Revenue Equivalence
Georg Nöldeke
Wirtschaftswissenschaftliche Fakultät, Universität Basel
Advanced Microeconomics, HS 11
Lecture 10
1/11
Revenue Comparison for Standard Auctions
Consider the expected revenue of the seller in the equilibria of
the first-price auction and the second price auction conditional
on the highest value being v:
In the first-price auction the seller’s revenue in this case is
certain and given by the bid of the buyer with the highest
value. This is, as we have seen, equal to the expectation of
the second highest value conditional on the highest value
being equal to v.
In the second-price auction the seller’s revenue is equal to
the bid of the buyer with the second highest value. This is,
as we have seen, the second highest value. Taking
expectations, the sellers expected revenue is equal to the
expectation of the second highest value conditional on the
highest value being equal to v.
Advanced Microeconomics, HS 11
Lecture 10
2/11
Revenue Comparison for Standard Auctions
We have thus established that conditional on the highest
value being v the first-price auction and the second-price
auction yield the same expected revenue.
As the distribution of the highest value does not depend on
the auction mechanism, the above argument shows the
following theorem.
Theorem (Revenue Equivalence for Standard Auctions)
In their Bayesian-Nash equilibria the first-price and
second-price auctions raise the same expected revenue for the
seller. By strategic equivalence, this is also the seller’s
expected revenue in the Dutch auction and the English auction.
Advanced Microeconomics, HS 11
Lecture 10
3/11
Revenue Comparison for Standard Auctions
Example: Expected revenue when values are uniformly
distributed
As we have seen, the equilibrium bidding function in the
first-price auction with uniformly distributed values is
N −1
v
v.
b̂v = v − =
N
N
Hence, conditional on the highest value being v, the
expected revenue of the seller is
N −1
v.
N
As the distribution function of the highest value is
F N (v) = vN the expected revenue is
Z 1
N −1
N −1
1
N −1
N
1−
v dv =
1−
=
.
N
N
N +1
N +1
0
Advanced Microeconomics, HS 11
Lecture 10
4/11
General Auctions and Incentive Compatibility
Continue to assume that the values vi of the bidders
i = 1, . . . , N are independent and identically distributed on
[0, 1] according to the distribution function F with density f .
Rather than restricting attention to the four standard
auctions, we can imagine other, more general auction
mechanisms which specify
a strategy set Si for each bidder i and for every strategy
profile s = (s1 , . . . , sN ) and bidder i = 1, . . . N:
the probability pi (s) that bidder i wins the object.
the payment ci (s) that bidder i has to make to the seller.
Advanced Microeconomics, HS 11
Lecture 10
5/11
General Auctions and Incentive Compatibility
Together with our assumptions about payoffs and types,
every auction mechanism defines a static game of
incomplete information.
Every Bayesian Nash equilibrium of any such game
induces
a probability p̄i (ri ) that bidder i with value ri wins the
auction.
an expected payment c̄i (ri ) that bidder i with value ri makes
to the seller.
When bidder i has value vi he could choose the strategy
si (ri ), resulting in a probability of winning equal to p̄i (ri )
and an expected payment of c̄i (ri ). The corresponding
expected payoff is:
ui (ri , vi ) = p̄i (ri )vi − c̄i (ri ).
Advanced Microeconomics, HS 11
Lecture 10
6/11
General Auctions and Incentive Compatibility
In a Bayesian Nash equilibrium, si (vi ) is the optimal choice
of bidder i with value vi , implying the following result.
Theorem (Incentive Compatibility)
Every Bayesian Nash equilibrium of every auction mechanism
is incentive compatible, that is
ui (vi , vi ) ≥ ui (ri , vi ) for all ri
holds for all bidders i and values vi .
As explained in the textbook, incentive compatibility is
equivalent to the condition that truthful reporting of types is
a Bayesian Nash equilibrium in a direct selling mechanism
in which each bidders’ strategy is an announcement of one
of his possible types.
Advanced Microeconomics, HS 11
Lecture 10
7/11
Incentive Compatibility and Revenue Equivalence
Incentive compatibility has very strong implications about the
outcomes that could possibly arise in the Bayesian Nash
equilibrium of some auction mechanism.
Incentive compatibility is equivalent to the statement that
for all bidders i and values vi , the “true value” vi solves the
problem
max p̄i (ri )vi − c̄i (ri ).
ri
Using the envelope theorem as in our derivation of the
equilibrium for the first-price auction, this implies:
Z vi
ui (vi , vi ) =
Advanced Microeconomics, HS 11
0
p̄i (ri )dri + ui (0, 0).
Lecture 10
8/11
Incentive Compatibility and Revenue Equivalence
As we also have
ui (vi , vi ) = p̄i (vi )vi − c̄i (vi )
we obtain the following relationship between the expected
payment of a bidder and the probability that he wins the
object (see Theorem 9.5 in the textbook).
Theorem
In every Bayesian Nash equilibrium of every auction
mechanism the following relationship holds for all bidders i and
values vi :
Z
vi
c̄i (vi ) = c̄i (0) + p̄i (vi )vi −
Advanced Microeconomics, HS 11
0
Lecture 10
p̄i (ri )dri .
9/11
Incentive Compatibility and Revenue Equivalence
The seller’s expected revenue R is given by the sum of the
bidders’ expected payments,
N
R=∑
i=1
Z
0
1
c̄i (vi ) f (vi )dvi ,
so that the previous result implies (see Theorem 9.6 in the
textbook):
Revenue Equivalence Theorem
If the Bayesian Nash equilibria of two different auction
mechanisms result in the same allocation of the object and the
same utility for all bidders with value 0, then they yield identical
expected revenue for the seller.
Advanced Microeconomics, HS 11
Lecture 10
10/11
Incentive Compatibility and Revenue Equivalence
Observe:
Revenue Equivalence for the four standard auctions is an
immediate consequence of the more general Revenue
Equivalence Theorem:
All four standard auctions result in the same allocation
(namely the efficient one) and bidders with value 0 obtain a
payoff of 0.
The Revenue Equivalence Theorem implies that in all four
auctions the seller’s expected revenue is the same, namely
R=N·
Z 1
c̄(v) f (v)dv,
0
where
c̄(v) = vF N (v) −
Z v
F N (r)dr.
0
Advanced Microeconomics, HS 11
Lecture 10
11/11