2
[n + 1][n + 2]
n
2
3
5
10
25
50
100
∞
EP n
.3333
.5000
.6667
.8182
.9231
.9608
.9802
1
.1667
.1000
0.0476
.0152
.0028
.0008
.0002
0
EP n +1 − EP n
Participation
Participation will be dependent upon
expected gains from winning and
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costs incurred from participation.
Lower costs leads to more participation and hence higher expected price.
It can be shown that
it may be beneficial to subsidise participation, if this can be done
selectively (i.e. only to some bidders);
but not if the subsidy has to be given to everyone.
Participation fee
If potential bidders know their valutions before deciding whether to participate,
then a participation fee will have the same effect as a reservation price.
If the participation fee is a, among n potential bidders only those will
pariticipate whose valuation v satisfies
[v − v ] F (v )
n −1
≥a
marginal participant wins only if no one else participates;
the fee reduces participation more the more potential participants there is.
The optimal fee balances the (marginal) increase in income from the fee from
the reduction in the selling price due to less competition
 1
uniform distribution: a =  
2
n
*
Reservation prices and participation fees – summary
As a general rule, the seller benefits from setting an effective reservation price
(or positive participation fee).
An effective reservation price leads to
more aggressive bidding, but
a positive probability of no sale.
The optimal reservation price balances the gain from more aggressive bidding
from the loss of reduced probability of a sale.
It may be beneficial to subsidise participation, but only if this can be done
selectively.
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Generalisations
More general assumptions about relationship between bidders valuations
correlated (or affiliated) values
common values
Experiment no 2
Value of object given by day in the year of lecturer’s (or someone in his family)
birth day.
Pay off = value - price
Selad-bid first-price auction, tre variants:
1)
Everybody symmetrically informed.
2)
One bidder is informed about month.
3)
One bidder is informed about date (i.e. becomes perfectly informed)
Open (English) auction
Common Value
General model:
each bidder receives a ’signal’ xi about the value of the object;
the value of the object to bidder i is v i ( x1,...., xn ) , i = 1,2,...,n.
Special cases
private values: v i = xi , with the xi ’s independently distributed
common value: v1 = v 2 = ... = v n = v .
Examples of (nearly) common value objects
oil fields
patents
objects with second-hand markets (shares, bonds, treasury bills etc)
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The Winner’s Curse
The winner is the bidder with the most optimistic estimate of the value!
Others’ bids provide information about value.
Optimal strategy (for a risk-neutral bidder) is to bid expected value, conditional
upon winning.
Bid lower
the better the information of others;
the more optimistic or aggressive the others are; and
the more risk averse you are.
Information
In the private-values case information about the object is of no importance for
bidding behaviour
although information about competitors’ valuations or strategies may affect
bidding.
With correlated (affiliated) valuations providing more information about the
object is always good:
No news is bad news!
reduces the Winner’s Curse
an open (ascending) auction provides more information and hence
reduces the Winner’s Curse
a Japanese auction provides more information than other open and
ascending formats.
Revenue comparison
Risk neutral bidders
Risk averse agents
Private values
Affiliated values
EP F = EP S = EP E = EP D
EP E > EP S > EP F = EP D
EP F = EP D > EP S = EP E
?
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Almost common values
Small asymmetries may have profound effects on bidding behaviour.
Assume it is common knowledge that one of the bidders has a slightly higher
valuation than the others:
example: v1 = v 2 = ... = v n −1 = v , v n = v + ε (ε small).
In an open ascending auction the others can only win at a price higher than
their valuation
they will only accept prices that gives a guaranteed gain in order not to be
hit by the Winner’s Curse;
if there are any participation costs at all, they will not participate.
In a sealed-bid auction the asymmetry only marginally affects bidding
behaviour
therefore, for the seller a sealed-bid first-price auction is better than an
English auction
Example: ’Toeholds and takeover battles’
Efficiency – summary
The object should be allocated to the bidder with the highest valuation.
In the common values case, from an efficiency point-of-view it does not matter
who wins.
More generally, the four standard auction formats considered above are
efficient only if
the potential bidders are ex ante symmetric, and
bid according to symmetric strategies.
Choice of auction format - summary
Price
risk
co-ordinated behaviour amond bidders
correlation (affiliation) of valuations
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asymmetries
Efficiency
Administrative costs
open versus sealed-bid
Simplicity
Conclusion
Not all auctions are the same!
What may appear to be small differences in design may lead to wildly different
outcomes.
‘The devil is in the details!’
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