Auctions
Many economic transactions are conducted through auctions
Microeconomics II
Auctions
Levent Koçkesen
treasury bills
art work
foreign exchange
antiques
publicly owned companies
cars
mineral rights
houses
airwave spectrum rights
government contracts
Also can be thought of as auctions
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takeover battles
queues
wars of attrition
lobbying contests
Levent Koçkesen (Koç University)
Auctions
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Auction Formats
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Auctions
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Auction Formats
Open bid auctions
1
ascending-bid auction
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⋆
2
Auctions also differ with respect to the valuation of the bidders
aka English auction
price is raised until only one bidder remains, who wins and pays the
final price
1
Private values
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descending-bid auction
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2
Levent Koçkesen (Koç University)
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aka Dutch auction
price is lowered until someone accepts, who wins the object at the
current price
2
Interdependent values
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Sealed bid auctions
1
first price auction
2
second price auction
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⋆
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highest bidder wins; pays her bid
each bidder knows only her own value and information that others have
do not affect her value
artwork, antiques, memorabilia
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bidders do not know the value of the object and have different private
information about that value
information that others have do have an affect on the value
special case: common values
oil field auctions, company takeovers
aka Vickrey auction
highest bidder wins; pays the second highest bid
Levent Koçkesen (Koç University)
Auctions
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Levent Koçkesen (Koç University)
Auctions
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Equivalent Formats
Independent Private Values
Open Bid
Sealed Bid
Single object for sale, n bidders
First Price
Bidder i assigns value Vi to the object
Vi is independently and identically distributed over [0, v] according to
distribution function F
Strategically Equivalent
Dutch Auction
English
Auction
Same Equilibrium
in Private Values
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Second Price
Bidders are risk neutral
We will study sealed bid auctions
Levent Koçkesen (Koç University)
F has a continuous density f = F ′ such that f (v) > 0 for all v ∈ [0, v]
Each bidder knows only her own valuation vi and that others’ values
are independently distributed according to F
Auctions
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Auctions as a Bayesian Game
Levent Koçkesen (Koç University)
Auctions
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Second Price Auctions
In this case player i pays p(b) = maxj6=i bj if she wins.
Set of players N = {1, 2, . . . , n}
Proposition
Type set Θi = [0, v], v > 0
βi (v) = v for all i is a weakly dominant strategy equilibrium.
Strategy set, Si = R+
Beliefs
Proof.
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Consider player i with value v and take any bid x < v. We have to show
that bidding v brings at least as much payoff as x for any bid profile b−i
and strictly more payoff for some b−i . Consider the bid profiles in which
Other bidders’ valuations are independent draws from F
Payoff functions: for any b ∈ Rn+ and vi ∈ Θi
(
vi −p(b)
m , if bi ≥ maxj6=i bj
ui (b, vi ) =
0,
if bi < maxj6=i bj
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p(b): price paid by the winner if the bid profile is b
m: Number of highest bidders
Strategy of player i, βi : Θi → R+
1
maxj6=i bj ≥ v > x: payoffs to both v and x are zero.
2
v > x > maxj6=i bj : both payoffs are equal to v − maxj6=i bj
3
v > maxj6=i bj > x: payoff to v is positive whereas the payoff to x is
zero.
4
v > maxj6=i bj = x: payoff to v is v − maxj6=i bj whereas the payoff
to x is (v − maxj6=i bj ) /m, for some m ≥ 2.
Exercise: Show that v weakly dominates any bid x > v.
Levent Koçkesen (Koç University)
Auctions
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Levent Koçkesen (Koç University)
Auctions
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First Price Auctions
Bayesian Equilibrium of First Price Auctions
2 bidders
F is uniform over [0, 1]
In this case player i pays bi if she wins.
Is there an equilibrium in which βi (v) = av for some a > 0?
Would you bid your value?
What happens if you bid less than your value?
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Optimal bid cannot be greater than a, since a is the highest possible
bid of the other player
You get a positive payoff if you win
But your chances of winning are smaller
Optimal bid reflects this tradeoff
Your expected payoff if you bid b ∈ [0, a]
(v − b)prob(you win) = (v − b)prob(b > aV2 )
Bidding less than your value is known as bid shading
Levent Koçkesen (Koç University)
Auctions
=(v − b)prob(V2 < b/a)
b
=(v − b)
a
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Bayesian Equilibrium of First Price Auctions
Levent Koçkesen (Koç University)
Auctions
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Bayesian Equilibrium of First Price Auctions
n bidders
If optimal bid is positive, FOC must hold
An equilibrium in which βi (v) = av for all i?
v
b v−b
=0⇒b=
− +
a
a
2
Your expected payoff if you bid b
(v − b)prob(you win)
Bidding zero yields zero payoff
Expected payoff to bid v/2
(v − b)prob(b > aV2 and b > aV3 . . . and b > aVn )
v2
2a
This is equal to
Bidding zero is optimal only for v = 0
(v−b)prob(b > aV2 )prob(b > aV3 ) . . . prob(b > aVn ) = (v−b)(b/a)n−1
βi (v) = v/2 is a Bayesian equilibrium
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Auctions
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Levent Koçkesen (Koç University)
Auctions
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Bayesian Equilibrium of First Price Auctions
Bayesian Equilibrium of First Price Auctions
First order condition:
−(b/a)n−1 + (n − 1)
Solving for b
βi (v) =
n−1
n v
βi (v) = β(v) and β is strictly increasing and differentiable?
β(0) = 0
v−b
(b/a)n−2 = 0
a
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n−1
v
b=
n
Suppose β(0) = b > 0
β(b) > b
Expected payoff to v = b
(b − β(b))prob(β(b) > β(V )) = (b − β(b))F (b) < 0
is a Bayesian equilibrium
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Auctions
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Bayesian Equilibrium of First Price Auctions
Levent Koçkesen (Koç University)
Equivalently
Expected payoff to b is
(v − b)F (β −1 (b))n−1 = (v − b)G(β −1 (b))
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d
(G(v)β(v)) = vG′ (v)
dv
Theorem (Fundamental Theorem of Calculus)
Let f : [a, b] → R and F : [a, b] → R be continuous on [a, b] and
F ′ (x) = f (x) for all x ∈ (a, b). Then
Z x
F (x) =
f (t)dt + F (a)
where G = F n−1 is the distribution function of the highest value
among n − 1 bidders.
Maximizing with respect to b yields the FOC
(v − b)
Auctions
G′ (β −1 (b))
− G(β −1 (b)) = 0
β ′ (β −1 (b))
a
By the fundamental theorem of calculus and the fact that β(0) = 0, we get
Z v
1
xG′ (x)dx
β(v) =
G(v) 0
Substituting b = β(v)
G(v)β ′ (v) + G′ (v)β(v) = vG′ (v)
This is only the necessary condition. Next we will show that β constitutes
an equilibrium.
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Auctions
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Auctions
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We can restrict ourselves to deviations b from β(v) that are in the
range of β.
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Proposition
b ≤ β(v) ⇒ ∃v : b = β(v) (since β is st’ly increasing and continuous)
... and b > β(v) cannot be a profitable deviation
Expected payoff of player i with type v who plays β(y)
Ui (y|v) = (v − β(y))G(y)
We want to show
Ui (v|v) ≥ Ui (y|v)
It has also been shown to be the unique equilibrium
Note that we can write this as
for all y ≥ 0
We have
∂
Ui (y|v) = (v − β(y))G′ (y) − β ′ (y)G(y)
∂y
From the FOC before: β ′ (y)G(y) = yG′ (y) − G′ (y)β(y)
Therefore
∂
Ui (y|v) = (v − y)G′ (y)
∂y
>0
for y < v
<0
Levent Koçkesen (Koç University)
β(v) = E[X|X < v]
where X is the random variable equal to the highest of n − 1 values.
We can also write it in terms of F
Rv
F (x)n−1 dx
β(v) = v − 0
F (v)n−1
for y > v
Auctions
β(v) < v for all v > 0: bid shading
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First price auctions: β(v) < v, highest value wins
Second price auctions: β(v) = v, highest value wins
If you were the seller which format would you use?
Doesn’t matter: they generate the same expected revenue
It turns out that this is a more general result:
Proposition (Revenue Equivalence Theorem)
Any auction with independent private values with a common distribution
in which
1
the number of the bidders are the same and the bidders are
risk-neutral,
2
the object always goes to the buyer with the highest value,
3
the bidder with the lowest value expects zero surplus,
yields the same expected revenue.
Levent Koçkesen (Koç University)
Auctions
There is a unique symmetric Bayesian equilibrium in strictly increasing and
differentiable strategies given by
Z v
1
xG′ (x)dx
β(v) =
G(v) 0
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Levent Koçkesen (Koç University)
Auctions
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