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Auction Seminar
Optimal Mechanism
Presentation by: Alon ReslerSupervised by: Amos Fiat
2
Review
Myerson Auction for distributions with strictly increasing virtual valuation:
1. Solicit a bid vector from the agents.2. Allocate the item to the bidder with the
largest virtual value , if positive, and otherwise, do not allocate.
3. Charge the winning bidder i, if any, the minimum value she could bid and still win, i.e.
( )i ib
1 max(0,{ ( )}i j j j ib
3
Review
we defined the virtual value of agent i to be:
We saw that when the bidders values drawn from independent distributions with increasing virtual valuations the Myerson auction is optimal, i.e.
it maximized the expected auctioneer revenue in Bayes-Nash equilibrium
)(
)(1:)(
ii
iiiii vf
vFvv
4
Review
5
Review
Characterization of BNE:
Let A be an auction for selling a single item, where bidder i’s value is drawn independently from .
If is a BNE, then for each agent i:
1. The probability of allocation is monotone increasing in
2. The utility is a convex function of with
3. The expected payment is determined by the allocation probabilities:
iV iF 1,..., n
( )i ia v iv( )i iu v iv
0
( ) ( )iv
i i iu v a z dz
0 0
( ) ( ) ( ) ( )i iv v
i i i i i i ip v v a v a z dz za z dz
6
Today issues:We will show a generalization
version of Myerson’s optimal mechanism, that doesn’t require that virtual valuations be increasing.
We will see one examples
7
Optimal MechanismOur goal now is to derived the general version of Myerson’s
optimal mechanism, that doesn’t require that virtual valuations be increasing.
We will a change in variables to quantile space, and define:
The payment function:
The allocation function:
Notice that:
ˆ( ) ( ( )) ( )a v a F v f v
)(: vFq )(:)( 1 qFqv
ˆ ( ) : ( ( ))p q p v q
ˆ( ) : ( ( ))a q a v q
8
Optimal MechanismGiven any and we
have:
0v 0 0( )q F v
( )dq f v dv( )q F v
0
0 0
0
ˆ ( ) ( ) ( )v
p q p v a v vdv 0 0
0 0
ˆ ˆ( ( )) ( ) ( ) ( )v q
a F v vf v dv a q v q dq
ˆ( ) ( ( )) ( )a v a F v f v
9
Optimal Mechanism
From this formula, we derive the expected revenue from this bidder.
Let Q be the random variable representing this bidder’s draw from the distribution in quantile space, i.e., Q=F(V) [Notice that ]. Then,
Reversing the order of integration, we get
01
0
0 0
ˆ ˆ[ ( )] ( ) ( )q
p Q a q v q dqdq
0 1Q
10
Optimal Mechanism
Where is called the revenue curve.
It represents the expected revenue to a seller from offering a price of v(q) to a buyer whose value V is drawn from F.
1 1
0
0
1 1
0 0
ˆ ˆ[ ( )] ( ) ( )
ˆ ˆ( )(1 ) ( ) ( ) ( )
q
p Q a q v q dq dq
a q q v q dq a q R q dq
( ) (1 ) ( )R q q v q
11
Optimal Mechanism Integrating by parts…
Explanation: R(1) = 0 and because is on we get,
iF
1ˆ(0) ( (0)) ( (0)) (0) 0a a v a F a
, io h
1
1
00
1
0
ˆ ˆ[ ( )] ( ) ( ) ( ) ( )
ˆ ˆ( ) ( ) [ ( ) ( )]
p Q R q a q a q R q dq
a q R q dq a Q R Q
12
Optimal MechanismTo summarizing, we proved a lemma:Consider a bidder with value V drawn from
distribution F, with Q=F(V). Then his expected payment in a BIC auction is
Where is the revenue curve.
( ) (1 ) ( )R q q v q
ˆ ˆ[ ( )] [ ( ) ( )] [ ( ) ( )]p Q a Q R Q a Q R Q
13
Optimal Mechanism
Lemma: let Then,
Proof:
From and we get:
( )q F v
1 ( )( ) '( )
( )
F vv v R q
f v
'( ) ((1 ) ( )) (1 ) ( )d d
R q q v q v q v qdq dq
1( ) ( )v q F q ( )q F v
1 ( )
( )
F vv
f v
14
Optimal Mechanism
As we discussed in David’s class (two weeks ago), allocation to the bidder with the highest virtual value (or equivalently, the largest –R’(q)) yields the optimal auction, provided that virtual valuations are increasing.
Hedva Observation:
Let be the revenue curve with
. Then is (weakly)
increasing if and only if R(q) is concave. (a function
is concave if it’s derivative is increasing ).
( ) (1 ) ( )R q q v q
( )q F v ( ) '( )v R q
15
Optimal Mechanism
To derived an optimal mechanism for the case when
R(q) is not concave envelope of
Definition: is the infimum over concave
functions such that for all
.
Passing from to is called ironing.
can also be interpreted as a revenue curve
when randomization is allowed.
Definition: The iron virtual value of bidder i with
value is:
( )R q
( )g q( ) ( )g q R q [0,1]q
( )R ( )R ( )R
( )iv q( ) ( )i i iv R q
( )R q
( )R q
16
Myerson auction with ironing
Now we can replace virtual values with ironed virtual
values to obtain an optimal auction even when virtual
valuations are not increasing.
Definition: The Myerson auction with ironing:
1. Solicit a bid vector b from the bidders.
2. Allocate the item to the bidder with the largest value
of , if positive, and otherwise
do not allocate.
3. Charge the winning bidder i, if any, her threshold bid,
the minimum value she could bid and still win.
( )i ib
17
Optimal Mechanism
Theorem: The Myerson Auction described above is optimal, i.e., it maximized the expected auctioneer revenue in Bayes-Nash equilibrium when bidders values are drawn from independent distribution.
Proof: The expected profit from a BIC auction isˆ ˆ( ) ( )( ( ))i i i i
i i
p Q a Q R Q
ˆ ( ) ( )i i i ii
a Q R Q
18
Proof cont.
(add and subtract
).
ˆ ˆ( )( ' ( )) '( ) ( ) ( )i i i i i i i i i ii i
a Q R Q a Q R Q R Q
ˆ ˆ( )( ' ( )) ( )( ( ))i i i i i i i ii i
a Q R Q a Q R Q
ˆ ˆ[ ( ) ( )] [ ( ) ( )]a Q R Q a Q R Q
19
Proof cont.
Consider choosing a BIC allocation rule to maximize the first term:
This is optimized by allocating to the bidder with the largest , if positive.
Moreover, because is concave, this is an increasing allocation rule and hence yields a dominant strategy auction.
( )
( )R
1ˆ ( ,..., )( ) ( ) .i n i ii
Q Q R Q
( ) ( )i i iv R q
20
Proof cont. Notice also that is constant in each interval of
non-concavity of , and hence in each such interval
is constant and thus .
Consider now the second term: In any BIC auction,
must be increasing and hence for all q.
But for all q and hence the second term is
non-positive. Since the allocation rule that optimized
the first term
has whenever , it ensures that
the second term is zero, which is best possible.
( )iR
( )R
( )ia q
( ) 0ia q
( )ia ( ) 0ia q
( ) 0a q ( ) ( )R q R q
( ) ( )R q R q
ˆ ˆ( )( ' ( )) '( ) ( ) ( )i i i i i i i i i ii i
a Q R Q a Q R Q R Q
21
The advantage of just one more bidder…
One of the downside of implementing the optimal
auction is that it requires that the auctioneer know
the distributions from which agents values are
drawn.
The following result shows that in instead of
knowing the distribution from which n independently
and identical distributed bidders are drawn, it
suffices to add just one more bidder into the
auction.
22
Theorem
Let F be a distribution for which virtual
valuations are increasing. The expected
revenue in the optimal auction with
independently and identical distributed
bidders with values drawn from F is upper
bounded by the expected revenue in a Vickrey
auction with n+1 independently and identical
distributed bidders with values drawn from F.
23
Proof
The optimal (profit-maximizing) auction that is
required to sell the item is the Vichrey auction (this
follows from the lemma we saw in previous class,
which says that for any auction, the expected profit is
equal to the expected virtual value of the winner).
Second, observe that one possible n+1 bidder
auction that always sells the item consist of, first,
running the optimal auction with n bidders, and then,
if the item is unsold, giving the item to the n+1–st
bidder for free.
24
First example:
War of Attrition
25
War of attrition (Reminder)
In Liran’s class we considered the war of
attrition: this single item auction allocates the
item to the player that bids the highest,
charges the winner the second-highest bid,
and charge all others players their bid.
Notice that in this auction the bidders decides
at the beginning of the game when to drop
out.
26
War of attrition (Reminder)
Formulas that we derived in Liran’s class that we will need for today: Let be a symmetric strictly increasing equilibrium strategy, Then:
Expected payment of an agent in a war-of-attrition auction in which all bidders use is
And
27
Dynamic war of attrition
A more natural model is for bidders to
dynamically decide when to drop out.
The last player to drop out is then the winner of
the item.
With two players this is equivalent to the model
discussed in Liran’s class:
The equilibrium strategy := how long a
player with value v waits before dropping out.
( )v
28
Dynamic war of attrition
satisfies:
Where is the hazard rate of
the distribution F. (it followed from the equilibrium we found in
Liran’s class:
).
( ) ( )( )
1 ( ) ( )
f w f wh w
F w F w
0 0
( )( ) ( )
1 ( )
v vf wv w dw wh w dw
F w
29
Dynamic war of attritionWe rederive this here without the use of
revenue equivalence. To this end, assume that the opponent plays
.The agent’s utility from playing when
his value is v is:
Differentiating with respect to w, we have:
( ) ( )w
0
( | ) ( ) ( ) ( ) ( ) ( )w
u w v vF w F w w z f z dz
( | )( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
u w vvf w f w w F w w w f w
wvf w F w w
30
Dynamic war of attritionFor this to be maximized at , we must
have
implying
w v
( ) ( ) ( )vf v F v v
0 0
( )( ) ( )
( )
v v f wv wh w dw w dw
F w
31
Dynamic war of attrition With three players strategy has two component,
and
.
For a player with value v, he will drop out at time if
no one else dropped out earlier.
Otherwise, if another player dropped out at time
, our player will continue until time
.
( , )y v( )v
( )v
( ) ( )y v ( ) ( , )y y v
32
Dynamic war of attritionThe case of two players applied at time y
implies that in equilibrium
Since the update density has the
same hazard rate as f.
Unfortunately there is no equilibrium once
there are three players…
( )1
( ) z y
f z
F y
( , ) ( )v
y
y v zh z dz
33
Dynamic war of attrition
To see this, suppose that players 2 and 3 are playing
and , and player 1 with value v plays
instead of . Then
Since with probability , the other two players
outlast player 1 and then he pays an additional
for naught.
And with probability for some
constant C, both of the others players drop out first.
( ) ( , ) ( )v
2( )F v
( ) ( )v v
( )v2 20 ( | ) ( | ) ( ) ( ( ) ( ))u v v u v v C F v v v
2
2( )v
v
f z dz C
34
Dynamic war of attrition
Thus, it must be that
And for any k we have:
(since is non-increasing function).
2 2( ) ( ) ( )v v C F v
( )F v
2 22 2
( ) ( )( ) ( )
C Cv k v k
F v k F v
35
Dynamic war of attrition Summing from k=0 to :
And hence:
Finally, we observe that is not equilibrium
since a player, knowing that the other players are going
to drop out immediately, will prefer to stay in.
/v
2( ) ( )v CF v v
( ) 0v
( ) 0v
2 2 2
( ) ( ) ( ) ( 2 ) ... ( ( 1) ) ( )
( ) (0) ( ) ( )
v vv v v v v v
vv CF v CF v v
36
Thanks for listening