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