auction theory

Auction Theory

Class 3 – optimal auctions

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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).

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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.

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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.

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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

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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

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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(

RR

RR

Probability that the buyer will

accept the priceThe payment for

the seller

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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…

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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?

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

revenue.

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

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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)(~

vf

vFvvv

121

1)(~

v

vvvv

)(~ 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.

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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.

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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)(~

vf

vFvvv

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)(~

vf

vFvvv

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:

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dvvQvQvvpv

a

iii '

)()'(')'(

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

a

ii ')'()()'(''

dvvfdvvQvQvb

a

v

a

ii


21

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

dvvfdvvQvQvvpEb

a

v

a

iii

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

dvvfdvvQdvvfvQvb

a

v

a

i

b

a

i

Let’s simplify this term….

Recall that:

Optimal auctions: proofFormula of integration

by parts:

22

dxxfdvvQb

a

x

a

i )()(

b

a

b

a

b

a

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

dxxFxQxFdvvQb

a

i

b

a

x

a

i )()()()(

dvvQxhx

a

i )()(

)()( xfxg

dxxFxQdvvQb

a

i

b

a

i )()(01)(

dxxQxFb

a

i )()(1

where

dxxfxFx

a )()(


23

dxxfdvvQxxQxpEb

a

x

a

iii )()()()]([

dxxfdvvQdxxfxxQb

a

x

a

i

b

a

i )()()()(

Let’s simplify this term…. dxxQxFdxxfxxQb

a

i

b

a

i )()(1)()(

dx

xf

xFxxfxQ

b

a

i

)(

)(1)()(

][ iplayerofvaluationvirtualE

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


24

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

n

iivv

n

iiivv vEvpE

nn1

,...,1

,...,~)(

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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)(~

vf

vFvvv

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.

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12)(~ vvv

0)2

1(~ 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)

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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

auction theory

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