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Econ 805
Advanced Micro Theory 1
Dan Quint
Fall 2007
Lecture 2 – Sept 6 2007
Today
Common auction formats
The Independent Private Values model
1
Common Auction Formats
and Strategic Equivalences
2
Dutch Auction
Auctioneer begins at a high price and lowers it until a
buyer claims the object at the current price
A slightly abstracted view: the price falls continuously
(on a clock) instead of in increments
In a literal sense, a bidder’s strategy can be thought
of as a choice of whether or not to buy at each price;
but for practical purposes, it can be reduced to a
decision of at what price to shout “mine!” if the item’s
still available
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“Sealed Tender” or “First-Price” Auction
Each bidder submits a sealed bid
The object goes to the bidder with the highest bid, at
that price
A strategy is simply a choice of how much to bid
4
Dutch Auction = First-Price Auction
If all the matters is who wins the object and how
much they pay, then the Dutch Auction and the FirstPrice Auction are equivalent
In each, a bidder’s strategy is reduced to picking a number;
the highest number wins, and pays that much
5
English (Ascending) Auction
Think of art auctions
Price begins low; auctioneer solicits bids at the next
price, keeps naming higher prices until no one is
willing to raise their bid
Or, bidders name their own prices until no one is
willing to outbid the high bidder – think of online
auctions without proxy bidding
High bidder pays what he bid
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Simplified English Auction (or Button
Auction)
Price begins low, rises continuously
At each price, bidders can remain active (hold down
a button) or drop out permanently
Bidders only know the current price (not who has
dropped out and at what price)
When the second-to-last bidder drops out, the last
man standing pays the current price
A bidder’s strategy can be reduced to choosing a
price at which to drop out if he hasn’t won
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Second-Price (or Vickrey) Auction
Each bidder submits a sealed bid
The object goes to the highest bidder, but the price
they pay is the second-highest bid
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Simplified English = Second-Price
Again, if we reduce the game to the question of who
wins and how much they pay, the Simplified English
Auction and Second Price Auction are equivalent
Strategies are reduced to picking a number
Highest number wins; payment is second-highest number
But the Simplified English Auction changes if bidders
can see who is still active at each price
If I’m unsure of the exact value of the object, I may revise
my estimate depending on how other bidders bid
Then strategies can no longer be reduced to picking a
single number, and the equivalence breaks down
9
All-Pay Auctions and Wars of Attrition
In an All-Pay Auction, bidders submit sealed bids, the
high bid wins the object, but everyone pays what they
bid
All-Pay Auctions are sometimes used to model lobbying,
attempts to buy political influence, and patent races – the
losers already made their contributions or incurred their
costs
War of Attrition is the same, but dynamic – like an allpay button auction where bidders can see who’s still
active
Great game for an undergrad game theory class – auction
off a $20 bill, highest bid wins, highest two bids both pay
what they bid
10
Multi-Unit Auctions with Unit Demand
Suppose there are k > 1 identical items for sale, but
each bidder can only have one
“Pay-as-bid” auction is like a first-price auction – the k
highest bidders win and pay their bids
Analog to the second-price auction is the “k+1st-price”
auction
Button auction works similarly, ends when the k+1st
bidder left drops out
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The Independent
Private Values Model
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Baseline model of an auction as a Bayesian
Game: Symmetric Independent Private Values
N > 1 bidders in an auction for a single object
Nature moves first, assigning each bidder a private
valuation vi for the object
Each bidder’s value vi is an independent draw from a
common probability distribution F
Each bidder knows his own value vi but not that of his
opponents
Bidder i’s payoff is vi – p if he wins, 0 if he loses,
where p is the price he pays for the object
Like the Cournot game, i’s payoff depends on j’s type only
through j’s action – this is what’s meant by “private values”13
Note all the implicit assumptions we’re
making
The number of bidders is fixed – there is no decision
over whether or not to participate
Each bidder knows his own valuation perfectly, does
not care what the other bidders think of the object
The bidders are symmetric ex-ante – valuations are
drawn from the same distribution, which is common
knowledge
Valuations are statistically independent
Bidders are risk-neutral
14
Auctions to sell versus auctions to buy
Suppose the government holds an auction for a
contract to provide some service
Bids are now offers to provide the service at a given price,
and the lowest bid wins
Where buyers were distinguished by their valuations for
winning their object, firms can be thought of as
distinguished by their cost of providing the service
So firm i’s payoffs would be p – ci, where p is the price
received, and all the same analysis goes through
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Solving for Equilibrium in the
First- and Second-Price Auctions
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Second-price (Vickrey) auctions in the
IPV world
Claim. In a second-price sealed-bid auction,
submitting a bid equal to your value is a weakly
dominant strategy
Proof. Let B be the highest of your opponents’ bids.
When B > v, you could only win the object at price B, for a
payoff of v – B < 0; bidding b = v gives you 0, which is as
good as you can do
When B < v, any bid b > B gives the same payoff, v – B > 0,
which is payoff from bidding b = v and the best you can do
When B = v, any bid gives the same payoff, 0
Corollary. Every bidder playing the strategy
bi(vi) = vi is a Bayesian Nash Equilibrium of the
second-price auction
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Similarly…
In a button auction, it’s a dominant strategy to drop out when the price
reaches your private value vi
Doesn’t matter if you can observe who’s already dropped out or not
In an open-outcry ascending auction…
Equilibrium strategies are not clear
But it is a dominant strategy to never bid above your private value vi, nor
to let the auction end at price below vi – d
So any equilibrium will involve the highest-value bidder winning (unless
the highest two are within d of each other), and paying within d of the
second-highest value
So with private values, as d gets small, second-price or button auctions
give approximately the same outcome as ascending auctions
Also similar is a first-price auction with proxy bidding, a la eBay
Bidders can name a maximum, then the computer raises their bid to the
minimum required to win until that maximum is reached
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Sadly, “everyone bids their value” is not the
only equilibrium of the second-price auction
Suppose bidder values were drawn from a
distribution with support [0,10]
The following is an equilibrium of the second-price
auction:
Bidder 1 bids 15 regardless of his type
All other bidders bid 0 regardless of their type
bi(vi)=vi is “nearly” the only symmetric equilibrium;
and it involves bidders playing a strict best-response
at nearly every type
19
First-price auctions in the symmetric
IPV world
We’ll look for “nice” equilibria:
Symmetric (bidders all play the same strategy)
Bids are increasing in valuations
Tomorrow, we’ll learn a trick that makes finding this
type of equilibrium much easier
Suppose such an equilibrium exists, and let
b : [0,V] R+ be the common bid function; then at a
given type v, b(v) must be a solution to
max x R+ (v – x) Pr(win | bid x, opponents bid b(-))
= max x R+ (v – x) Pr(b(vj) < x " j i)
= max x R+ (v – x) Pr(vj < b-1(x) " j i)
= max x R+ (v – x) FN-1(b-1(x))
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If there is a symmetric, increasing
equilibrium in a first-price auction…
maxx (v x)F N 1 (b1 ( x))
b(v) must solve
0 F
N 1
0 F
N 1
b ' (v ) F
N 1
dF N 1 1
(b ( x)) (v x)
(b ( x))(b 1 )' ( x)
dv
1
d
(v) (v b(v)) F N 1 (v) / b' (v)
dv
b(v) F
(b-1)’ = 1/b’
x = b(v) in equilibrium
d N 1
d N 1
(v ) b (v ) F (v ) v F (v )
dv
dv
d
d N 1
N 1
b (v ) F (v ) v F (v )
dv
dv
N 1
First-order condition
so integrating from 0 to v,
v
(v) sd ( F N 1 (s))
0
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So if there is a “nice” equilibrium, it must be
b(v) = 0v sd(FN-1(s)) / FN-1(v)
What is this?
Well, if a random variable y has cumulative
distribution G with positive support, then
0v s dG(s) / G(v) = E(y | y < v)
And FN-1(v) is the cumulative distribution function of
the highest of N-1 independent draws from F
So if we let v1 and v2 refer to the highest and secondhighest valuations in a symmetric IPV model, then
b(v) = E(v2 | v1 = v)
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Now here’s where it gets cool…
In the symmetric equilibrium of the second-price auction, the
price paid is v2, so the seller’s expected revenue is simply E(v2)
In the symmetric, increasing equilibrium in the first-price
auction (if it exists),
The bidder with the highest value wins
If the highest value is v, the winner pays E(v2 | v1 = v)
So the seller’s expected revenue is
E v1 E(v2 | v1) = E(v2)
So the seller’s expected revenue is the same in both auctions!
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And similarly…
In the first-price auction…
A bidder with type v expects to win with probability FN-1(v), and to
pay b(v) = E(v2 | v1 = v) when he wins
So his expected payoff is FN-1(v) [ v – E(v2 | v1 = v) ]
In the second-price auction…
A bidder with type v expects to win whenever he has the highest
value (v1 = v), and to pay v2 when he wins
So his expected payment, conditional on winning, is E(v2 | v1 = v)
And so his expected payoff is FN-1(v) [ v – E(v2 | v1 = v) ]
So each type of bidder gets the same expected payoff in the
two auctions
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This turns out not to be a fluke
This is exactly what we’ll prove more generally next class:
With independent private values, any two auctions in
which, in equilibrium,
the player with the highest value wins the object, and
any player with the lowest possible type gets expected payoff of 0
will give the same expected payoff to each type of each
player, and the same expected revenue to the seller
So, (a) this is pretty interesting, and (b) once we’ve
proven this, we can use it to calculate equilibrium
strategies much more easily
25