1

22

problem set 11

from Binmore’sFun and Games.

p. 563 Exs. 35, (36)p. 564. Ex. 39

3

Auctions

• Set of bidders 1,2….n • The states of nature: Profiles of valuations

(v1,v2,…..vn), Each is informed

about his own valuation only.

• Given a profile (v1,v2,…..vn), the probability of having a profile (w1,w2,…..wn), s.t. vi ≥ wi is F(v1)F(v2)…...F(vn). Where F() is a cummulative distribution function.

first & second price auctions with independent private valuations

iv v v

4

Auctions

• Actions: Set of possible (non negative) bids • Payoffs: In a state (v1,v2,…..vn), if player i’s bid is

the highest and there are m such bids he gets [vi-P(b)]/m . If there are higher bids he gets 0.

• P(b) is what the winner pays when the profile of bids is b. It is the highest bid in a first price auction, and the second highest bid in a second price auction.

first & second price auctions with independent private valuations

5

Auctions, Nash Equilibria

• In a second price (sealed bid) auction, bidding the true value is a weakly dominating strategy.

• If the highest bid of the others is lower than my valuation I can only win by bidding my valuation.

• If the highest bid of the others is higher than my valuation I can possibly win by lowering my bid to my valuation.

first & second price auctions with independent private valuations

Hence, truth telling is a Nash equilibrium (there may be others)

6

Auctions, Nash Equilibria

• In a first price (sealed bid) auction, bidding the true value is not a dominating strategy: It is better to bid lower when the highest bid of the others is lower than my valuation.

first & second price auctions with independent private valuations

A simple case:

First price (sealed bid) auction with 2 bidders, where the valuations are uniformly distributed

between [0,1]

7

Auctions, Nash EquilibriaA simple case:

First price (sealed bid) auction with 2 bidders, where the valuations are uniformly distributed between [0,1]

There is an equilibrium in which all types bid

half their valuation: b(v) = ½v

Assume all types of player 2 bid as above.

If player 1 bids more than ½ he certainly wins (v-b).

i.e. if player 2’s valuation is lower than 2b. This happens with probability 2b.

If player 1 bids b < ½, he wins if player 2’s bid is lower than b,

In this case his expected gain is 2b(v-b).

8

for v > ½ the payoff function is:

Auctions, Nash EquilibriaA simple case:

First price (sealed bid) auction with 2 bidders, where the valuations are uniformly distributed between [0,1]

½v b

2b(v-b)

½

v-b

v

The maximum is at b = ½v

9

for v < ½ the payoff function is somewhat different but the maximum is as before at b = ½v

Auctions, Nash EquilibriaA simple case:

First price (sealed bid) auction with 2 bidders, where the valuations are uniformly distributed between [0,1]

½v b

2b(v-b)

½

v-b

v

i.e. each type of player 1 wants to use the same

strategy

There is an equilibrium in which all types bid

half their valuation: b(v) = ½v

10

Auctions, Nash EquilibriaA simple case:

First price (sealed bid) auction with 2 bidders, where the valuations are uniformly distributed between [0,1]

½v b

2b(v-b)

½

v-b

v

i.e. each type of player 1 wants to use the same

strategy

There is an equilibrium in which all types bid

half their valuation: b(v) = ½v

11

Auctions, Nash EquilibriaA simple case:

First price (sealed bid) auction with 2 bidders, where the valuations are uniformly distributed between [0,1]

There is an equilibrium in which all types bid

half their valuation: b(v) = ½v

A player with valuation v, bids v/2 and will pay it if his valuation is the highest, this happens with probability v, i.e. he expects to pay v2/2

12

Auctions, Nash Equilibria

second price (sealed bid) auction. (2 bidders)

• Each player’s valuation is independently drawn from the uniform distribution on [0,1]

• A player whose valuation is v, will bid v. • He wins with probability v, and expects to pay

v2/2.

v 2

0

vsds

2

Now consider this simple case for a second price auction

same expected payoff as in the first price auction

13

AuctionsFirst price (sealed bid) auction with n bidders. Valuations are independently drawn from the cumulative distribution F( ).

let be the bid of player whose valuation is .iβ v i v

assume that (a symmetric equilibrium)

and that is an increasing function of .

iβ v β v

β v v

We look for a symmetric equilibrium

v

b

β(v)

v

b

β-1(b)

The inverse function

-1

-1

dβ b 1=

db β β b

-1 -1 -1β β b = b β β b β b = 1

14

assume all players use the bid funtion b = β v .

Pr a player whose valuation is and who bids expects

to earn (All other bids )

v - b

v

b

b

.if h a is vplayer w aluationill bi isd -1 b β b

the probability that one player's valuation is

is

-1

-1

β

F β

b

b

.

the probability that n -1 players' valuation are

s i

alln-1

-1 -1β b F β b

15

assume all players use the bid funtion b = β v .

Pra player whose valuation is and who bids expects

to earn (All other ) bids

v b

- b bv

.if his a player valuatio will b n isid -1 b β b

the probability that one player's valuation is

is

-1

-1

β b

F β b

.

the probability that n -1 players' valuation are

is

alln-1

-1 -1β b F β b

a player whose valuation is and who bids expects

to earn n-1

-1v - b F β b

v b

16

a player whose valuation is and who bids expects

to earn n-1

-1v - b F β b

v b

he should choose his bid to maximize his expected

gain.

b

17

max n-1

-1v - b F β bb

n-2-1 -1

n-1-1

-1

v - b n - 1 F β b F β b- F β b + = 0

β β b

now if the bidding function is an equilibrium

then should be the solution of the

above equation.

β

b = β v

substitute in the equation abo e. vb = β v

18

n-2-1 -1

n-1-1

-1

v - n - 1 F β F β- F β + = 0

β β

b b bb

b

n-2n-1 v - n - 1 F v F v

- F v +β v

= 0β v

n-1 n-2

n-2

β v F v + n - 1 F v F v =β v

v= n - 1 F v F v

n-1β v F v

n-1 n-2vβ v F v n - 1 F x F x dx

vx

19

n-1 n-2v

β v F v n - 1 F x F x dxv

x

n-1F x

n-1 n-1v

β v F v x F x dxv

20

n-1 n-1v

β v F v x F x dxv

integration by parts :

n-1 n-1 n-1v

β v F v v F v F x dxv

n - 1

n - 1

vF x dx

v

F vβ v v

v

v

n-1F x

β v v dxF v

Is this an increasing function of

v ???

Is this an increasing function of

v ???

21

The Optimality of Auctions

• A seller sells an object whose value to him is zero, he faces two buyers.

• The seller does not know the value of the object to the buyers.

• Each of the buyers has the valuation 3 or 4 with probability p, 1-p (respc.)

•The seller wishes to design a mechanism that will yield the highest possible expected payoff.

An example:

?

22

The Optimality of Auctions

Consider the ‘first best’ case:

If the seller can identify the buyer's type

he could earn : 2 23p + 4(1 - p )

the probability that at least one buyer values the object at 4

2= 4 - p

the probability that both buyers value the object at 3

23

The Optimality of Auctions

Posted Prices (take it or leave it offer)

the probability that at least one buyer values the object at 4

Posting the price the seller will earn 3, 3.

Posting the price the seller will earn 24, 4(1 - p ).

only 3,4

24

The Optimality of Auctions

Posted Prices (take it or leave it offer)

Posting the price the seller will earn 3, 3.

Posting the price the seller will earn 24, 4(1 - p ).

If then <2 20 < p < 1 3, 4(1 - p ) 4 - p .

the first best

If then 2p < 1/2 3 < 4(1 - p )

25

The Optimality of Auctions

Second price auction

Truth telling is an equilibrium :The expected payoff is

2 24 1 - p + 3 1 - 1 - p

the probability that both buyers value the object at 4.

2= 3 + 1 - p

if then

if th en

2

22

p < 1 3 < 3 + 1 - p

2< p < 1 4 1 - p < 3 + 1 - p

5

Posted price 4.

26

The Optimality of Auctions

Modified Second price auctionThe buyers are restricted to bid only .

The winner pays the of the two bids.average

3,4

Truth telling is an equilibrim.

It is optimal for

(the player with a low valuation ) to bid .

L

3 3

Player H (High) :If he bids 4 he wins against L and gains :

1 1 14 - 3+4 = , i.e. he gains p.

2 2 2

27

The Optimality of AuctionsModified Second price auction

Truth telling is an equilibrim.It is optimal for

(the player with a low valuation ) to bid .

L

3 3

Player (High) :If he bids he wins against and gains :

, i.e. he gains .

H4 L

1 1 14 - 3 + 4 = p

2 2 2

If he bids he wins against with probability

and earns

31

L p2

1 11 p = p.

2 2

28

The Optimality of Auctions

Modified Second price auction

Truth telling is an equilibrim.

The buyers are restricted to bid only .

The winner pays the of the two bids.average

3,4

The seller expects to earn :

2 214 1 - p + 3 + 4 2p 1 - p + 3p

2= 4 - p

if then2

0 < p < 1 3 + 1 - p < 4 - p

Second price auction

if then 21 < p 1 4 1 - p < 4 - p4

Posting price 4