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Common Auction Formats
Open ascending price or English auctionI Auctioneer begins by calling out low price and raises it in small
incrementsI Auction ends when there is only one remaining bidder
Open descending price or Dutch auctionI Descending counterpart to English auction
Common Auction Formats
First-Price, Sealed Bid AuctionI Bidders submit bids in sealed envelopesI At a pre-determined time, auctioneer opens all envelopes and
ranks bidsI Highest bidder obtains object and pays his bid amount
Second-Price, Sealed Bid AuctionI Same as first-price sealed bid auction except that highest bidder
obtains object and pays second highest bid amount
Common Auction Formats
All-Pay auctionI Bidders submit bids (open or closed)I At a pre-determined time, auctioneer opens all envelopes and
ranks bidsI Every bidder pays what they bid regardless of whether or not
they have the highest bidI Highest bidder obtains objectI Examples: Elections, almost any kind of contest or sports, R&D,
wars, lobbying
Auctions
All auctions can be interpreted as allocation mechanisms with thefollowing ingredients:
An allocation ruleI who gets the object
A payment ruleI how much every bidder pays
Private Valuations
Each bidder knows her own valuation v
i
, but not other bidders’valuationsThe distribution of i ’s valuation v
i
is common knowledge:
F
i
(x) = Pr (vi
< x)
Private Valuations
Let b
i
: [0, 1] ! <+ denote i ’s bidding functionI We will also use b
i
to denote i ’s actual bid
If all bidders have the same valuation distribution F , then theywill be using the same bidding function b
They might, however, submit different bids, depending on theirprivately observed valuation
I If b (v) = v
2
, then bidder with valuation 4 will submit $2. Bidderwith valuation 10 will submit $5
I Even if bidders are symmetric in the bidding function they use,they can be asymmetric in the actual bid they submit.
First-Price Auctions
b
i
< v
i
I “Bid shading”: no point in bidding b
i
� v
i
Assume v
i
⇠ U (0, 1) 8i
We will be looking for equilibrium bidding functions of the formb
i
(vi
) = a · vi
, a 2 (0, 1)I We will prove later when we look at the general case (without
the assumption of v
i
⇠ U (0, 1)) that the only symmetricequilibrium bidding function in FPA under uniform valuedistribution is indeed of this form
FPA with 2 bidders
EU
i
(bi
|vi
) = Pr (bi
> b
j
) (vi
� b
i
)
= Pr (bi
> av
j
) (vi
� b
i
)
= F
✓
b
i
a
◆
(vi
� b
i
)
=b
i
a
(vi
� b
i
)
)
b
i
= arg maxx
EU
i
(x |vi
)
=v
i
2
FPA with N bidders
EU
i
(bi
|vi
) = Pr (bi
> max {b�i
}) (vi
� b
i
)
=Y
j 6=i
Pr (bi
> b
j
) (vi
� b
i
)
=
F
✓
b
i
a
◆�
N�1
(vi
� b
i
)
=
b
i
a
�
N�1
(vi
� b
i
)
)
b
i
= arg maxx
EU
i
(x |vi
)
=N � 1
N
v
i
FPA with N bidders
Optimal bidding function: b (v) = N�1
N
v
Bid shading diminishes as N increases
Optimal bidding function increases in N
FPA - Generalization
Arbitrary distribution of v
i
b
i
= arg maxx
Pr (bi
> max {b�i
}) (vi
� b
i
)
= arg maxx
Y
j 6=i
F
j
�
b
�1
j
(x)�
(vi
� x)
= arg maxx
⇥
F
�
b
�1 (x)�⇤
N�1
(vi
� x)
FOC:
�⇥
F
�
b
�1 (x⇤)�⇤
N�1
+ (N � 1)
⇥⇥
F
�
b
�1 (x⇤)�⇤
N�2
f
�
b
�1 (x⇤)� 1
b
0 (b�1 (x⇤))(v
i
� x
⇤) = 0
FPA - Generalization
In symmetric equilibrium, b
i
(vi
) = b (vi
) = x
⇤, )
b
i
=1
F
N�1 (vi
)
ˆvi
0
udF
N�1 (u)
= v
i
� 1F
N�1 (vi
)
ˆvi
0
F
N�1 (u) du
When F (v) = v , b
i
= N�1
N
v
i
FPA - Generalization
Theorem
If N bidders have independent private values drawn from the common
distribution F , then bidding
b
i
(vi
) =1
F
N�1 (vi
)
ˆvi
0
udF
N�1 (u)
is the symmetric NE of a first-price, sealed-bid auction.
FPA with risk-averse bidders
Utility function is concave in income: u (c) = c
↵, ↵ 2 (0, 1)Expected utility of bidding:
EU
i
(bi
|vi
) = Pr (bi
= max {b1
, . . . , bN
}) (vi
� b
i
)↵
FPA with risk-averse bidders
Consider 2 bidders with v
i
⇠ U (0, 1) 8i
b
i
= arg maxx
x
a
(vi
� x)↵
=v
i
1 + ↵
When ↵ = 1 (risk-neutral), b
i
= vi2
FPA with risk-averse bidders
Optimal bidding function: b (v) = v
1+↵
Bid shading is ameliorated as bidders’ risk aversion increases
Optimal bidding function with risk-averse bidders
FPA with risk-averse bidders
Intuition: for a risk-averse bidder:I the positive effect of slightly lowering his bid, arising from
getting the object at a cheaper price, is offset by...I the negative effect of increasing the probability that he loses the
auction.
Ultimately, the bidder’s incentives to shade his bid arediminished.
Second-price Auctions
b
i
(vi
) < v
i
EU
i
= Pr (bi
> h
i
) (vi
� h
i
)
< Pr (vi
> h
i
) (vi
� h
i
)
Second-price Auctions
b
i
(vi
) > v
i
EU
i
= Pr (bi
> h
i
) (vi
� h
i
)
= Pr (vi
> h
i
) (vi
� h
i
)� Pr (vi
< h
i
< b
i
) (hi
� v
i
)
< Pr (vi
> h
i
) (vi
� h
i
)
Second-price Auctions
Bidding b
i
(vi
) = v
i
is a weakly dominant strategy.Alternative argument:
I Since you pay h
i
when you win, the optimal strategy is one thatguarantee you will win when v
i
> h
i
and you will lose whenv
i
< h
i
I This strategy is to bid v
i
The result is unaffected byI number of bidders N
I their risk-aversion preferencesI their valuation distributions
F bidders need not be symmetric: Fi (vi ) can differ from Fj (vj)
Second-price Auctions
Theorem
If N bidders have independent private values, then bidding one’s
value is the unique weakly dominant bidding strategy for each bidder
in a second-price, sealed-bid auction.
Dutch Auctions
Bidder decides at what price to raise hand.If wins, bidder pays the price at which he/she raises hand.Dutch auction is equivalent to FPA.
Theorem
If N bidders have independent private values drawn from the common
distribution F , then raising one’s hand when the price reaches
1F
N�1 (vi
)
ˆvi
0
udF
N�1 (u)
is the symmetric NE of a Dutch auction.
English Auctions
Bidder decides at what price to drop out.Winner pays the price at which last remaining competitor dropsout, i.e. winner pays the second highest bid.English auction is equivalent to SPA.
Theorem
If N bidders have independent private values, then dropping out when
the price reaches one’s value is the unique weakly dominant bidding
strategy for each bidder in an English auction.
Revenue Comparisons
In First-price and Dutch auctions, bidders bid less than theirvaluations but the seller receives the highest bid.In Second-price and English auctions, bidders bid their valuationsbut the seller receives the second highest bid.Which formats have the highest expected revenue?
I Let v
[1]N
⌘ max {v1
, . . . .vN
} and v
[2]N
⌘ maxn
{v1
, . . . .vN
}\ v
[1]N
o
I When v
[1]N
� v
[2]N
, FPA can yield more revenue than SPAI When v
[1]N
, v [2]N
are close, SPA can yield more revenue than FPA
Revenue Comparisons
Distribution of v
[1]N
F
[1]N
(v) = F
N (v)
Distribution of v
[2]N
F
[2]N
(v) = Prn
v
[2]N
v
o
Revenue Comparisons
The eventn
v
[2]N
v
o
can occur in one of two distinct (mutuallyexclusive) ways
1 All valuations below v : v
i
v 8i
2N � 1 valuations below v and 1 valuation above v
I This event can occur in N different ways1 v1 > v and vi v 8i 6= 12 v2 > v and vi v 8i 6= 23 . . .
Revenue Comparisons
Hence,
F
[2]N
(v) = Prn
v
[1]N
v
o
+N
X
i=1
Pr {v1
> v and v
j
v 8j 6= i}
= F
N (v) + N (1 � F (v)) F
N�1 (v)
= NF
N�1 (v)� (N � 1) F
N (v)
Revenue Comparisons
Assuming v is distributed according to F (v) on [0, 1], theexpected revenue of FPA
E
⇥
r
FPA
⇤
=
ˆ1
0
b (v) dF
[1]N
(v)
=
ˆ1
0
1F
N�1 (v)
ˆv
0
udF
N�1 (u)
�
dF
N (v)
= N (N � 1)ˆ
1
0
ˆv
0
uF
N�2 (u) f (u) du
�
f (v) dv
= N (N � 1)ˆ
1
0
ˆ1
u
uF
N�2 (u) f (u) f (v) dvdu
= N (N � 1)ˆ
1
0
uF
N�2 (u) f (u) (1 � F (u)) du
Revenue Comparisons
The expected revenue of SPA:
E
⇥
r
SPA
⇤
=
ˆ1
0
vdF
[2]N
(v)
= N (N � 1)ˆ
1
0
vF
N�2 (v) f (v) (1 � F (v)) dv
= E
⇥
r
FPA
⇤
Revenue Comparisons
When v
i
⇠ U (0, 1),
E
⇥
r
FPA
⇤
=
ˆ1
0
b (v) dF
[1]N
(v) =
ˆ1
0
✓
N � 1N
v
◆
Nv
N�1 (v)
= E
⇥
r
SPA
⇤
=
ˆ1
0
vdF
[2]N
(v) = N (N � 1)ˆ
1
0
v
N�1 (1 � v) dv
=N � 1N + 1
Revenue Equivalence Principle
Theorem
Suppose that values are independently and identically distributed and
all bidders are risk-neutral. Then any symmetric and increasing
equilibrium of any standard auction, such that the expected payment
of a bidder with value zero is zero, yields the same expected revenue
to the seller.
An auction is called standard if the rules of the auction dictatethat the person who bids the highest is awarded the object
Revenue Equivalence Principle
Proof.
Let P (v) be the equilibrium expected payment by a bidder withvalue v . Now consider bidder i . Suppose other bidders are followingthe equilibrium bidding strategy b (v). Bidder i needs to decidewhether to bid b (z) instead of the equilibrium bid b (v
i
)
Bidder i solvesmax
z
{G (z) v
i
� P (z)}
, where G (z) ⌘ F (z)N�1
Revenue Equivalence Principle
Proof. (Cont.)
FOC)g (z) v
i
� P 0 (z) = 0
In symmetric equilibrium,
P 0 (vi
) = g (vi
) v
i
) P (vi
) =
ˆvi
0
ug (u) du
= E
h
v
[1]�i
�
�
�
v
i
> v
[1]�i
i
Pr⇣
v
i
> v
[1]�i
⌘
, where v
[1]�i
denotes the highest value among v�i
.Since the equilibrium expected payment function P (v) does notdepend on any particular auction format, we prove the theorem.
Revenue Equivalence Principle
Example (FPA)
Since the equilibrium expected payment isP (v) = E
h
v
[1]�i
�
�
�
v
i
> v
[1]�i
i
Pr⇣
v
i
> v
[1]�i
⌘
and the winning bidder payswhat she bids, the equilibrium bidding strategy must be
b (vi
) = E
h
v
[1]�i
�
�
�
v
i
> v
[1]�i
i
=1
G (vi
)
ˆvi
0
udG (u)
⌘ 1F
N�1 (vi
)
ˆvi
0
udF
N�1 (u)
Efficiency in Auctions
The object is assigned to the bidder with the highest valuation.I Otherwise, the outcome of the auction cannot be efficient, since
there exist alternative reassignments that would still improvewelfare.
I FPA, SPA, Dutch and English auctions are hence efficient, sincethe player with the highest valuation submits the highest bid andwins the auction.
I Lottery auctions are not necessarily efficient.
Reference
The lecture slides draw from the following sources
Jehle, G. A. and P. J. Reny. 2011. “Advanced MicroeconomicTheory,” Prentice Hall, 3e.Krishna, V. 2009. “Auction Theory,” Academic Press, 2e.Munoz-Garcia, F. 2015. “Advanced Microeconomic Theory II,”lecture notes at http://cahnrs-cms.wsu.edu/ses/people/Munoz/Pages/default.aspx