Econ 805
Advanced Micro Theory 1
Dan Quint
Fall 2009
Lecture 2
Today
 Common auction formats
 The Independent Private Values model
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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
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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
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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

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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
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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
 F is common knowledge

 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”
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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
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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 (where d is the minimum bid
increment)
 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; and it’s the equilibrium we’ll
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focus on
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…
max x (v  x) F N 1 (b1 ( 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
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In Case We Have Time,
A Few Slides on Second-Order
Stochastic Dominance
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When is one probability distribution less
risky than another?

Two random variables X and Y with the same
mean, with distributions F and G

Three conditions to consider:
1. “Every risk-averse utility maximizer prefers X to Y”, i.e.,
E u(X)  E u(Y) for every nondecreasing, concave u,
or - u(s) dF(s)  - u(s) dG(s)
(also called SOSD)
2. “Y is a mean-preserving spread of X”, or “Y = X + noise”:
$ r.v. Z s.t. Y =d X + Z, with E(Z|X) = 0 for every value of X
3. For every x,
-x F(s) ds  -x G(s) ds

Rothschild-Stiglitz (1970): 1  2  3
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What does this tell us?

Risk-averse buyers greatly impact auction design –
changes equilibrium strategies – we’ll get to that in
a few lectures (Maskin and Riley)

Risk-averse sellers have less impact – equilibrium
strategies are the same, all that changes is seller’s
valuation of different distributions of revenue

Claim. With symmetric IPV, a risk-averse seller
prefers a first-price to a second-price auction
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Proof: we’ll show revenue in second-price
auction is MPS of revenue in first-price


Recall that revenue in a second-price auction is v2, and
revenue in a first-price auction is E(v2 | v1)
Let X, Y, and Z be random variables derived from bidders’
valuations, as follows:







X = g(v1)
Z = v2 – g(v1)
Y = v2
where g(t) = 0t s dFN-1(s) / FN-1(t) = E(v2 | v1 = t)
Note that Y = X + Z, and
E(Z | X=g(t)) = E(v2 | v1 = t) – E(v2 | v1 = t) = 0
So Y is a mean-preserving spread of X, so any risk-averse
utility maximizer prefers X to Y
But X is the revenue in the first-price auction, and Y is the
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revenue in the second-price auction – Q.E.D.
A cool proof SOSD 
“-x F(s) ds  -x G(s) ds everywhere”
 We’ll use the “extremal method” or “basis function method”
 We’ll rewrite our generic (increasing concave) function u(s) as a
positive sum of basis functions
u(s) = - w(q) h(s,q) dq
with w(q)  0, where these basis functions are themselves increasing
and concave
 Then we’ll show that X SOSD Y if and only if
- h(x,q) dF(x)  - h(y,q) dG(y)
for all the basis functions
 (“Only if” is trivial, since h(s,q) is increasing and concave; “if” just
involves multiplying this inequality by w(q) and integrating over q)
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A cool proof SOSD 
“-x F(s) ds  -x G(s) ds everywhere”
 We’ll do the special case of u twice differentiable. Our basis functions
will be a constant, a linear term, and the functions
h(x,q) = min(x,q)
 Claim is that
u(x) = a + bx + 0 (-u’’(q)) h(x,q) dq
 Note that -u’’(q) is nonnegative, since u is concave
 To see the equality, integrate by parts, with db = -u’’ dq, a = h:
a db
= a b – b da
= –h(x,q)u’(q)|q=- – - –u’(q) 1q<x dq
= –xu’() + constant + -x u’(q) dq
 Since X and Y have the same mean,
- (a+bx) dF(x)  - (a+by) dG(y)
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A cool proof SOSD 
“-x F(s) ds  -x G(s) ds everywhere”
 So all that’s left is to determine when
- h(s,q) dF(s)  - h(s,q) dG(s)
 Integrate by parts: u = h(s,q), dv = dF(s), LHS becomes
h(,q) F() – h(-q) F(-) – -F(s) hs(s,q) ds
= q – 0 – -F(s) 1s<q ds
= q – -qF(s) ds
 Similarly, the right-hand side becomes q – -qG(s) ds
 So Es~F h(s,q)  Es~G h(s,q)

-qF(s) ds  -qG(s) ds
 So X SOSD Y if and only if this holds for every q
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