a. sealed second-bid auction 2 b. sealed high-bid auction: a … · 2016. 12. 8. · a. sealed...

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© John Riley December 2, 2016

Auctions

A. Sealed second-bid auction 2

B. Sealed high-bid auction: a general approach 10

C. Comparing the two common auctions 21

D. Optimal efficient auctions 41

E. Reserve prices 51

F. Shifts in probability mass without changing the mean 72

G. Exercises 82

83 pages

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A: Sealed second-bid auction

Bidder {1,..., }i I I has a value [ , ]i i i i .

In this case it is easy to confirm that it is an equilibrium for each buyer to bid his true value. i.e. the

profile of strategies,

( )i i iB , 1,...,i I

are mutual best responses.

*

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A. Sealed second-bid auction

Bidder {1,..., }i I I has a value [ , ]i i i i .

In this case it is easy to confirm that it is an equilibrium for all buyers to bid their true values.

i.e. the “profile” of strategies,

( )i i iB , 1,...,i I

are mutual best responses.*

Consider buyer i . Define im to be the

maximum of the other buyers’ bids.

Instead of bidding ( )i i iB suppose

that buyer i bids ib .

*With private information, mutual best response strategies are called Bayesian Nash Equilibrium

(BNE) strategies

’s value bids below her value

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Case (i) im x

Buyer 1 wins in both cases and pays im

***

’s value ’s bid maximum of other bids

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Case (i) im x

Buyer 1 wins in both cases and pays im

Case (ii) i im

Buyer i loses in both cases and pays nothing

**



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Case (i) im x

Buyer i wins in both cases and pays im

Case (ii) i im


Case (iii) i ix m

Buyer i loses, but would have won with

a bid of i and paid

im with a profit of i im .

*




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Case (i) im x

Buyer i wins in both cases and pays im

Case (ii) i im


Case (iii) i ix m

Buyer i but would have won with

a bid of i and paid

im with a profit of i im .

Thus buyer i never gains and may do worse by bidding less than his value.




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An almost identical argument shows that buyer i does not gain by bidding ib either.

Exercise: Prove this statement

It follows that ( )i i iB is a best response

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Dominant strategies

Remark: We have actually proved a stronger result. We defined

1 1 1{ ,..., , ,...., }i i i Im Max b b b b .

If buyers bid their values, then 1 1 1{ ,..., , ,...., }i i i Im Max .

However we did not use this assumption. Regardless of how smart or stupid the other buyers are, we

have shown that it is a best response for buyer i to bid his value. Such a strategy is called a dominant

strategy.

Dominant strategy equilibrium

1( ,..., )Is s is a dominant strategy equilibrium if the strategy of each player is a dominant strategy.

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B. Sealed high-bid auctions: A general approach

Consider once more the sealed high bid auction with two bidders. The set of possible values is

[ , ] . The probability that buyer i ’s value is less than i is ( )iF . (This is the cumulative

distribution function for i .)

Given the symmetry of the model we seek a symmetric equilibrium bid function that maps values into

sealed bids.

( )i ib B .

**

Equilibrium bid function

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Auctions: A general approach


[0, ] . The probability that buyer i ’s value is less than i is ( )iF . (This is the cumulative



sealed bids.

( )i ib B .

If a buyer wins his payoff is i b . Therefore a buyer with

a higher value has a higher payoff to increasing his win

probability.

*


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sealed bids.

( )i ib B .

*


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sealed bids.

( )i ib B .

If a buyer wins her payoff is i b . Therefore a buyer with

a higher value has a higher payoff to increasing her win

probability.

Intuitively, it follows that the equilibrium bid of a buyer with a

higher value will be higher. We will simply assume that this is

the case and then check at the end that this assumption is satisfied.


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Equilibrium win probability

Proposition: In a sealed high-bid auction with I buyers, the equilibrium win probability of buyer i

with value is 1ˆ ˆ( ) ( )IW F .

Proof: The probability that buyer i wins is the

probability that no other buyer bids higher.

Since ( )B is strictly increasing, this is the probability

that every other buyer has value below .

The probability that a particular buyer has a lower

value is ˆ( )F . Therefore the probability that all 1I

other buyers have lower values is 1ˆ( )IF .

QED

Buyer 1 bidding wins if all other

buyers’ values are below .

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Properties of the equilibrium payoff and bid function

(i) ( ) 0U

This is true since the bid function is strictly increasing and so ( ) 0W .

****

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(i) ( ) 0U


(ii) The equilibrium win probability ˆ( ) 0W for all .

The bid function is strictly increasing, thus all other buyers bid lower than ˆ( )B with strictly positive

probability.

***

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(i) ( ) 0U




probability.

(iii) ˆ ˆ( )B

Appealing to (ii), if ˆ ˆ( )B the buyer has a strictly negative payoff if she wins. This cannot be a best

response as it is strictly better to not bid or bid ( )B and win with zero probability.

**

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(i) ( ) 0U




probability.

(iii) ˆ ˆ( )B

Appealing to (ii), if ˆ ˆ( )B the buyer has a strictly negative payoff if he wins. This cannot be a best


(iv) ˆ( )B for all

Suppose ˆ( )B . A buyer with the minimum value, , can then bid ˆ( )B with a profit of ˆ( )B

if she wins. From (ii) the probability of winning is strictly positive. Therefore the expected payoff is

strictly positive. But this violates (i).

*

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(i) ( ) 0U




probability.

(iii) ˆ ˆ( )B

Appealing to (ii), if ˆ ˆ( )B the buyer has a strictly negative payoff if he wins. This cannot be a best


(iv) ˆ( )B

Suppose ˆ( )B . A buyer with the minimum value can then bid ˆ( )B with a profit of ˆ( )B if

she wins. From (ii) the probability of winning is strictly positive. Therefore the expected payoff is

strictly positive. But this violates (i).

By hypotheses ( )B is continuous and we have shown that ˆ ˆ( )B . Therefore

( )B .

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Equilibrium marginal payoff ( )U

Let ( )U be the equilibrium payoff:

( ) ( )( ( )) ( ) ( ) ( )U W B W W B (*)

Since ( ) 0U , it follows that if we can solve for the equilibrium marginal payoff

( )U

then we can integrate and solve for ( )U .

Since ( )B , we can then use (*) to solve for the bid function.

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We now prove the following key result.

Proposition: In a sealed high-bid auction the equilibrium marginal payoff is

( ) ( )U W

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Deviation from equilibrium by buyer 1

Suppose buyer 1 deviates from her

equilibrium strategy for some or all values

other than .

1 1( ) ( )DB B for some 1 [ , ]

ˆ ˆ( ) ( )DB B .

**

Buyer 1 deviates and always bids

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other than .

1 1( ) ( )DB B for some 1 [ , ]

ˆ ˆ( ) ( )DB B .

Since ( )B is a best response, the payoff to

the deviation cannot be higher.

In the lower figure 1 1( ) ( )DU U

and the graphs coincide at .

* (*)


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other than .

1 1( ) ( )DB B for some 1 [ , ]

ˆ ˆ( ) ( )DB B .





It follows that the slopes are equal at .

ˆ ˆ( ) ( )DU U (*)


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Simple deviation from equilibrium by buyer 1

Suppose buyer 1 bids 1ˆ( ) ( )DB B .

i.e. regardless of her value, she submits the bid that is

a best response when her value is .

Thus her win probability is always ˆ( )W and her

expected payoff is

1 1 1ˆ ˆ ˆ ˆ ˆ( ) ( )( ( )) ( ) ( ) ( )DU W B W W B

Thus 1( )DU is a linear function of 1 with slope ˆ( )W .

The slopes of 1( )U and

1( )DU are equal at .

Therefore

ˆ ˆ ˆ( ) ( ) ( )DU U W .

QED


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Remark: Convexity of the payoff function

We have shown that the equilibrium win probability

1( ) ( )IW F

We have also shown that the equilibrium marginal payoff is

( ) ( )U W .

Since ( )W is strictly increasing, ( )U is strictly increasing and so ( )U is strictly convex.

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Examples

1. Two buyers [0,1] , ( ) , 0F

The win probability is ( ) ( )W F . Therefore the equilibrium marginal payoff is

( )U .

We integrate to obtain

11( )

1U K

.

*

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Examples

2. Two buyers [0,1] , ( ) , 0F

The win probability is ( ) ( )W F . Therefore the equilibrium marginal payoff is

( )U .

We integrate to obtain

11( )

1U K

.

Since a buyer with a value of zero has an equilibrium payoff of zero, 0K .

Also

1( ) ( )( ( )) ( ( ) ( )U W B B B .

Therefore

1 11( ) ( )

1U B

and so 1

( )1

B

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3. Uniform distribution with a reserve price

There are two buyers. Values are uniformly distributed on [0,100] . The reserve price (minimum

accepted bid) is r .

Anyone with a value below r will not bid. Then any bidder wins with strictly positive probability. It

follows that a buyer with a value greater than r has a strictly positive expected payoff.

Note that

( ) ( )( ( ))U W B r since ( ) 1W and ( )B r

It follows that the equilibrium payoff must have a limit of zero as approaches the reserve price r .

Thus the boundary condition is ( )U r r .

For all higher values

( ) ( ) ( )100

U W F

.

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Therefore

2

( )200

U K

. Note that 2

( ) 0200

rU r K .

Therefore

2 2 2 ( )

( ) ( )( ( ))200 200 100 100

r BU F B

.

Hence

2 2 2

( )2 2 2

r rB

.

Remark 1: Note that the higher is the reserve price, the higher is the equilibrium bid of a buyer with a

value above r .

Remark: If you check you will find that near the reserve price ( ) 0B . Buyers with values close to

the reserve price are almost indifferent between their equilibrium bid and bidding the reserve price.

This is a general result.

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C. Comparing the two common auctions

Consider the sealed second-bid auction. Every buyer has a value in [ , ] and the c.d.f. of buyer

i ’s distribution of values is ( )iF . In the dominant strategy equilibrium buyers bid their values.

***

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Therefore, if buyer bids ( )i iB , conditional upon i being the highest value, her expected

payment is the expectation of the maximum of the other values

**

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Rather than write down the formula for this expected payment we will simply define

( )iR

to be the equilibrium expected payment by a buyer with value i .

*

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Rather than write down the formula for this expected payment we will simply define

( )iR

to be the equilibrium expected payment by a buyer with value .

The equilibrium expected payoff in the sealed second-bid auction is the win probability multiplied by

the value of the item minus the expected payment

( ) ( ) ( )S i i i iU W R (*)

Note that the win probability of a buyer with value is zero and so ( ) 0SU .

Also, just as in the sealed high bid auction, the equilibrium win probability is 1( ) ( )I

i iW F .

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As for the sealed high-bid auction we seek to characterize the equilibrium marginal payoff.

We will show that it is the same for the two auctions.

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other than .

1 1( ) ( )DB B for some 1 [ , ]

ˆ ˆ( ) ( )DB B .

**


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other than .

1 1( ) ( )DB B for some 1 [ , ]

ˆ ˆ( ) ( )DB B .





*


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other than .

1 1( ) ( )DB B for some 1 [ , ]

ˆ ˆ( ) ( )DB B .





It follows that the slopes are equal at .

ˆ ˆ( ) ( )DU U (*)


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Simple deviation from equilibrium by buyer 1

Suppose buyer 1 bids 1ˆ( ) ( )DB B .

i.e. regardless of her value, she submits the bid that is

a best response when her value is .

Thus her win probability is always ˆ( )W and her

expected payoff is

1 1ˆ ˆ( ) ( ) ( ))DU W R

Thus 1( )DU is a linear function of 1 with slope ˆ( )W .

The slopes of 1( )SU and

1( )DU are equal at .

Therefore

ˆ ˆ ˆ( ) ( ) ( )S DU U W .

QED


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Since the equilibrium payoff of a buyer with the lowest value is zero in both the sealed high-bid and

sealed second-bid auction and the equilibrium marginal payoffs are the same it follows that the

equilibrium payoff functions, ( )iU and ( )S iU are equal.

*

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Since the equilibrium payoff of a buyer with the lowest value is zero in both the sealed high-bid and

sealed second-bid auction and the equilibrium marginal payoffs are the same it follows that the

equilibrium payoff functions, ( )iU and ( )S iU are equal.

But

1( ) ( ) ( ) ( ) ( )I

S i i i S i i i S iU W R F R

and

1( ) ( )( ( )) ( ) ( ) ( ) ( )I

H i i i H i i i H i i i H iU W B W R F R

Thus the equilibrium expected payments by buyers are the same in the two auctions.

Then the expected seller revenue is also the same in the two auctions.

Proposition: Buyer and revenue equivalence of the sealed high-bid and second-bid auctions

If buyer’s values are independently and identically distributed according to a continuous distribution,

the equilibrium expected payoff of the buyers are the same for every value and the expected revenue

of the seller is the same.

This was first proved by Vickrey (1962) for the two buyer case with uniformly distributed values.

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D. General revenue equivalence theorem for efficient auctions

It is easy to think up other auctions. The winning bidder might have to pay some weighted average of

the two highest bids. Or the seller might keep both the highest and the second highest bids (or all of

the bids submitted.)

The two common auctions share an important feature. The equilibrium allocation of the item is to the

buyer who values it the most. This maximizes the value of the item to society so is the Pareto efficient

allocation. (Often we simply say the “efficient allocation”.)

Of all efficient auction schemes, which generated the greatest expected revenue?

We show below that the two common auctions are revenue maximizing among all efficient auctions.

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Choosing a selling scheme

The seller designs a selling scheme with a symmetric equilibrium. In the selling scheme each buyer must

chooses a strategy from strategy set S selected by the seller.The auction rules describe the payoff

1( ,..., )Iu s s associated with every feasible “strategy profile” 1( ,..., )Is s .

**

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Equilibrium buyer payoff

A buyer with value i has an expected payoff

i iu w r where w is his win probability and r is his

expected payment.

Let ( ), ,i is i I be the symmetric equilibrium strategy. Taking the expectation over all other

buyers we let ( )iU be buyer 'i s equilibrium expected payoff when his value is i , we let ( )iW be

his equilibrium win probability and let ( )iR be his expected payment to the seller. Then

( ) ( ) ( )i i i iU W R

*

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Equilibrium buyer payoff

A buyer with value i has an expected payoff

i iu w r where w is his win probability and r is his

expected payment.

Let ( ), ,i is i I be the symmetric equilibrium strategy. Taking the expectation over all other

buyers we let ( )iU be buyer 'i s equilibrium expected payoff when his value is i , we let ( )iW be

his equilibrium win probability and let ( )iR be his expected payment to the seller. Then

( ) ( ) ( )i i i iU W R

Efficient selling scheme

To be an efficient action, buyer i must be the winner if her value is higher than all other values and

be a loser if someone else has a higher value.

We assume that there are I buyers and that each buyer’s value is in the interval [ , ] .

Buyer i ’s value is continuously distributed with c.d.f. ( )iF

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Since the probability of tying high values is zero for continuous distributions the win probability in an

efficient auction of buyer i with value i is

( ) Pr{ ,for all }i j iW j i .

Then

1( ) ( )I

i iW F

**

*

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Then

1( ) ( )I

i iW F

Therefore, in an efficient selling scheme we can rewrite the equilibrium expected payment as follows:

1( ) ( ) ( ) ( ) ( )I

i i i i i i iU W R F R .

**

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Then

1( ) ( )I

i iW F


1( ) ( ) ( ) ( ) ( )I


Arguing exactly as in the examination of the sealed second-bid auction it follows that the equilibrium

marginal payoff is

1( ) ( ) ( )IU W F

*

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Then

1( ) ( )I

i iW F


1( ) ( ) ( ) ( ) ( )I


Arguing exactly as in the examination of the sealed second-bid auction it follows that the equilibrium

marginal payoff is

1( ) ( ) ( )IU W F

Also

1( ) ( ) ( ) ( )IU F R R .

Thus raising ( )R lowers the payoff to all buyers.

Since the expected benefit is unchanged, this raises the payment by all buyers. The seller then

maximizes revenue by choosing a selling scheme for which ( ) 0R .

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Thus the two common auctions are revenue maximizing efficient auctions.

Class discussion: What are some other revenue maximizing efficient auctions?

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E. Setting reserve prices (exclusion of low value buyers)

Consider the sealed second price auction with a reserve price (i.e. minimum acceptable bid) .

The best response of all buyers with valuations below r is to not bid.

Since there is a strictly positive probability of being the only bidder, a buyer with a value above the

reserve is strictly better off bidding.

A buyer with value r is indifferent between entering and staying out.

The dominant strategies of the entering buyers is for them to bid their values.

We begin with the case of one buyer. Without a reserve price the sellers revenue is zero.

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Review: The model

The buyer’s value is a draw from a distribution

with values on the interval [ , ] .

The c.d.f. is ( )F .

The slope of the c.d.f. is the

probability density function p.d.f. ( )f

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The objective:

Solve for the change in expected revenue if the minimum price set by the seller (the “reserve price”)

is raised from to

The method:

We will show that the change in expected revenue

can be written as follows:

1 2R H H p

where 1H and

2H are functions of the parameters.

1 2 ( )p

R H H

*

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The objective:

Solve for the change in expected revenue if the minimum price set by the seller (the “reserve price”)

is raised from to

The method:

We will show that the change in expected revenue

can be written as follows:

1 2R H H p .

where 1H and

2H are functions of the parameters

1 2 ( )p

R H H

*Therefore

1 2 ( )R p

H H

.

Let approach zero. Note that in the limit p

becomes the slope of the c.d.f. ( )

pf

. So

1 2( ) ( )MR H H f

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The one buyer model

Three possibilities

1. The buyer’s value is less than .

Pr{ } ( )p F

2. The buyer’s value is between and

Pr{ } p

3. The buyer’s value is greater than .

Pr{ } q

We will take a limit letting and hence 0p .

Then 1 ( )q p F

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The effect of raising the minimum price from to with one buyer

Three cases.

1. The buyer’s value, , is in the white zone

so the item is never sold.

2. The value is in the cross-hatched zone

so the item is sold only at the initial price.

3. The value is In the solid shaded zone so

the item is still sold but at the new price.

The revenue changes are shown in the top figure.

The probabilities of the three outcomes are shown in the bottom figure.

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The change in expected revenue is

R p q

R pq

Note that ( )p F . In the limit 0p

So in the limit 1 ( )q F

Taking the limit,

( ) ( ) 1 ( )MR f F

Exercise: Values uniformly distributed on [ , ]

(a) Show that if 0 then the profit-maximizing reserve price is 12

.

(b) Show that if 12

then the profit-maximizing reserve price is .

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Raising the reserve price in a sealed second price auction with two buyers

The seller raises the reserve price

from to

The joint probabilities of the nine

different outcomes are indicated.

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Proposition: In the two buyer sealed second price auction, the marginal expected revenue from

raising the reserve price is

( ) 2 ( )[ ( ) 1 ( )]MR F f F

Corollary: In the I buyer sealed second price auction, the marginal expected revenue from raising

the reserve price is

1( ) ( ) [ ( ) 1 ( )]IMR IF f F

The proof is almost identical.

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The method:

Step 1: Show that an upper bound for the change in expected revenue can be written as follows:

2

1 2 3 4 ( )R H H p H p H p

2 2 2

1 2 3 4( ) ( )( ) ( ) ( )p p p

H H H H

***

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The method:


2

1 2 3 4 ( )R H H p H p H p

2 2 2

1 2 3 4( ) ( )( ) ( ) ( )p p p

H H H H

Then

2

1 2 3 4( ) ( ) ( )R p p p

H H H H

**

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The method:


2

1 2 3 4 ( )R H H p H p H p

2 2 2

1 2 3 4( ) ( )( ) ( ) ( )p p p

H H H H

Then

2

1 2 3 4( ) ( ) ( )R p p p

H H H H

Taking the limit as 0 , the third and fourth terms disappear.

Therefore

1 2 1 2( ) ( )dp

MR H H H H fd

.

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The method:


2

1 2 3 4 ( )R H H p H p H p

2 2 2

1 2 3 4( ) ( )( ) ( ) ( )p p p

H H H H

Then

2

1 2 3 4( ) ( ) ( )R p p p

H H H H

Taking the limit as 0 , the third and fourth terms disappear.

Therefore

1 2 1 2( ) ( )dp

MR H H H H fd

.

Step 2: Show that a lower bound can be written as follows:

2

1 2 4 ( )R H H p H p

So in the limit the upper and lower bounds are the same.

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Case 1: If both buyers have values

above , then the selling price is

unaffected.

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Case 2: If one is above and

the other between and , then the

selling price rises from the second value

to . The gain is therefore between

zero and .

The probability is 2q p .

Therefore the change in expected

revenue lies between

zero and 2q p .

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Case 3: If both are between the two

reserve prices then the sale is again

lost. The probability of this event is

2( )p

so the contribution to expected

revenue is between 22( )p

and 22( )p .

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Case 4: If one value is above and the

other below , then the revenue rises by

.

The probability is

2pq

Expected revenue thus changes by

2 pq . (4)

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Case 5: If one value is between the

two reserve prices and the other is

below then the sale is lost.

The probability of this event is

2p p

So the expected revenue is reduced by

2p p (5)

Case 6: If both values are below

there is no change in revenue.

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Lower bound for R .

Combining the results for cases (ii)-(v), we have the following lower bound to the change in expected

revenue.

2[2 2 ] ( )R pq p p p

Therefore

2[2 2 ] ( )R p p

pq p

Combining the results for cases (ii)-(v), we have the following upper bound to the change in expected

revenue.

2[2 2 ] [( ) 2 ]R pq p p p q p

Therefore

2[2 2 ] [( ) 2 ]R p p p

pq p q

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In the limit only the first two terms are non-zero. Since the first two terms are the same, the upper

and lower bounds are the same in the limit. Therefore

( ) 2 2 2 ( ) 2 ( )[1 ( ) ( )]dp dp

MR pq p p q F F fd d

QED

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Comparison of auction with reserve prices

Note that in the sealed second-bid auction with reserve price

(i) For , ( )W , the win probability of a buyer with value is zero.

(ii) The equilibrium payoff of a buyer with value , ( ) 0U

(iii) For , ( )W is the probability that all other buyers have lower values.

By our general argument,

1( ) ( ) ( )IU W F .

We can then integrate to solve for ( )U . The boundary condition ( ) 0U determines the constant

of integration.

Class Discussion: Does it follow that the expected revenue maximizing reserve price is the same in a

sealed high-bid auction?

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F. Comparing two buyer auctions with the same mean

We will use the following on the next slide:

( ) ( ) ( ) ( )dR

R R MR q dq dqdq

(*)

If R is revenue, the increase in R as output increases

is the area under the marginal revenue curve.

Mean of a finite distribution

1

( )S

s s

s

x x

.

Mean of a continuous distribution on [ , ]

( )xf x dx

.

For a symmetric distribution 12( ) .

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Proposition: Geometry of the mean

The area under the c.d.f. on [ , ] is where and are the maximum and mean values.

Proof: [ ( )] ( ) ( )d

F F fd

.

Therefore

( ) [ ( )] ( )d

f F Fd

Integrate:

( ) [ ( )] ( )d

f d F d F dd

Appealing to (*) on the previous slide,

[ ( )] ( ) ( )d

F d F Fd

Therefore ( )F d

The integral is the area of the dotted region. The sum of the dashed and dotted areas is 1 .

QED

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Proposition: For a two buyer auction, if the means of two distributions with values on [ , ] are the

same then the equilibrium payoff of a buyer with the maximum value is the same.

Proof:

( ) ( ) ( ) ( )U U v dv W v dv F v dv

.

*

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Proposition: For a two buyer auction, if the means of two distributions with values on [ , ] are the

same then the equilibrium payoff of a buyer with the maximum value is the same.

Proof:

( ) ( ) ( ) ( )U U v dv W v dv F v dv

.

Corollary: For a two buyer auction the maximum bid is the mean

( )B

Proof:

In the sealed high-bid auction,

( ) ( )( ( ))U F B .

Since ( ) 1F ,

( ) ( )U B .

Appealing to the proposition above, ( )B

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Effect of shifting probability mass away from the tails

without changing the mean of a symmetric distribution

Note that under symmetry

12

( ) ( )F F .

Given the shift in probability mass towards the mean,

( ) ( ),F F

and

( ) ( ),F F

See the bottom figure.

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Since ( ) ( ),F F

it follows that

( ) ( ) ( ) ( ),U F F U

Since ( ) ( ) 0U U and ( )U increases faster

than ( )U it follows that

( ) ( ),U U (*)

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Since ( ) ( ),F F

It follows that

( ) ( ) ( ) ( ),U F F U

We have shown that ( ) ( )U U

Since ( )U increases faster than ( )U when

, it follows that

( ) ( ),U U (**)

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Combining (**) and (*)

( ) ( ),U U

Unless a buyer has the minimum or maximum value,

he or she is strictly worse off after the

shift in probability mass away from the tails.

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Implications for the equilibrium bid function

We have shown that the maximum bid ( )B

Proposition: A shift in the distribution away for the tails

of the distribution increases bids for values satisfying

Proof:

( ) ( )( ( ))U F B and ( ) ( )( ( ))U F B

Therefore

( )( )

( )

UB

F

and

( )( )

( )

UB

F

.

We have shown that ( ) ( )U U if .

Also ( ) ( )F F if

It follows that ( ) ( )B B if .

Therefore

( ) ( )B B if .

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Final remarks:

Thus a buyer bids strictly more if his value lies on the interval [ , ) and probability mass is shifted

towards the mean.

The maximum value is below the mean with probability 0.25 therefore the winning bid is higher with

high probability.

As more probability mass is shifted towards the mean the equilibrium bid continues to rise for values

at or above the mean.

In the limit all of the probability mass is at the mean. There is no longer any private information. Each

buyer knows that both values are equal to . The unique equilibrium is for both buyers to bid their

values.

In class we saw that in the simple case of a uniform distribution the equilibrium bid function is

12

( )B . The above results show that for any single peaked density function, bids will be higher

either close to or above the mean.

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G. Exercises

Exercise 1: Uniform distribution on the interval [ ,1 ]

( ) , [ ,1 ]F

For what values of does it pay to set a reserve price?

Exercise 2: All pay auction

In an all-pay sealed-bid auction the seller keeps all of the bids.

(a) Use a proof similar to that for the sealed second-bid auction to show that seller revenue is the

same in the all-pay auction as in the sealed second-bid auction.

(b) Solve for the equilibrium bidding strategy if there are 2 buyers and the c.d.f is 2( ) 2F .

Exercise 3: Revenue equivalence of the common auctions with two identical items for sale when

each buyer wishes to purchase one unit.

Sealed bid auction. The two highest bidders win an item and each pays his bid.

Sealed third bid auction: The two highest bidders win an item and each pays the third highest bid.

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(a) Show that in the sealed third bid auction it is a dominant strategy for the buyers to bid their

values.

(b) Extend the arguments above to establish the revenue equivalence of the two auctions.

a. sealed second-bid auction 2 b. sealed high-bid auction: a … · 2016. 12. 8. · a. sealed...

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