Auction Theory

Class 3 – optimal auctions

1

Optimal auctions• Usually the term optimal auctions stands for revenue

maximization.

• What is maximal revenue?– We can always charge the winner his value.

• Maximal revenue: optimal expected revenue in equilibrium.– Assuming a probability distribution on the values.– Over all the possible mechanisms.– Under individual-rationality constraints (later).

2

Next: Can we get better revenue?• Can we achieve better revenue than the 2nd-price/1st

price?

• If so, we must sacrifice efficiency. – All efficient auction have the same revenue….

• How?– Think about the New-Zealand case.

3

4

5

Vickrey with Reserve Price• Seller publishes a minimum (“reserve”) price R.

• Each bidder writes his bid in a sealed envelope.

• The seller:– Collects bids– Open envelopes.

• Winner: Bidder with the highest bid, if bid is above R.

Otherwise, no one wins.Payment: winner pays max{ 2nd highest bid, R}

Still Truthful? Yes. For bidders, exactly like an extra bidder bidding R.

6

Can we get better revenue?• Let’s have another look at 2nd price auctions:

0 10

1

1 wins

2 wins

x

1 wins and pays x(his lowest winning bid)

x v1

v2

7

R

Can we get better revenue?• I will show that some reserve price improve revenue.

v10 1

0

1

v2 1 wins

2 wins

Revenue increased

Revenue increased

When comparing to the 2nd-price auction with no reserve price: Revenue loss here (efficiency loss too)

R

8

Can we get better revenue?

• Gain is at least 2R(1-R) R/2 = R2-R3

• Loss is at most R2 R = R3

0 10

1

1 wins

2 winsWe will be here with

probability R(1-R)

Average loss is R/2

When R2-2R3>0, reserve price of R is beneficial.(for example, R=1/4)

We will be here with

probability R2

Loss is always at

most R

9

v1

v2

Reservation priceLet’s see another example:

How do you sell one item to one bidder?– Assume his value is drawn uniformly from [0,1].

• Optimal way: reserve price. – Take-it-or-leave-it-offer.

• Let’s find the optimal reserve price:E[revenue] = ( 1-F(R) ) × R = (1-R) ×R

R=1/2 021)1(

RRRR

Probability that the buyer will

accept the priceThe payment for

the seller

10

Back to New Zealand• Recall:

Vickrey auction.Highest bid: $100000. Revenue: $6.

• Two things to learn:– Seller can never get the whole pie.

• “information rent” for the buyers.– Reserve price can help.

• But what if R=$50000 and highest bid was $45000?

• Of the unattractive properties of Vickrey Auctions:– Low revenue despite high bids.– 1st-price may earn same revenue, but no explanation needed…

11

Optimal auctions: questions.• Is indeed Vickrey auctions with reserve price achieve

the highest possible revenue?

• If so, what is the optimal reserve price?

• How the reserve price depends on the number of bidders?– Recall:

for the uniform distribution with 1 bidder the optimal reserve price is ½. What is the optimal reserve price for 10 players?

12

Optimal auctions• So auctions with the same allocation has the same

revenue.

• But what is the mechanism that obtains the highest expected revenue?

13

Virtual valuations• Consider the following transformation on the value

of each bidder:

– This is called the virtual valuation.– Like bidders’ values: The virtual valuation is when a

player wins and zero otherwise.

• Example: the uniform distribution on [0,1]– Recall: f(v)=1, F(v)=v for every v

14

)()(1)(~

vfvFvvv

1211)(~

vvvvv

)(~ vv

Optimal auctions• Why are we interested in virtual valuations?

• Meaning: for maximizing revenue we will need to maximize virtual values.– Allocate the item to the bidder with the highest virtual value.

• Like maximizing efficiency, just when considering virtual values.

15

A key insight (Myerson 81’):

In equilibrium, E[ revenue ] = E[ virtual valuation ]

Optimal auctions• An optimal auction allocates the item to the bidder

with the highest virtual value.– Can we do this in equilibrium?

• Is the bidder with the highest value is the bidder with the highest virtual value?– Yes, when the virtual valuation is monotone non-

decreasing. – And when values are distributed according to the same F– Therefore, Vickrey with a reserve price is optimal.

• Will see soon what is the optimal reserve price.

16

Optimal auctions• Bottom line:

The optimal auction allocates the item to the bidder with the highest virtual value, and this is a truthful mechanism when is non-decreasing.– Vickrey auction with a reserve price.

• Remark: distribution for which the virtual valuation is non-decreasing are called Myerson-regular.– Example: for the uniform distribution

is Myerson-regular.17

)()(1)(~

vfvFvvv

12)(~ vvv

Optimal auctions: proof

where the virtual valuations is:

(Note: this theorem does not require that the virtual valuation is Myerson-monotone.)

18

)()(1)(~

vfvFvvv

A key insight (Myerson 81’):

In equilibrium, E[revenue] = E[virtual valuation]

Calculus reminder: Integration by parts

19

b

a

b

a

b

a

dydxxgxhdxxgxhdxxgxh )()(')()()()(

')()( xgxh

)()('')()()(')( xgxhxgxhxgxh

dxxgxhxgxhdxxgxh )()(')()()(')(

Integrating:

And for definite integral (אינטגרל מסויים):

)(')()()(' xgxhxgxh

Optimal auctions: proof• We saw:

consider a truthful mechanism where the probability of a player that bids v’ to win is Qi(v).

Then, bidder i’s expected payment must be:

• The expected payment of bidder i is the average over all his possible values:

20

dvvQvQvvpv

aiii

'

)()'(')'(

')'()'()]'([ dvvfvpvpEb

aii ')'()()'('

'

dvvfdvvQvQvb

a

v

aii

Optimal auctions: proof

21

')'()()'(')]'(['

dvvfdvvQvQvvpEb

a

v

aiii

')'()(')'()'(''

dvvfdvvQdvvfvQvb

a

v

ai

b

ai

Let’s simplify this term….

Recall that:

Optimal auctions: proofFormula of integration

by parts:

22

dxxfdvvQb

a

x

ai )()(

b

a

b

a

b

a

dydxxgxhdxxgxhdxxgxh )()(')()()()(

dxxFxQxFdvvQb

ai

b

a

x

ai )()()()(

dvvQxhx

ai )()(

)()( xfxg

dxxFxQdvvQb

ai

b

ai )()(01)(

dxxQxFb

ai )()(1

where

dxxfxFx

a )()(

Optimal auctions: proof

23

dxxfdvvQxxQxpEb

a

x

aiii )()()()]([

dxxfdvvQdxxfxxQb

a

x

ai

b

ai )()()()(

Let’s simplify this term…. dxxQxFdxxfxxQb

ai

b

ai )()(1)()(

dxxfxFxxfxQ

b

ai

)(

)(1)()(

][ iplayerofvaluationvirtualE

Taking out a factor of Qi(x)f(x)

Optimal auctions: proof

24

]~[)]([ ,...,,..., 11 ivviivv vEvpEnn

Expected payment of

bidder I

Expected virtual valuation of

player i

n

iivv

n

iiivv vEvpE

nn1

,...,1

,...,~)(

11

Expected revenue Expected virtual valuation

Optimal auctions• Bottom line:

The optimal auction allocates the item to the bidder with the highest virtual value, and this is a truthful mechanism when is non-decreasing.

• The auction will not sell the item if the maximal virtual valuation is negative.– No allocation 0 virtual valuation.

• The optimal auction is Vickrey with reserve price p such that

25

)()(1)(~

vfvFvvv

0)(~ pv

Optimal auctions: uniform dist.• The virtual valuation:

• The optimal reserve price is ½:

• The optimal auction is the Vickrey auction with a reserve price of ½.

26

12)(~ vvv

0)21(~ v

Remarks• Reservation price is independent of the number of

bidders– With uniform distribution, R=1/2 for every n.

• With non-identical distributions (but still statistically independent), the same analysis works– Optimal auction still allocate the item to the bidder with

the highest virtual valuation.– However, Vickrey+reserve-price is not necessarily the

optimal auction in this case.• (it is not true anymore that the bidder with the highest value is the bidder

with the highest virtual value)

27

Summary: Efficiency vs. revenuePositive or negative correlation ?

• Always: Revenue ≤ efficiency– Due to Individual rationality.More efficiency makes the pie larger!

• However, for optimal revenue one needs to sacrifice some efficiency.

• Consider two competing sellers: one optimizing revenue the other optimizing efficiency.– Who will have a higher market share?– In the longer terms, two objectives are combined. 28