Applications of Myerson’s Lemma
Professors Greenwald and Oyakawa
2017-02-15
In these notes, we find direct, DSIC mechanisms for the following
auction scenarios:
• single-item,
• k-Vickrey,
• Sponsored Search.
To do this, we first compute the welfare-maximizing allocation. Then
we apply Myerson’s Lemma to compute payments which grant us
the DSIC property. This involves two steps:
1. Argue that the allocation rule is monotone.
2. Use Myerson’s payment formula,
pi ( vi , v −i ) = vi xi ( vi , v −i ) −
Z v
i
0
xi (z, v −i ) dz,
(1)
to compute DSIC payments.
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Single-Item Auction
Though this application is quite easy, it is a useful sanity check to
ensure that the theory checks with the results we found in the singleitem auction—namely, that we achieve a DSIC mechanism by charging the winner the second-highest bid.
1.1 Welfare Maximization
Recall that welfare is the quantity
∑ xi vi
i
over all bidders i. When the allocation vector x is a one-hot vector1 ,
this quantity is maximized by awarding the good to
arg max vi ,
i
the bidder with the highest valuation for the good.
Zero in all entries except one (the
winner), for which the value is 1.
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applications of myerson’s lemma
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1.2 Monotonicity
The good is awarded to the highest bidder. If some bidder i bids bi
and wins, bidding bi + e cannot possibly result in a loss, so xi (bi +
e) ≥ xi (bi )2 ; if i loses, then the same inequality automatically holds
because xi (bi + e) cannot be less than 0. We have shown that the
allocation rule is monotone.
1.3 Payments
Observe that if xi = 0, then pi = 0. Therefore, only the winner of the
auction will make a payment to the auctioneer. The payment of the
winner is computed as follows. Let b∗ denote the second-highest bid.
Z v
i
pi ( vi , v −i ) = vi xi ( vi , v −i ) −
xi (z, v −i ) dz,
0
Z b∗
Z v
i
= vi · 1 −
0 dz +
1 dz
0
b∗
= vi − ( vi − b ∗ )
= b∗ .
(We split up the integral in this way because the allocation for bidding less than the second-highest bid is 0, while the allocation of
bidding more is 1.) This shows that to make the mechanism DSIC, we
charge winner of the auction the second-highest bid.
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k-Vickery Auction
In this auction setting, there are k identical items and n < k bidders,
each of whom have a private, nonzero valuation vi for exactly one
copy of the good.
2.1
Welfare Maximization Problem
The welfare is given by
∑ xi vi ,
i
where x ∈ {0, 1}n and |x | = k. This is achieved by setting the nonzero
entries of x to precisely the entries corresponding to the k largest
bids. Therefore, we award the k items to the k highest bidders.
We plot the allocation function in Figure 1.
2.2
Monotonicity
In the following analysis, we fix all bids except for that of bidder i. In
this case, the necessary and sufficient condition for i being allocated
Since the other bids are fixed, we often
omit the full vector to the argument of
xi , instead just zeroing-in on bidder i’s
bid.
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applications of myerson’s lemma
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Figure 1: Bidder i’s allocation function
for a given v −i .
xi ( vi , v −i )
1
0
vi
b∗
0
is bidding more than b∗ , the kth highest bid among (among bidders
other than i). If bi < b∗ , then xi (bi ) = 0 so increasing the bid cannot
possibly result in lower allocation; if bi > b∗ , then xi (bi + e) =
xi (bi ) = 1.
2.3 Payments
Since the condition for being allocated is simply bidding higher than
b∗ , this will look similar to the payment calculation for the singleitem auction.
Z v
i
pi ( vi , v −i ) = vi xi ( vi , v −i ) −
xi (z, v −i ) dz,
0
Z b∗
Z v
i
= vi · 1 −
0 dz +
1 dz
0
b∗
= vi − ( vi − b ∗ )
= b∗ .
In this auction scenario, we obtain a DSIC mechanism by awarding
the goods to the k highest bidders and charge each winner the kth
highest bid.
The payment is precisely the shaded region in Figure 2.
Figure 2: Bidder i’s payment for a given
v −i .
xi ( vi , v −i )
1
0
0
b∗
vi
applications of myerson’s lemma
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Sponsored Search
In this setting, there are n bidders (online advertisers) who are competing for one of k positions on a search page’s sponsored links. For
each slot j, there is a probability that a user will click on a link at slot
j. This is called the click-through-rate (CTR). For slot j, the CTR is α j ,
where
α1 ≥ α2 ≥ · · · ≥ α k .
If bidder i receives slot j, we say that xi = α j . This is our first example
of “partial” allocation, not necessarily 0 or 1.
3.1 Welfare Maximization Problem
We wish to maximize the quantity
∑ xi vi ,
i
where x contains each of the values α1 , . . . , αk exactly once, and all
other entries are 0.
This is optimized by matching α j with the jth highest bidder for
1 ≤ j ≤ k. That is, we award the jth slot to the jth highest bidder.
Figure 3 shows bidder i’s allocation as a function of her bid. (For
example, if she bids between b j and b j−1 , her allocation is α j .)
Figure 3: Bidder i’s allocation function.
3.2 Monotonicity
Fix some bid bi and consider the resulting allocation xi (bi ). xi (bi )
and xi (bi + e) can each be one of k + 1 possible values: any α j for
applications of myerson’s lemma
1 ≤ j ≤ k, or 0. If xi (bi ) = 0, then we automatically have that
xi (bi + e) ≥ xi (bi ). If xi (bi ) = αs and xi (bi + e) = αt , then it must be
that t ≤ s, since by the allocation rule, higher bidders get higher slots.
Since the CTRs are decreasing, then αt ≥ αs , and so xi (bi + e) ≥ x (bi ).
3.3 Payments
Recall that if xi = 0, then pi = 0, so we assume that i has a nonzero
allocation. In particular, if, in absence of bidder i, the slots 1 . . . k
are allocated to bidders with bids b1 , . . . , bk , let us assume that b j ≤
vi ≤ b j−1 . In this case, if i bids truthfully, then she will receive slot j,
pushing bidders j, j + 1, . . . , k − 1 down a slot and evicting bidder k
from slot k.
pi ( vi , v −i ) = vi xi ( vi , v −i ) −
"Z
= vi · α j −
bk
0
Z v
i
0
xi (z, v −i ) dz,
0 dz +
Z b
k −1
bk
αk dz +
Z b
k −2
bk −1
αk−1 dz + · · · +
Z b
j
b j +1
α j+1 dz +
Z v
i
bj
#
α j dz
= v i · α j − ( bk − 1 − bk ) α k + ( bk − 2 − bk − 1 ) α k − 1 + · · · + ( b j − b j + 1 ) α j + 1 + ( v i − b j ) α j
= b j α j − ( bk − 1 − bk ) α k + ( bk − 2 − bk − 1 ) α k − 1 + · · · + ( b j − b j + 1 ) α j + 1 .
j +1
= b j α j − ∑ ( bt − 1 − bt ) α t
t=k
While this expression might not be intuitive by itself, a picture
may help. It is the shaded region in Figure 4.
Figure 4: Bidder i’s payment for bidding in [b j , b j−1 ]. (Image courtesy of
Zechen Ma.)
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