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
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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)
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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]
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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).
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ???
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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:
?
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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
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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
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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 )
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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
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p < 1 3 < 3 + 1 - p
2< p < 1 4 1 - p < 3 + 1 - p
5
Posted price 4.
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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
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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
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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