Advanced Microeconomics
The revelation principle and mechanism design
Harald Wiese
University of Leipzig
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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The revelation principle and mechanism design
1
Introduction
2
First-price auction
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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The revelation principle and mechanism design
Introduction
By mechanism design, we understand the problem of setting up a
Bayesian game so as to bene…t the principal.
Example: …rst-price auction versus second-price auction
Problem: players do not, in general, “tell the truth” (e.g., bid
according to willingness to pay)
Revelation principle: we can restrict attention to mechanisms where
players “tell the truth” or “reveal their own type”
Thus, the revelation principle helps to …nd the best mechanism by
restricting the set of candidate mechanisms.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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The revelation principle and mechanism design
Nobel Prize
In 2007, the Sveriges Riksbank Prize in Economic Sciences in Memory of
Alfred Nobel was awarded to the US economists
1/3 Leonid Hurwicz (University of Minnesota),
1/3 Eric S. Maskin (Institute for Advanced Study, Princeton), and
1/3 Roger B. Myerson (University of Chicago)
for having laid the foundations of mechanism design theory.
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Advanced Microeconomics
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The revelation principle and mechanism design
Example: …rst-price auction and half-price auction
In the …rst-price auction, every bidder acts strategically because a bid
does not only in‡uence the chance of getting the object but also the
price the bidder has to pay in case of winning the auction.
We found the equilibrium strategy combination s = (s1 , s2 ) de…ned
by
s1
: [0, 1] ! R+ ,
s2
: [0, 1] ! R+ ,
t1
and
2
t2
t2 7 ! s 2 ( t2 ) = .
2
t1 7 ! s 1 ( t1 ) =
In that combination, every player uses the half-bid strategy.
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Advanced Microeconomics
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The revelation principle and mechanism design
Example: …rst-price auction and half-price auction
Half-price auction:
the object goes to the highest bidder but
the payment is set at half the announced willingness to pay
Construction (using the …rst-price auction and the half-bid strategies):
The half-bid strategies have every bidder halve his type, ai = t2i .
Under the …rst-price auction, the successful bidder pays ai = t2i .
The half-price auction tries to achieve the same outcome for players
who tell the truth. Thus, the factor 12
stemming from the players’strategic behavior
is brought into the half-price auction by requiring that the successful
bidder pays a2i , only.
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Advanced Microeconomics
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The revelation principle and mechanism design
Example: …rst-price auction and half-price auction
Assume truthful bidding in the half-price auction:
idi : [0, 1] ! [0, 1] , ti 7! idi (ti ) = ti .
Bidder with highest willingness to pay obtains the object:
t1 t2
,
| {z }
comparison of the player’s
truthful announcements
under the half-price auction
t2
t1
.
|2 {z 2}
comparison of the player’s
strategic announcements
under the …rst-price auction
The successful bidder i pays ti /2:
in the …rst-price auction, by understating the willingness to pay.
in the half-price auction, by truthful announcement.
Is (id1 , id2 ) a Bayesian equilibrium of the half-price auction? Yes ...
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Advanced Microeconomics
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The revelation principle and mechanism design
Example: …rst-price auction and half-price auction
s is an equilibrium of the …rst-price auction.
Assume s1 = id1 were not a best response to s2 = id2 in the half-price
auction. Then there is a type t1 and an action (announced type)
a1 6= t1 such that
t1
t1
|
a1
t1
τ (ft2 : a1 > t2 g) >
2
{z
}
|
t1
2
untruthful bid
a1
τ
2
a1
t2
t2 :
>
2
2
>
t1
τ (ft2 : t1 > t2 g) or
{z
}
truthful bid
t1
2
τ
t2 :
t1
t2
>
2
2
Contradiction to s1 being a best response to s2 in the …rst-price
auction.
(id1 , id2 ) is a Nash equilibrium in the half-price auction.
Harald Wiese (University of Leipzig)
Advanced Microeconomics
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The revelation principle and mechanism design
Example: …rst-price auction and half-price auction
In line with the revelation principle
we can restrict attention to mechanisms where players “tell
the truth” or “reveal their own type”
the results from the …rst-price auction
who gets the object
what price does the auctioneer get
are also obtained by the half-price auction
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Advanced Microeconomics
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