Advanced Microeconomics II

Auction Theory

Jiaming Mao

School of Economics, XMU

Introduction

Auction is an important allocaiton mechanismI EbayI ArtworkI Treasury bondsI Air waves

Introduction

Common Auction Formats

Open ascending price or English auctionI Auctioneer begins by calling out low price and raises it in small

incrementsI Auction ends when there is only one remaining bidder

Open descending price or Dutch auctionI Descending counterpart to English auction

Common Auction Formats

First-Price, Sealed Bid AuctionI Bidders submit bids in sealed envelopesI At a pre-determined time, auctioneer opens all envelopes and

ranks bidsI Highest bidder obtains object and pays his bid amount

Second-Price, Sealed Bid AuctionI Same as first-price sealed bid auction except that highest bidder

obtains object and pays second highest bid amount

Common Auction Formats

All-Pay auctionI Bidders submit bids (open or closed)I At a pre-determined time, auctioneer opens all envelopes and

ranks bidsI Every bidder pays what they bid regardless of whether or not

they have the highest bidI Highest bidder obtains objectI Examples: Elections, almost any kind of contest or sports, R&D,

wars, lobbying

Auctions

All auctions can be interpreted as allocation mechanisms with thefollowing ingredients:

An allocation ruleI who gets the object

A payment ruleI how much every bidder pays

Private Valuations

Each bidder knows her own valuation v

i

, but not other bidders’valuationsThe distribution of i ’s valuation v

i

is common knowledge:

F

i

(x) = Pr (vi

< x)

Private Valuations

v ⇠ U (0, 1)

Private Valuations

Let b

i

: [0, 1] ! <+ denote i ’s bidding functionI We will also use b

i

to denote i ’s actual bid

If all bidders have the same valuation distribution F , then theywill be using the same bidding function b

They might, however, submit different bids, depending on theirprivately observed valuation

I If b (v) = v

2

, then bidder with valuation 4 will submit $2. Bidderwith valuation 10 will submit $5

I Even if bidders are symmetric in the bidding function they use,they can be asymmetric in the actual bid they submit.

First-Price Auctions

b

i

< v

i

I “Bid shading”: no point in bidding b

i

� v

i

Assume v

i

⇠ U (0, 1) 8i

We will be looking for equilibrium bidding functions of the formb

i

(vi

) = a · vi

, a 2 (0, 1)I We will prove later when we look at the general case (without

the assumption of v

i

⇠ U (0, 1)) that the only symmetricequilibrium bidding function in FPA under uniform valuedistribution is indeed of this form

FPA with 2 bidders

EU

i

(bi

|vi

) = Pr (bi

> b

j

) (vi

� b

i

)

= Pr (bi

> av

j

) (vi

� b

i

)

= F

✓

b

i

a

◆

(vi

� b

i

)

=b

i

a

(vi

� b

i

)

)

b

i

= arg maxx

EU

i

(x |vi

)

=v

i

2

FPA with 2 bidders

Bid shading in half

Optimal bidding function with N = 2 bidders

FPA with N bidders

EU

i

(bi

|vi

) = Pr (bi

> max {b�i

}) (vi

� b

i

)

=Y

j 6=i

Pr (bi

> b

j

) (vi

� b

i

)

=

F

✓

b

i

a

◆�

N�1

(vi

� b

i

)

=

b

i

a

�

N�1

(vi

� b

i

)

)

b

i

= arg maxx

EU

i

(x |vi

)

=N � 1

N

v

i

FPA with N bidders

Optimal bidding function: b (v) = N�1

N

v

Bid shading diminishes as N increases

Optimal bidding function increases in N

FPA - Generalization

Arbitrary distribution of v

i

b

i

= arg maxx

Pr (bi

> max {b�i

}) (vi

� b

i

)

= arg maxx

Y

j 6=i

F

j

�

b

�1

j

(x)�

(vi

� x)

= arg maxx

⇥

F

�

b

�1 (x)�⇤

N�1

(vi

� x)

FOC:

�⇥

F

�

b

�1 (x⇤)�⇤

N�1

+ (N � 1)

⇥⇥

F

�

b

�1 (x⇤)�⇤

N�2

f

�

b

�1 (x⇤)� 1

b

0 (b�1 (x⇤))(v

i

� x

⇤) = 0

FPA - Generalization

In symmetric equilibrium, b

i

(vi

) = b (vi

) = x

⇤, )

b

i

=1

F

N�1 (vi

)

ˆvi

0

udF

N�1 (u)

= v

i

� 1F

N�1 (vi

)

ˆvi

0

F

N�1 (u) du

When F (v) = v , b

i

= N�1

N

v

i

FPA - Generalization

Theorem

If N bidders have independent private values drawn from the common

distribution F , then bidding

b

i

(vi

) =1

F

N�1 (vi

)

ˆvi

0

udF

N�1 (u)

is the symmetric NE of a first-price, sealed-bid auction.

FPA with risk-averse bidders

Utility function is concave in income: u (c) = c

↵, ↵ 2 (0, 1)Expected utility of bidding:

EU

i

(bi

|vi

) = Pr (bi

= max {b1

, . . . , bN

}) (vi

� b

i

)↵

FPA with risk-averse bidders

Consider 2 bidders with v

i

⇠ U (0, 1) 8i

b

i

= arg maxx

x

a

(vi

� x)↵

=v

i

1 + ↵

When ↵ = 1 (risk-neutral), b

i

= vi2

FPA with risk-averse bidders

Optimal bidding function: b (v) = v

1+↵

Bid shading is ameliorated as bidders’ risk aversion increases

Optimal bidding function with risk-averse bidders

FPA with risk-averse bidders

Intuition: for a risk-averse bidder:I the positive effect of slightly lowering his bid, arising from

getting the object at a cheaper price, is offset by...I the negative effect of increasing the probability that he loses the

auction.

Ultimately, the bidder’s incentives to shade his bid arediminished.

Second-price Auctions

Let h

i

⌘ maxj 6=i

{bj

}

EU

i

(bi

|vi

, hi

) = Pr (bi

> h

i

) (vi

� h

i

)

Second-price Auctions

b

i

(vi

) = v

i

EU

i

= Pr (vi

> h

i

) (vi

� h

i

)

Second-price Auctions

b

i

(vi

) < v

i

EU

i

= Pr (bi

> h

i

) (vi

� h

i

)

< Pr (vi

> h

i

) (vi

� h

i

)

Second-price Auctions

b

i

(vi

) > v

i

EU

i

= Pr (bi

> h

i

) (vi

� h

i

)

= Pr (vi

> h

i

) (vi

� h

i

)� Pr (vi

< h

i

< b

i

) (hi

� v

i

)

< Pr (vi

> h

i

) (vi

� h

i

)

Second-price Auctions

Bidding b

i

(vi

) = v

i

is a weakly dominant strategy.Alternative argument:

I Since you pay h

i

when you win, the optimal strategy is one thatguarantee you will win when v

i

> h

i

and you will lose whenv

i

< h

i

I This strategy is to bid v

i

The result is unaffected byI number of bidders N

I their risk-aversion preferencesI their valuation distributions

F bidders need not be symmetric: Fi (vi ) can differ from Fj (vj)

Second-price Auctions

Theorem

If N bidders have independent private values, then bidding one’s

value is the unique weakly dominant bidding strategy for each bidder

in a second-price, sealed-bid auction.

Dutch Auctions

Bidder decides at what price to raise hand.If wins, bidder pays the price at which he/she raises hand.Dutch auction is equivalent to FPA.

Theorem

If N bidders have independent private values drawn from the common

distribution F , then raising one’s hand when the price reaches

1F

N�1 (vi

)

ˆvi

0

udF

N�1 (u)

is the symmetric NE of a Dutch auction.

English Auctions

Bidder decides at what price to drop out.Winner pays the price at which last remaining competitor dropsout, i.e. winner pays the second highest bid.English auction is equivalent to SPA.

Theorem

If N bidders have independent private values, then dropping out when

the price reaches one’s value is the unique weakly dominant bidding

strategy for each bidder in an English auction.

Revenue Comparisons

In First-price and Dutch auctions, bidders bid less than theirvaluations but the seller receives the highest bid.In Second-price and English auctions, bidders bid their valuationsbut the seller receives the second highest bid.Which formats have the highest expected revenue?

I Let v

[1]N

⌘ max {v1

, . . . .vN

} and v

[2]N

⌘ maxn

{v1

, . . . .vN

}\ v

[1]N

o

I When v

[1]N

� v

[2]N

, FPA can yield more revenue than SPAI When v

[1]N

, v [2]N

are close, SPA can yield more revenue than FPA

Revenue Comparisons

Distribution of v

[1]N

F

[1]N

(v) = F

N (v)

Distribution of v

[2]N

F

[2]N

(v) = Prn

v

[2]N

v

o

Revenue Comparisons

The eventn

v

[2]N

v

o

can occur in one of two distinct (mutuallyexclusive) ways

1 All valuations below v : v

i

v 8i

2N � 1 valuations below v and 1 valuation above v

I This event can occur in N different ways1 v1 > v and vi v 8i 6= 12 v2 > v and vi v 8i 6= 23 . . .

Revenue Comparisons

Hence,

F

[2]N

(v) = Prn

v

[1]N

v

o

+N

X

i=1

Pr {v1

> v and v

j

v 8j 6= i}

= F

N (v) + N (1 � F (v)) F

N�1 (v)

= NF

N�1 (v)� (N � 1) F

N (v)

Revenue Comparisons

Assuming v is distributed according to F (v) on [0, 1], theexpected revenue of FPA

E

⇥

r

FPA

⇤

=

ˆ1

0

b (v) dF

[1]N

(v)

=

ˆ1

0

1F

N�1 (v)

ˆv

0

udF

N�1 (u)

�

dF

N (v)

= N (N � 1)ˆ

1

0

ˆv

0

uF

N�2 (u) f (u) du

�

f (v) dv

= N (N � 1)ˆ

1

0

ˆ1

u

uF

N�2 (u) f (u) f (v) dvdu

= N (N � 1)ˆ

1

0

uF

N�2 (u) f (u) (1 � F (u)) du

Revenue Comparisons

The expected revenue of SPA:

E

⇥

r

SPA

⇤

=

ˆ1

0

vdF

[2]N

(v)

= N (N � 1)ˆ

1

0

vF

N�2 (v) f (v) (1 � F (v)) dv

= E

⇥

r

FPA

⇤

Revenue Comparisons

When v

i

⇠ U (0, 1),

E

⇥

r

FPA

⇤

=

ˆ1

0

b (v) dF

[1]N

(v) =

ˆ1

0

✓

N � 1N

v

◆

Nv

N�1 (v)

= E

⇥

r

SPA

⇤

=

ˆ1

0

vdF

[2]N

(v) = N (N � 1)ˆ

1

0

v

N�1 (1 � v) dv

=N � 1N + 1

Revenue Equivalence Principle

Theorem

Suppose that values are independently and identically distributed and

all bidders are risk-neutral. Then any symmetric and increasing

equilibrium of any standard auction, such that the expected payment

of a bidder with value zero is zero, yields the same expected revenue

to the seller.

An auction is called standard if the rules of the auction dictatethat the person who bids the highest is awarded the object

Revenue Equivalence Principle

Proof.

Let P (v) be the equilibrium expected payment by a bidder withvalue v . Now consider bidder i . Suppose other bidders are followingthe equilibrium bidding strategy b (v). Bidder i needs to decidewhether to bid b (z) instead of the equilibrium bid b (v

i

)

Bidder i solvesmax

z

{G (z) v

i

� P (z)}

, where G (z) ⌘ F (z)N�1

Revenue Equivalence Principle

Proof. (Cont.)

FOC)g (z) v

i

� P 0 (z) = 0

In symmetric equilibrium,

P 0 (vi

) = g (vi

) v

i

) P (vi

) =

ˆvi

0

ug (u) du

= E

h

v

[1]�i

�

�

�

v

i

> v

[1]�i

i

Pr⇣

v

i

> v

[1]�i

⌘

, where v

[1]�i

denotes the highest value among v�i

.Since the equilibrium expected payment function P (v) does notdepend on any particular auction format, we prove the theorem.

Revenue Equivalence Principle

Example (FPA)

Since the equilibrium expected payment isP (v) = E

h

v

[1]�i

�

�

�

v

i

> v

[1]�i

i

Pr⇣

v

i

> v

[1]�i

⌘

and the winning bidder payswhat she bids, the equilibrium bidding strategy must be

b (vi

) = E

h

v

[1]�i

�

�

�

v

i

> v

[1]�i

i

=1

G (vi

)

ˆvi

0

udG (u)

⌘ 1F

N�1 (vi

)

ˆvi

0

udF

N�1 (u)

Efficiency in Auctions

The object is assigned to the bidder with the highest valuation.I Otherwise, the outcome of the auction cannot be efficient, since

there exist alternative reassignments that would still improvewelfare.

I FPA, SPA, Dutch and English auctions are hence efficient, sincethe player with the highest valuation submits the highest bid andwins the auction.

I Lottery auctions are not necessarily efficient.

Reference

The lecture slides draw from the following sources

Jehle, G. A. and P. J. Reny. 2011. “Advanced MicroeconomicTheory,” Prentice Hall, 3e.Krishna, V. 2009. “Auction Theory,” Academic Press, 2e.Munoz-Garcia, F. 2015. “Advanced Microeconomic Theory II,”lecture notes at http://cahnrs-cms.wsu.edu/ses/people/Munoz/Pages/default.aspx