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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