Auction Theory: Equilibrium Analysis
Georg Nöldeke
Wirtschaftswissenschaftliche Fakultät, Universität Basel
Advanced Microeconomics, HS 11
Lecture 9
1/13
Strategic Equivalence
Given the same vector of bids (b1 , . . . , bN ) the first-price
(sealed-bid) auction and the Dutch auction produce the
same material outcome:
The highest bidder wins and pays his own bid – all other
bidders pay nothing.
Given the same vector of bids (b1 , . . . , bN ) the second-price
(sealed-bid) auction and the English auction produce the
same material outcome:
The highest bidder wins and pays the second highest bid –
all other bidders pay nothing.
Consequently, we may restrict our equilibrium analysis to
the first-price and the second-price auction as the Dutch
auction is strategically equivalent to the first-price auction
and the English auction is strategically equivalent to the
second price auction.
Advanced Microeconomics, HS 11
Lecture 9
2/13
Strategic Equivalence
Remark: As explained in the textbook, the relationship
between the second-price auction and the English auction
is more subtle when one considers the possibility of
conditioning ones bid on the quitting behavior of the other
bidders.
Advanced Microeconomics, HS 11
Lecture 9
3/13
Equilibrium in the Second Price Auction
A strategy for bidder i in the second price auction is given
by a bidding function bi : [0, 1] → [0, ∞), specifying a bid
bi (vi ) for every possible type vi of bidder i.
The following argument shows that the bidding function
bi (vi ) = vi specifies a weakly dominant strategy for every
type of every bidder:
If bidder i with value vi bids bi < vi , his payoff is unaffected if
the highest bid of the other bidders is strictly lower than bi
or higher than vi . In all other cases his payoff is strictly
higher if he bids vi rather than bi .
If bidder i with value vi bids bi > vi , his payoff is unaffected if
the highest bid of the other bidders is lower than vi or
strictly higher than bi . In all other cases his payoff is strictly
higher if he bids vi rather than bi .
Advanced Microeconomics, HS 11
Lecture 9
4/13
Equilibrium in the Second Price Auction
Theorem (Second-Price Auction Equilibrium)
Bidding one’s value is the unique weakly dominant bidding
strategy for each bidder in a second-price auction. In particular,
the bidding functions given by bi (vi ) = vi for i = 1, . . . , N are a
Bayesian Nash equilibrium of the second-price auction.
Observe:
As the bidding functions are strictly increasing, it is always
the bidder with the highest value that wins the auction and
obtains the object. In particular, the outcome of the auction
is efficient.
Equilibrium strategies are independent of beliefs and thus
easy to determine.
Advanced Microeconomics, HS 11
Lecture 9
5/13
Equilibrium in the First Price Auction: Derivation
As in the second price auction, a strategy for bidder i in the
first price auction is given by a bidding function
bi : [0, 1] → [0, ∞), specifying a bid bi (vi ) for every possible
type vi of bidder i.
Throughout the following we will only search for symmetric
Bayesian Nash equilibria in which all bidders use the
same, strictly increasing and differentiable bidding function
b̂(·) satisfying b̂(0) = 0.
It can be shown that in the symmetric auction environment
we consider there are no other Bayesian Nash equilibria.
Recall that we have assumed that all bidders’ values are
independently distributed on [0, 1] according to the
distribution function F(·) with density f (·).
Advanced Microeconomics, HS 11
Lecture 9
6/13
Equilibrium in the First Price Auction: Derivation
Suppose all bidders but bidder i bid according to the strictly
increasing bidding function b̂(·).
For every type of bidder i, bidding b > b̂(1) is not a best
response, so in the following we will only consider bids
0 ≤ b ≤ b̂(1).
The probability that bidder i wins with such a bid b is given
by the probability that all other bidders have values below
v̂(b), where v̂(b) is the value satisfying b̂(v̂(b)) = b.
Consequently, the expected payoff for bidder i with value v
from bidding b is:
U(v, b) = F(v̂(b))N−1 (v − b)
and the bidding function b̂(·) describes a symmetric
equilibrium if and only if
U(v, b̂(v)) = max U(v, b) holds for all 0 ≤ v ≤ 1.
b∈[0,b̂(1)]
Advanced Microeconomics, HS 11
Lecture 9
7/13
Equilibrium in the First Price Auction: Derivation
While the condition
U(v, b̂(v)) = max U(v, b) holds for all 0 ≤ v ≤ 1.
b∈[0,b̂(1)]
can be used to derive the bidding function b̂(·), there is a
cleverer way to do so:
Bidding b̂(v) is optimal for a bidder with valuation v if and
only if there does not exists r ∈ [0, 1] such that the bidder
prefers to bid b̂(r), which would result in the expected
payoff
u(r, v) = F(r)N−1 v − b̂(r)
Hence, we can rewrite the equilibrium condition as
u(v, v) = max u(r, v) holds for all 0 ≤ v ≤ 1.
r∈[0,1]
Advanced Microeconomics, HS 11
Lecture 9
8/13
Equilibrium in the First Price Auction: Derivation
From the envelope theorem this version of the equilibrium
condition implies
du(v, v)
= F N−1 (v) ⇒ u(v, v) =
dv
Z v
F N−1 (x)dx.
0
From the definition of u(v, v) we know
u(v, v) = F N−1 (v) v − b̂(v) .
Equating the two expressions and solving for b̂(v):
b̂(v) = v −
Advanced Microeconomics, HS 11
1
Z v
F N−1 (v)
0
Lecture 9
F N−1 (x)dx.
(1)
9/13
Equilibrium in the First Price Auction: Derivation
Equation (1) identifies a unique candidate for a symmetric
Bayesian Nash equilibrium.
To prove the following result, the following facts have to be
confirmed:
b̂(·) is increasing.
b̂(0) = 0 holds.
Equation (1) is not only necessary, but sufficient.
Theorem (First-Price Auction Equilibrium)
The first-price auction has a unique symmetric Bayesian Nash
equilibrium in which all N bidders use the bidding function
b̂(v) = v −
Advanced Microeconomics, HS 11
1
Z v
F N−1 (v)
0
F N−1 (x)dx.
Lecture 9
10/13
Equilibrium in the First Price Auction: Comments
The formula for the equilibrium bidding function b̂(·) given
above looks different from the one in the textbook . . .
. . . but it is the same.
Verification for the case of the uniform distribution (see
Example 9.1 in the textbook):
1
Z v
F N−1 (x)dx
F N−1 (v) 0
Z v
1
= v − N−1
xN−1 dx
v
0
1
vN
= v − N−1 ·
v
N
N −1
v
= v− =
v.
N
N
b̂(v) = v −
Advanced Microeconomics, HS 11
Lecture 9
11/13
Equilibrium in the First Price Auction: Comments
Observe:
In contrast to the second-price auction, bidders do not bid
their “true value”, but “shade their bids”:
v > b̂(v) = v −
1
Z v
F N−1 (v)
0
F N−1 (x)dx for all v > 0.
As in the second-price auction, the bidder with the highest
value wins the object. Hence, the resulting allocation of the
object is identical to the one in the equilibrium of the
second-price auction.
While bids are lower than in the second-price auction, the
seller obtains the highest rather than the second highest
bid. Hence, at first glance it is unclear whether the seller
obtains higher revenues in the first-price or the
second-price auction.
Advanced Microeconomics, HS 11
Lecture 9
12/13
Equilibrium in the First Price Auction: Interpretation
The expression
b̂(v) = v −
1
Z v
F N−1 (v)
0
F
N−1
(x)dx = v −
Z v N−1
F
(x)
0
F N−1 (v)
dx.
has a simple interpretation:
F N−1 (·) is the distribution function of the highest value of
the other bidders.
F N−1 (·)/F N−1 (v) it the distribution function of the highest
value of the other bidders conditional on this value being
smaller than v.
b̂(v) is thus the expectation of the highest value of the other
bidders conditional on this value being smaller than v.
In the unique symmetric equilibrium of a first-price auction,
each bidder bids the expectation of the second-highest bidder’s
value conditional on winning the auction.
Advanced Microeconomics, HS 11
Lecture 9
13/13