Second-Price Sealed-Bid Auctions
Professors Greenwald and Oyakawa
2017-02-01
We introduce the second-price, sealed-bid auction. This auction format
requires auction winners to pay the second highest bid. We go over the
strategic consequences of this payment rule.
1
The Second-Price, Sealed-Bid Auction
In a second-price, sealed-bid auction for one good, each bidder i ∈
N submits a “sealed-bid” (a secret number) to the auctioneer, so
that no bidder knows what any other has bid. Each bidder i has
valuation vi for the good, and does not know the valuations of the
other bidders. We assume valuations follow the IPV model. The
auctioneer selects the bidder with the highest bid, i∗ ∈ arg maxi∈ N bi ,
as the winner, allocates to the winner with probability xi∗ (bi∗ , b−i∗ ) =
1 (after a tie-breaking decision is made, if necessary), and allocates
with probability xi (bi , b−i ) = 0 to all other bidders i 6= i∗ ∈ N.
1 The winner of the auction, i ∗ , is charged the second highest bid,
pi∗ (bi∗ , b −i∗ ) = b(n−1) , and all other bidders i 6= i∗ ∈ N are charged
pi (bi , b −i ) = 0.
The second-price, sealed-bid auction is also called the Vickrey
auction, named after Nobel laureate William Vickrey.
1.1
Computational Complexity
With n bidders, determining the winner takes O(n) time, as this is
the complexity of an arg max function. Given the winner, we can
determine payments in O(n) time, as we can do this by invoking an
arg max on all bids aside from the winner. Therefore, this auction can
be run in polynomial time.
2
A Strategy for the Second-Price, Sealed-Bid Auction
Reasoning about the optimal strategy for this scenario is, fortunately,
easier than in the first-price auction setting. Each bidder i ∈ N can
bid one of three ways:
• bi < v i
• bi = v i
• bi > v i .
The utility of each case is summarized in Figure 1.
Ex-ante, each bidder in arg maxi∈ N bi
is allocated with probability
1/|arg maxi∈ N bi |
1
second-price sealed-bid auctions
Bid
Outcome
Win
Lose
bi < v i
>0
=0
bi = v i
≥0
=0
2
Figure 1: Bidder i’s payoff for bidding
one of three ways.
bi > v i
≤ 0 if pi ≥ vi , < 0 if pi > vi
=0
We can see that bidding bi > vi isn’t as good as bidding either
bi < vi or bi = vi . What’s left is to reason between bi < vi or bi = vi .
Notice that if
bi < b ( n − 1 ) ≤ v i ,
then i loses and gains 0 utility. However, there exists an e ∈ R≥0 such
that
b ( n − 1 ) ≤ bi + e ≤ v i ,
so i would have the opportunity to gain non-negative utility. Thus, it
should be preferred to bid vi . In fact, since it is always optimal to bid
ones true valuation.
We can describe all of this graphically. First notice that
vi xi (bi , b −i ) − pi (bi , b −i ) = (vi − pi (bi , b −i )) xi (bi , b −i ).
(1)
If bidder i wins, she will receive value vi xi (bi , b −i ) = vi :
Figure 2: The value bidder i receives if
she wins.
x i ( bi , b − i )
1
0
bi
vi
0
Payments are either less than vi , equal to vi , or greater than vi :
x i ( bi , b − i )
1
0
Figure 3: Possible levels of payment. If
bidder i wins, one of the three payment
outcomes will occur.
x i ( bi , b − i )
1
0
pi vi
bi
0
0
vi pi
bi
The difference between the shaded regions is the winners utility.
We can also describe this by plotting utility as a function of payment, as in Figure 4 and Figure 5. Notice that utility is at least zero
when v ≥ p, and at most zero when v ≤ p. Thus, one can conclude
that bidding truthfully is optimal.
second-price sealed-bid auctions
Figure 4: The utility of bidder i if the
price, the second highest bid, is smaller
than v, as a function of what she bids.
Utility
v−p
0
p
v
Bid
Figure 5: The utility of bidder i if the
price, the second highest bid, is larger
than v, as a function of what she bids.
Utility
3
0
v−p
v
p
Bid
Theorem 2.1. Placing bid bi = vi is a dominant strategy in the secondprice, sealed-bid auction.
Since bidding truthfully is utility maximizing, the second-price
auction is dominant strategy incentive compatible (DSIC), where,
by incentive compatible, we mean that truthful bidding is an optimal strategy. Furthermore, notice that if everyone knew everyone
else’s valuations, there would not be a way for bidders to change
their strategy and increase utility (without colluding). Thus, bidding
truthfully is an ex-post dominant strategy.
second-price sealed-bid auctions
2.1 Total Welfare
Summing over the utility of the bidders and the auctioneer gives us
the total welfare of our agents:
"
# "
#
∑ v i x i ( bi , b − i ) − p i ( bi , b − i )
i∈ N
+
∑ p i ( bi , b − i )
i∈ N
=
∑ v i x i ( bi , b − i ) .
i∈ N
(2)
In a Bayes-Nash equilibrium, the bidder with the highest bid also
values the good the most:

1
if vi ≥ v j , ∀ j ∈ N
(3)
x i ( bi , b − i ) =
0
otherwise.
This allocation scheme maximizes total welfare, so in a Bayes-Nash
equilibrium, the second-price, sealed-bid auction is a welfare maximizing auction.
3
Revenue
Assume that all bidders are symmetric, where there valuations are
drawn independently from some distribution F. Let v = min T, and
v = max T. Using what we know about order statistics, we can now
derive the expected revenue of the second-price, sealed-bid auction.
Lemma 3.1. In a second-price, sealed-bid auction, the total expected revenue generated is
R2 =
Z v
v
xn(n − 1) F ( x )n−2 (1 − F ( x )) f ( x ) dx
(4)
Proof. For n iid draws from distribution F with density f , the distribution of the kth smallest value is
n!
(5)
f X( k ) ( x ) =
( F ( x ))k−1 (1 − F ( x ))n−k f ( x ).
( k − 1) ! ( n − k ) !
Plug in k = n − 1 to get
f X( k ) ( x ) =
=
n!
( F ( x ))(n−1)−1 (1 − F ( x ))n−(n−1) f ( x )
((n − 1) − 1)!(n − (n − 1))!
(6)
n!
( F ( x ))n−2 (1 − F ( x ))1 f ( x )
( n − 2) ! (1) !
= n(n − 1) F ( x )n−2 (1 − F ( x )) f ( x ).
(7)
(8)
With this, we can write the expected value of the second highest
order statistic:
R2 =
Z v
v
xn(n − 1) F ( x )n−2 (1 − F ( x )) f ( x ) dx.
(9)
4
second-price sealed-bid auctions
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