Standard Auctions
Dirk Bergemann
20th January 2009
Dirk Bergemann
Standard Auctions
A General Framework
A seller with an indivisible item for sale, zero cost
I bidders: i = 1; :::; I
Each bidder i has private information vi 2 Vi .
Given the pro le v = (vi ; v i ), bidder i's valuation is
ui (vi ; v i ) if he gets the item and zero otherwise.
I V is F (v ). After
The prior distribution over V
i=1 i
knowing one's own vi , bidder i forms the posterior
distribution of others' valuation payoff as Fi (v i jvi ).
All bidders and seller have von-Neuman Morgenstern
expected and transferable utility function.
Dirk Bergemann
Standard Auctions
A General Framework (Cont.)
Si : (pure) Strategy space for bidder i (the amount i can bid
in auction)
Let Pi (s1 ;
; sI ) :
bidder i wins.
I S
i=1 i
! RI+ be the probability that
For any strategy pro le (si ; s i ), 0
I
X
Pi (si ; s i )
1:
i=1
Let Ti (s1 ;
; sI ) : Ii=1 Si ! RI be monetary payment that
bidder i transfers to seller (no matter i wins or not).
Ti (si ; s i ) can even be negative.
Dirk Bergemann
Standard Auctions
Categories
Private value model: if for all vi ; vi0 ,
vi = vi0 ) ui (vi ; v i ) = ui vi0 ; v i :
Common value model: if the above condition is violated.
Independent value model: if v = (vi ; v i ) is independently
I
Y
drawn. So, f (v ) =
fi (vi ) :
i=1
Symmetric case: if fi (v ) = fj (v ) for any i and j.
Throughout the lecture, we will focus on the independent,
symmetric and private value model in which all vi s are i.i.d.
drawn from a common distribution.
We also assume that all bidders and seller are risk neutral.
Hence, given the strategy pro le (si ; s i ), bidder i's payoff
is vi Pi (si ; s i ) Ti (si ; s i ).
Dirk Bergemann
Standard Auctions
Four Standard Auctions
First Price Auction (High-bid Auction)
buyers simultaneously submit bids
the highest bidder wins (tie broken by ip coin)
winner pays bid (losers pay nothing)
Pi (si ; s i ) =
8
< 1
:
1
K
0
if si > sj ; 8j 6= i
if si ties for highest with K
otherwise
Ti (si ; s i ) =
Dirk Bergemann
si
0
if i wins
:
otherwise
Standard Auctions
1 others :
Four Standard Auctions (Cont.)
Dutch Auction (Open Descending Auction)
Auctioneer starts with a high price and continuously lowers
it until some buyer agrees to buy at current price
the highest bidder wins (tie broken by ip coin)
Pi (si ; s i ) =
8
< 1
:
1
K
0
if si > sj ; 8j 6= i
if si ties for highest with K
otherwise
Ti (si ; s i ) =
si
0
1 others :
if i wins
:
otherwise
This is the same as the case in FPA.
Dutch Auction and First Price Auction are strategically
equivalent.
Dirk Bergemann
Standard Auctions
Four Standard Auctions (Cont.)
Second Price Auction (Vickrey Auction)
same rules as FPA except that winner pays second highest
bid
proposed in 1961 by William Vickrey
Pi (si ; s i ) =
8
if si > sj ; 8j 6= i
< 1
1
if
s
ties
for
highest with K
i
: K
otherwise
0
1 others :
maxj6=i si
if i wins
:
0
otherwise
Recall that in SPA, it is a (weakly) dominant strategy for
each bidder to bid truthfully, i.e., si = vi .
Ti (si ; s i ) =
Dirk Bergemann
Standard Auctions
Four Standard Auctions (Cont.)
English Auction (Ascending Price Auction)
buyers announce bids, each successive bid higher than
previous one
the last one to bid the item wins at what he bids
As long as the current price p is lower than vi , bidder i has
a chance to get positive surplus. He will not drop out until p
hits vi .
Only when anyone else drops out before bidder i, i.e.,
p = maxj6=i vj can he win by paying p, the second highest
valuation.
This shows that English Auction and Second Price Auction
are equivalent.
Dirk Bergemann
Standard Auctions
Independent & Private Value Model
Let vi be agent i's valuation (type).
vi v F ( ) on [0; 1], i.i.d. and symmetric across all bidders.
Bidder i's payoff is vi
and zero otherwise.
ti if he wins the auction and pays ti
We now discuss the equilibrium features in SPA and FPA.
Dirk Bergemann
Standard Auctions
Order Statistics
Given a random variable vei , we know that
F (v ) = Pr vei v .
Suppose we sample n times i.i.d. and get ve1 ; ve2 ; : : : ; ven .
Denote the maximum ve(1) = max ve1 ; ve2 ; : : : ; ven .
Pr ve(1)
n
v = [F (v )]
Dirk Bergemann
Standard Auctions
Order Statistics (Cont.)
Denote the second maximum
ve(2) = max ve1 ; ve2 ; : : : ; ven n ve(1) .
Pr ve(2) v = Pr ve(1) v + Pr ve(2)
n
n 1
= [F (v )] + n [F (v )]
[1 F (v )]
v < ve(1)
From an individual bidder's perspective, the probability that
his valuation (v ) is the highest amounts to:
G (v ) = Pr [all the others' valuations are no more than v ]
= [F (v )]n
1
:
Dirk Bergemann
Standard Auctions
Second Price Auction
Recall: in SPA, it is a (weakly) dominant strategy for each
bidder to bid truthfully, i.e., bi (vi ) = vi .
Buyer's expected payoff is
Z v
SPA
(v ) =
(v
x) G (x) dx
0
=
=
(v
Z v
x) G (x)jx=v
x=0
+
G (x) dx:
0
Dirk Bergemann
Standard Auctions
Z
0
v
G (x) dx
Second Price Auction (Cont.)
Seller's expected revenue is
Z 1
n
SPA
e
R
= E v(2) =
xd [F (x)]n + n [F (x)]n
0
|
{z
=n
Z
=n(n 1)[F (x)]n
1
x [1
0
=n
Z
1
x [1
F (x)] (n
|
F (x)] dG (x)
0
n
=n
Z
2
1) [F (x)]n
{z
=d[F (x)]n
Integrating
=
by part
1
nx [1
1
[1
x
F (x)
1
0
Dirk Bergemann
2
f (x) dx
}
=dG(x)
F (x)] G (x) jx=1
x=0
xf (x)] G (x) dx
F (x)
G (x) dF (x)
f (x)
Standard Auctions
o
F (x)]
}
[1 F (x)]f (x)dx
1
0
Z 1
[1
First Price Auction
Recall: in FPA the highest bidder wins and pays his bid
uniform tie breaking rule
If an individual bidder's valuation is vi and he bids bi , his
expected payoff is:
i
(vi ; bi ) = (vi
b i (bi ) ;
bi ) Q
b i (bi ) is the probability of winning if he bids bi and
where Q
each other bidder j (6= i) bids according to bj ( )
bid function is a mapping from valuation space into positive
real line bi : [0; 1] ! R+
We consider the symmetric and pure strategy where the
amount of a bid only depends on one's valuation.
Dirk Bergemann
Standard Auctions
First Price Auction (Cont.)
Conventionally, we use FOC and SOC to solve out the
optimal bi for a bidder, i.e.
FOC :
@
i
(vi ; bi )
= (vi
@bi
b 0 (bi )
bi ) Q
i
b i (bi ) = 0
Q
This gives a best response for type vi against bj , j 6= i, an
ODE (ordinary differentiable equation) of Qi ( ).
Let's make some guess rst
Guess A: b (v ) is monotone.
Guess B: b (v ) is differentiable.
Dirk Bergemann
Standard Auctions
Equilibrium Bidding Function in FPA
Theorem
In the independent, private value, symmetric environment, the
unique equilibrium strategy is
Z v
G (x)
dx:
b (v ) = v
0 G (v )
Observe that
Underbidding: b (v ) < v , but the gap v
as n increases (Competition effect).
b (v ) decreases
b (v ) is strictly increasing in v :
g(v ) R v G(x)
b 0 (v ) = G(v
) 0
G(v ) dx > 0 for any v > 0.
Dirk Bergemann
Standard Auctions
Equilibrium Bidding Function in FPA: Proof (Cont.)
In equilibrium, bi = b (vi ), vi = ' (bi ) and
'0 (bi ) = 1=b0 (vi ):
(vi
b (vi ))
G0 (vi )
= G (vi ) :
b0 (vi )
Rearranging terms,
vi G0 (vi ) = b (vi ) G0 (vi ) + b0 (vi ) G (vi ) =
d
[b (vi ) G (vi )] :
dvi
Rv
Integrating, 0 i xG0 (x) dx = b (vi ) G (vi ).
LHS
be computed
as
R vi can
R
R vi
0 (x) dx = vi xdG (x) = v G (v )
xG
i
i
0
0
0 G (x) dx.
We thus achieve the goal.
Dirk Bergemann
Standard Auctions
Equilibrium Allocation in FPA
Ef ciency: the bidder who values the item most wins the
auction.
A bidder's ex ante payoff is
FPA
(v ) = [v
=
Z
Prob. of
b (v )]
=
winning
| {z }
Z
0
v
G (x)
dx G (v )
G (v )
=G(v )
v
G (x) dx:
0
Bidder's ex ante payoff in FPA is identical to that in SPA.
Dirk Bergemann
Standard Auctions
Equilibrium Allocation in FPA (Cont.)
Seller's expected revenue is
Z 1
FPA
R
= E b ve(1) =
v
Z
0
=n
Z
vG (v )
0
=n
Z
1
0
Z
=n
|
v
=
0
Z
1
Rv
0
v
v
0
{z
Z
0
=nG(v )dF (v )
G (x) dx dF (v )
}
xdG(x)
xdG (x) dF (v ) = n
0
G (x)
dx d [F (v )]n
G (v )
|
{z
}
Z
1
x [1
F (x)] dG (x)
0
= nx [1 F (x)] G (x) jx=1
x=0
Z 1
n
[1 F (x) xf (x)] G (x) dx
0
1
x
1
F (x)
G (x) dF (x) = R SPA
f (x)
Dirk Bergemann
Standard Auctions
Payoff and Revenue Equivalence in FPA and SPA
Seller's expected revenue in FPA is identical to that in SPA.
We thus show the payoff and revenue equivalence
between FPA and SPA under the independent, private
value setting.
Such equivalence can also be generalized to the case if
two auctions (i) assign the item to the bidder of the highest
value and (ii) leave the bidders of the lowest value zero
surplus. (See our lecture note on "Optimal Auction".)
Dirk Bergemann
Standard Auctions
Optimal Mechanism
So far we have studied bidder' bidding strategies while
taking the auction rule as given, compute and compare the
expected revenue for seller and payoff for bidders. You can
gure out any other auction rule and solve it from the very
beginning.
We don't know whether a speci c auction can extract
revenue as much as possible for the seller.
We need to explore the optimal mechanism where the
seller's ex ante revenue is maximized and how to design a
speci c auction to achieve such goal. This will be covered
in our ongoing lectures on "Optimal Auction".
Dirk Bergemann
Standard Auctions