Post on 10-Apr-2018
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Auctions
Chapter 11
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Sealed-Bid Auctions with
Complete Information
2-bidder auctions as matrix games
The 3 principles of bidding
The relationship of auction equilibrium toBertrand equilibrium
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Bidding Principle 1
Never overbid. As a strategy,
overbidding is dominated bybidding zero.
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Bidding Principle 2
If you know you are the high-bidder
in a first-price auction, you shouldalways shave your bid.
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z1 = 2 and z2 = 1
When, b1 = $2 and b2 = $1,
u1 = 2 - 2 = 0 and u2 = 0
When, b1 = $1 and b2 = $1,
u1 = .5 v (2 - 1) + .5 v 0 = 0.5 andu2 = .5 v (1 - 1) + .5 v 0 = 0
When, b1 = $0 and b2 = $1,
u1 = 0 and u2 = 1 - 1 = 0
First-Price auction, $1 bidding
interval: deriving the payoffs
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When, b1 = $2 and b2 = $0,
u1 = 2 - 2 = 0 and u2 = 0
When, b1 = $1 and b2 = $0,u1 = 2 - 1 = 1 and u2 = 0
When, b1 = $0 and b2 = $0,
u1 = .5
v(2 - 0) + .5
v0 = 1 and
u2 = .5 v (1 - 0) + .5 v 0 = 0.5
First-Price auction, $1 bidding
interval: deriving the payoffs
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First-Price auction, $1 bidding interval
0, 0
0.50, 0
0, 0
Player 2
Player 1b2= $1
1, 0
0, 0 1, 0.50
b2= $0
b1= $2
b1= $0
b1
= $1
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First-Price auction, $1 bidding interval:
strategy for player 1
0, 0
0.50, 0
0, 0
Player 2
Player 1b2= $1
1, 0
0, 0 1, 0.50
b2= $0
b1= $2
b1= $0
b1
= $1
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First-Price auction, $1 bidding interval:
strategy for player 2
0, 0
0.50, 0
0, 0
Player 2
Player 1b2= $1
1, 0
0, 0 1, 0.50
b2= $0
b1= $2
b1= $0
b1
= $1
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First-Price auction, $1 bidding interval:
three pure strategy equilibria
0, 0
0.50, 0
0, 0
Player 2
Player 1b2= $1
1, 0
0, 0 1, 0.50
b2= $0
b1= $2
b1= $0
b1
= $1
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First-Price auction, $0.5 bidding
interval
0.5, 0
0.5, 0
1.5, 0
Player 2
Player 1b2= $1
1, 0
0, 0 0.75, 0.25
1, 0
b2= $0.5 b2= $0
b1= $1.5
b1= $0.5
b1
= $1
0.5, 0 0.5, 0
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First-Price auction, $0.5 bidding
interval: strategy for player 1
0.5, 0
0.5, 0
1.5, 0
Player 2
Player 1b2= $1
1, 0
0, 0 0.75, 0.25
1, 0
b2= $0.5 b2= $0
b1= $1.5
b1= $0.5
b1
= $1
0.5, 0 0.5, 0
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0.5, 0
0.5, 0
1.5, 0
Player 2
Player 1b2= $1
1, 0
0, 0 0.75, 0.25
1, 0
b2= $0.5 b2= $0
b1= $1.5
b1= $0.5
b1
= $1
0.5, 0 0.5, 0
First-Price auction, $0.5 bidding
interval: strategy for player 2
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First-Price auction, $0.5 bidding
interval: three pure strategy equilibria
0.5, 0
0.5, 0
1.5, 0
Player 2
Player 1b2= $1
1, 0
0, 0 0.75, 0.25
1, 0
b2= $0.5 b2= $0
b1= $1.5
b1= $0.5
b1
= $1
0.5, 0 0.5, 0
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First-Price auction, bidding interval I:z1 u z2 and I is very small
z1 - z2 - I, 0
Player 2
Player 1b2= z2
0, 0
b2= z2 + I
b1= z2 + I
b1= z2
b1= z2 - I
z1 - z2, 0
(z1 - z2 - I)/2,I/2
z1 - z2 - I, 0
(z1 - z2)/2, 0
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First-Price auction, bidding interval I:strategy for player 1
z1 - z2 - I, 0
Player 2
Player 1b2= z2
0, 0
b2= z2 + I
b1= z2 + I
b1= z2
b1= z2 - I
z1 - z2, 0
(z1 - z2 - I)/2,I/2
z1 - z2 - I, 0
(z1 - z2)/2, 0
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First-Price auction, bidding interval I:strategy for player 2
z1 - z2 - I, 0
Player 2
Player 1b2= z2
0, 0
b2= z2 + I
b1= z2 + I
b1= z2
b1= z2 - I
z1 - z2, 0
(z1 - z2 - I)/2,I/2
z1 - z2 - I, 0
(z1 - z2)/2, 0
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First-Price auction, bidding interval I:two pure strategy equilibria
z1 - z2 - I, 0
Player 2
Player 1b2= z2
0, 0
b2= z2 - I
b1= z2 + I
b1= z2
b1= z2 - I
z1 - z2, 0
(z1 - z2 - I)/2,I/2
z1 - z2 - I, 0
(z1 - z2)/2, 0
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First-price auction as a market
SupplyP
Q
z2
z1
Demand
Market equilibrium,
Auction equilibrium
1 2
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Bidding Principle 3
If you have the high valuation in a
first-price auction, you should bid asclose to the second-highest bidder as
the bidding interval allows.
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Second-Price Auctions
Two types of sealed-bid auctions, first-price and second-price
Second-price auctions remove theincentive to underbid
Unique solution in true values
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Second-price auction, $1 bidding interval
-0.5, -1
Player 2Player 1
b2 = $3
0, 1
b2 = $2 b2 = $1 b2 = $0
b1
= $3
b1 = $2
b1 = $0
b1 = $1 0, 0 0, 0 0.5, 0 2, 0
0, 1 1, 0.50, 1
2, 01, 0
2, 01, 00, 0
0, -1 0, -0.5
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Bidding Principle 4
In a second-price auction, always bid
true value.
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Individual Private Value
Auctions
The private value assumption
The probability of having the highest
value
Shaving ones bid for optimal expectedpayoff
Monotonic bidding functions
Perfect competition as the limit of anauction
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Three monotonic bid functions
b1
100
z140 80 100
b1 = z1
0
b1 = z1/2
b1 = z1
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Uniform distribution of valuations
Probability
0.01
z140 80 1000
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EV1 =p1(win) (z1 - b1) +p1(lose) v 0 =p1(win) (z1 - b1)
Suppose,p1(win) = kb1
With two bidders, EV1 = kb1(z1 - b1)
First Order Condition, xEV1/xb1 = 0 0 = kb1 v -1 + k(z1 - b1) b1(z1) = z1/2
With three bidders,p(1 wins against two rivals)
=p(1 wins against rival 1) v p(1 wins against rival 2)=p1(win)
2 = (kb1)2
Individual Private Value Auctions
with risk-neutral players
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With n bidders,p(1 wins against n - 1 rivals) = (kb1)n-1
@ EV1 = (kb1)n-1 (z1 - b1)
First Order Condition, xEV1/xb1 = 0 0 = (n - 1) k(kb1)n-2 (z1 - b1) + (kb1)
n-1 v -1
0 = (n - 1) z1 + n b1 b1(z1) = (n -1) z1/ n
Taking limit at n p w, b1(z1) = z1
Individual Private Value Auctions
with risk-neutral players
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When both players are risk-averse,
Eu1 =p1(win) v money = kb1 v (z1 - b1)
First Order Condition,xEV
1/xb
1 = 0 0 = kb1 v .5 v (z1 - b1)-.5 v -1 + k(z1 - b1)
b1(z1) = 2z1/3
When both players are risk-seeking,
Eu1 =p1(win) v (money)2 = kb1 v (z1 - b1)2
First Order Condition, xEV1/xb1 = 0 0 = kb1 v 2 v (z1 - b1) v -1 + k(z1 - b1)
2
b1(z
1) =z
1/3
Individual Private Value Auctions with
risk-averse or risk-seeking players
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Auctioning off Failed Thrifts
The sequel to depositors versus S&L
RTC auction procedures, especially the
generation of information
A statistical bidding function
Mitigating the damage to the Treasury of
the S&L crisis
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Common Value Auctions
Auctions where the underlying value isunknown
Transforming a signal into conditionalexpected value
The bid shaving principle for common
value auctions The winners curse
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Generating a noisy signal, table of
maximum values
1
Bidder 1
signalz2 = 1
2 3 4 65
2 2 3 4 65
3 3 3 4 65
4 4 4 4 65
5 5 5 5 65
6 6 6 6 66
z2 = 2 z2 = 3 z2 = 4 z2 = 5 z2 = 6
z1 = 1
z1 = 2
z1 = 3
z1 = 4
z1 = 5
z1 = 6
Bidder 2
signal
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Common-value auctions with two
bidders
Each bidderi can receive either the signal z1 = 1 or
z1 = 2 with probability 1/2 and the true value of the
item at auction, v = (z1 + z2)/2
When z1" z2, b1(z1)*" b2(z2)*
When both bidders received the signal zi= 1, there
are two equilibria:
b(1)* = (b1(1)*, b2(1)*) = (.80, .80) andb(1)* = (.90, .90)
When z1 = 2, z2 = 2 with probability 1/2 and z2 = 1
with probability 1/2 andb
1(z
1) = 0.875 + 0.25z
1
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Common-value auctions, two-bidders
E(v|z1)E(v|z1) = 1.75 + 0.5z1
4.75
z11 2 3 4 5 6
2.25
b1(z1) = 0.875 + 0.25z1
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-0.25, -0.25
Player 2
Player 1b2(2)= $1.7 b2(2)= $1.6 b2(2)= $1.5
b1(2)= $1.7
b1(2)= $1.5
b1(2)= $1.6
0.05, 0 0.05, 0
0.05, 0.050, 0.05
0, 0.05 0, 0.15
0.15, 0
0.125, 0.125
Common-Value auction, z1 = 2 is a
maximum signal
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-0.25, -0.25
Player 2
Player 1b2(2)= $1.7 b2(2)= $1.6 b2(2)= $1.5
b1(2)= $1.7
b1(2)= $1.5
b1(2)= $1.6
0.05, 0 0.05, 0
0.05, 0.050, 0.05
0, 0.05 0, 0.15
0.15, 0
0.125, 0.125
Common-Value auction, z1 = 2 is a
maximum signal: strategy for player 1
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Common-Value auction, z1 = 2 is a
maximum signal: strategy for player 2
-0.25, -0.25
Player 2
Player 1b2(2)= $1.7 b2(2)= $1.6 b2(2)= $1.5
b1(2)= $1.7
b1(2)= $1.5
b1(2)= $1.6
0.05, 0 0.05, 0
0.05, 0.050, 0.05
0, 0.05 0, 0.15
0.15, 0
0.125, 0.125
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Common-Value auction, z1 = 2 is a maximum
signal: three pure strategy equilibria
-0.25, -0.25
Player 2
Player 1b2(2)= $1.7 b2(2)= $1.6 b2(2)= $1.5
b1(2)= $1.7
b1(2)= $1.5
b1(2)= $1.6
0.05, 0 0.05, 0
0.05, 0.050, 0.05
0, 0.05 0, 0.15
0.15, 0
0.125, 0.125
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Bidding for Offshore Oil
Government leases of
The common value assumption in offshore
oil
Positive industry returns, some negativecompany returns
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Appendix. Auctions in the
Laboratory
Individual private value auctionexperiments
Common value auction experiments
Evidence in the winners curse
Lingering laboratory mysteries
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The bid function for a risk-neutral player,
bi(z
i) = 0.67 z
i
The bid function that best fits the data,b
i(z
i) = a + bz
iwhere a" 0, 1 " b " 0
Break-even bidding,
bi(zi) = zi
Bidding function of a typical subject
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Bidding function of a typical subject
bi
zi
a"0
0.67 b1 0
bi = a + bzi
bi = 0.67zi
bi = zi