Graduate Microeconomics II
Lecture 10: Mechanism Design and Auctions
Patrick Legros
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Outline
Mechanism Design
The Revelation Principle
A Generalized Envelop Theorem
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Outline
Mechanism Design
The Revelation Principle
A Generalized Envelop Theorem
Auctions
Types of Auctions
Revenue Equivalence
Winnner’s curse
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Mechanism Design
General approach to “institutional” design, with or without
asymmetry of information. Idea:
I
agents communicate with a “center”
I
center aggregates messages received
I
center takes a decision
Exemples
I
market settings
I
auctions
I
procurement
I
collective decision making
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Mechanism design
Basic decision problem
I
N agents
I
each agent has type ti ∈ Ti
I
the prior probability of a state t = (t1 , · · · , tN ) is p(t)
I
agents are Bayesian: upon knowing their type they form a
belief pi (t−i |ti )
I
a decision d ∈ D has to be made and agents have utility
ui (d, t) from decision d
I
if d(t) is a state contingent decision the expected utility of
agent i with type ti is
X
U(d, ti ) =
pi (t−i |ti )ui (d, t)
t−i
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Mechanism design
Mechanism
A mechanism is a pair (M, g ) where M = ×N
i=1 Mi and g : M → D.
A mechanism generates a Bayesian game. Action for i is mi ∈ Mi
and a behavioral strategy is a map σi : Ti → Mi .
Given strategies σi the expected payoff of agent i with type ti is
X
p(t−i |ti )u(g (σ(t), t)
t−i
If σ is an equilibrium of the Bayesian game generated by (M, g ),
then we say that the decision rule d(t) = g (σ(t)) is implemented
by the mechanism (M, g ).
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Mechanism design
Direct revelation games
A special case of mechanism is that of direct revelation games
when Mi = Ti for each i.
In a direct revelation game there is truth-telling if “telling the
truth” is an equilibrium, that is if the strategy σi (ti ) = ti is an
equilibrium.
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Mechanism Design
The Revelation Principle
Consider a mechanism (M, g ) and an equilibrium σ of the
corresponding Bayesian game. There exists a direct revelation
mechanism (T , h) such that truth-telling is an equilibrium and
such that for each t the outcome is the same as in the initial
game; that is g (σ(t)) = h(t).
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d
{Ti }
{σi }
D
g
{Mi }
d
{Ti }
{σi }
id
D
g
{Mi }
h =g ◦σ
{Ti }
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Mechanism design
“Proof” of the RP
With one agent: if σ is optimal in state t, then for all t 0 ,
u(g (σ(t)), t) ≥ u(g (σ(t 0 ), t)
(1)
Let h = g ◦ σ and suppose that the agent does not want to tell the
truth in the direct revelation game. Then there exist t and t 0 such
that
u(h(t 0 ), t)
>
u(h(t), t)
⇔
0
u(g (σ(t )), t)
>
u(g (σ(t)), t)
which contradicts (1).
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Mechanism Design
A Generalized Envelop Theorem
Screening problems are examples of mechanism design (where the
center is a firm offering a wage).
A buyer-seller transaction is another example of mechanism design:
the seller commits to a menu (quality,price). Since the buyer will
self-select, that is choose the element of the menu that is optimal
given his type, this is equivalent to having the firm commit to a
direct revelation mechanism where the quality and price are a
function of the type announced by the buyer.
Another example is an auction, where there are multiple potential
buyers and (to simplify) a single object for sale by a (monopoly)
seller.
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Auctions
Types of Auctions
English auction auctioneer calls a low price and raises it by small
increments as long as there are at least two bidders
still competing. Auction stops when there is only one
bidder left.
Dutch auction auctioneer calls a “high” price and starts decreasing
it. The first bidder to indicate his interest wins and
pays the corresponding price.
Sealed-bid bidders write a bid in a sealed envelop; the highest
bidder wins. Two variants:
I First price: the winner pays his bid to the seller
I Second price: the winner pays the bid of the
second highest bid.
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Auctions
Valuations
Private value the bidder knows the value of the object: his
valuation does not depend on the “types” of other
bidders
Common value a bidder has some information about the value of
the object but the other bidders’ information is
relevant for his valuation.
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Auctions
Efficiency, seller’s revenue
Total surplus (if risk neutrality): difference between the valuation
of the winner and the valuation of the seller.
Efficiency : highest valuation bidder gets the object. Is this
consistent with revenue maximization by the seller?
A way to make sense of this is to take a mechanism design
approach and characterize the revenue maximizing mechanism for
a seller.
A DRM corresponding to an auction specifies for a given profile of
types t that has been sent (i) the probability p(t) that bidder i
gets the object and (ii) the price x(t) that this bidder will pay to
the seller.
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Auctions
Revenue Equivalence
Under private values and independent types (with distribution f ),
the utility of a bidder is then
Z
Y
U(ti ) =
(p(t)u(ti ) − x(t))
f (tj )dtj
t−i
j6=i
= P(ti )u(ti ) − X (ti )
where
Z
P(ti ) =
p(t))
t−i
and
j6=i
Z
X (ti ) =
x(t)
t−i
Y
(f (tj )dtj )
Y
f (tj )dtj
j6=i
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Auctions
Incentive compatibility
Hence in a DRM, imposing truth-telling for the other agents, we
are back to a one agent problem!
Incentive compatibility requires that for all ti , ti0 ,
U(ti ) ≥ P(ti0 )u(ti ) − X (ti0 )
and replicating the reasoning we made previously for screening
problems we conclude that
I
X and P are increasing in ti
I
hence are differentiable
I
moreover, U̇(ti ) = P(ti )
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Auctions
Revenue equivalence
We get the “envelop result”:
Z
ti
U(ti ) = U(tmin ) +
P(τ )dτ
tmin
Moreover the expected revenue to the seller is
R =
=
Z X
N
t i=1
N Z
X
i=1
xi (t)
N
Y
f (ti )dt
i=1
X (ti )f (ti )dti
ti
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Auctions
Revenue equivalence
It follows form this that if two auction forms :
1. give the same level of expected utility to each bidder when he
has his lowest valuation
2. is efficient in the sense that the object goes to the highest
valuation bidder
then the two auctions give the same level of expected revenue to
the seller.
True if: risk neutrality, independent valuations (could have
distributions of types that are dependent).
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Examples
Order statistic
(n)
Let F1 (t) be the distribution of the first order statistic when there
(n)
are n bidders and F2 (t) that of the second order statistic. Clearly,
(n)
F1 (t) = F (t)n
(n)
F2 (t) = F (t)n + nF (t)n−1 (1 − F (t))
= nF (t)n−1 − (n − 1)F (t)n−1
(n−1)
= nF1
(n)
(t) − (n − 1)F1 (t)
Note that
(n)(t)
f2
= n(n − 1)(1 − F (t))F (t)n−2 f (t)
(n−1)
= n(1 − F (t))f1
(t)
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Auctions
Example 1: second price sealed bid
2 bidders with private values and independent types that are
identically distributed. Each bidder has type t uniformly
distributed on [0, 1]: F (t) = t.
If bids are b1 , b2 the winner is 1 if b1 > b2
A (weakly) dominant strategy to bid his valuation: b(ti ) = ti .
Hence for a vector of types t the price paid is equal to the second
highest type.
Type t wins if the other bidder has type less than t: this happens
with probability F (t); the winning bidder pays in this case the
second highest bid. Hence the expected payment of type t is
Z t
f (x)
P(t) = F (t) ×
x
dt
F
(t)
0
| {z }
conditional density
2
= t /2
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Auctions
Example 2: first price
same assumptions as in the previous example. In a first price
auction, the winner is the highest type and pays b(t). Let β(ti ) be
the equilibrium bid (symmetric equilibrium). By IC, we know that
β is increasing, hence has an inverse β −1 (b).
Bidder 1 wins if and only if he has a higher type. The probability
that he wins with a bid of b is F (β −1 (b)) and his utility is
F (β −1 (b)) × (t − b) = β −1 (b) × (t − b)
The FOC leads to
1
β 0 (β −1 (b))
(t − b) − β −1 (b) = 0
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and since the optimum must be reached at β −1 (b) = t, we have
tβ 0 (t) + β(t) = t
The solution is
t
2
The expected payoff of type t is then
β(t) =
t2
2
which is indeed exactly equal to the expected payment of the
bidder in the second price auction.
F (t) × (t − β(t)) =
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Auctions
Winner’s curse
Consider a common value setting: agent i receives a signal that is
represented by a r.v. Si . His valuation is vi = v (S1 , · · · , SN ),
strictly increasing in the signals.
Often assume that signal is unbiased: E [Si |V = v ] = v .
Consider bidder 1. Upon receiving his signal S, his estimate of the
valuation is E [V |S1 = s].
Consider a sealed bid first price auction with symmetric equilibrium
β(s). Then upon winning, bidder 1 realizes that all other bidders
have received a signal lower than s.
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Let Y1 be the highest bid among the N − 1 other bidders. Upon
learning that he is the winner is now
E [V |S1 = s, Y1 < s] < E [V |S1 = s]
Winning brings bad news: winner’s curse. Bidder has to shave his
bid relative to his estimate of the valuation based on his own signal.
Example: Si = V + ˜, where ˜ is white noise. Then
E [Si |V = v ] = v : signal is unbiased. However,
E [max(Si )|V = v ] > max E [Si |V = v ] = v
Winner’s curse more pronounced when the number of bidders is
larger.
An interesting link that enables you to “get a feel” for the winner’s
curse: Applet for the winner’s curse
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