CS 573: Algorithmic Game Theory Lecture date: April 11, 2008
Instructor: Chandra Chekuri Scribe: Hannaneh Hajishirzi
Contents
1 Sponsored Search Auctions 1
1.1 VCG Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Do search engines use VCG? . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 GFP and GSP Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 GSP is not truthful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Equilibrium Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 More issues in sponsored search auctions . . . . . . . . . . . . . . . . . . . . . 6
1 Sponsored Search Auctions
The static model for the keyword auctions is as follows:
 n bidders/advertisers.
 k slots (k is xed apriori).
 ij as a click through rate (CTR) of the bidder i if placed in slot j.
 vi is the value of the bidder i for a click. vi is independent of the slot position.
In this static model we have the following assumptions:
 Bidders prefer a higher slot to a lower slot (i.e., for every bidder i, i1  i2  : : :  ik) .
 vi (value of the bidder i) is independent of the slot position.
In particular, allocation (winner determination) problem is assigning bidders to slots to nd the
optimal allocation which maximizes the utility. In the current setting, it is a single parameter
problem given that each bidder has a value vi per click, and click through rates ij are public
knowledge. In what follows we examine dierent mechanisms for this allocation problem. We rst
start with VCG mechanism.

1.1 VCG Mechanism
Let xij be a f0; 1g variable that indicates whether slot j is assigned to the bidder i or not. Then,
the winner determination problem is to solve the following integer linear program:
max
nX
i=1
vi
kX
j=1
ijxij
s.t.
kX
j=1
xij  1 8i  n
nX
i=1
xij  1 8j  k
xij 2 f0; 1g 8i  n;8j  k
Note that we had earlier seen this matching problem, but there vij could be dierent from vij0 .
Here, it is a single parameter problem because vij = viij and ijs are public knowledge. As we
saw before, the matching problem has integral value even if we relax the integrality constraints, so
we can replace the constraint xij 2 f0; 1g with xij  0 in the above linear program. We can in fact
solve the following linear program:
max
nX
i=1
vi
kX
j=1
ijxij
s.t.
kX
j=1
xij  1 8i  n
nX
i=1
xij  1 8j  k
xij  0 8i  n; 8j  k
The dual of this linear program has the following form:
min
kX
j=1
pj +
nX
i=1
ui (1)
s.t. ui + pj  ijvi 8i  n; 8j  k (2)
ui; pj  0 8i  n; 8j  k (3)
The intuition is that we set the prices of the slots and let the bidders decide what to do. Finally,
we have a Walrassian equilibrium with prices given by optimal dual values.
The Mechanism is truthful and ecient. If CTRs are independent of the bidder (i.e., ij = j ,
8i  n) then the assignment can be computed easily. We just assign slots in decreasing order of
bids: assign top slot to the top bidder and so on.
1.1.1 Do search engines use VCG?
There are several reasons that search engines do not use VCG for keyword auctions:

1. VCG requires solving a computational problem which needs to be done online for every search
and is expensive.
2. VCG can lead to a low revenue for the auctioneer, in this case the search engine. For example,
suppose that all the values vi are the same for all the bidders and ijs are also the same.
Everybody get their slot for the k + 1st bid and it is not benecial for the search engine.
3. VCG can cause problems to the advertisers as illustrated by the following example:
Example 1.1 There are 2 bidders and 2 slots given in the following setting:
Bidder 1: 11 = 0:8 12 = 0:8
Bidder 2: 12 = 0:15 22 = 0:1
VCG will assign slot 1 to bidder 2 and slot 1 to bidder 1 even if v1  v2. Bidder 1 does not
like it because he pays a lots of money and will not go to the top. Furthermore, it is hard to
add branding to the model.
1.2 GFP and GSP Mechanisms
Here, we show some mechanisms used for actual search engines.
Yahoo! used a generalized rst price (GFP) auction until 2004.
Denition 1.2 (GFP mechanism) Let b1; : : : ; bn be the bids. The GFP mechanism is as follows:
1. Sorts bidders according to the bids b1; : : : ; bn.
2. Assigns slots according to the order (assign top slot to the highest bidder and so on).
3. Change bidder i according to his bid.
This mechanism is not truthful, but has a Bayesian Nash equilibrium in static setting. Most current
search engines now a days use a generalized second price (GSP) auction.
Denition 1.3 (GSP mechanism) Let w1; : : : ; wn be the weights on bidders which are static and
independent of the bids b1; : : : ; bn. The GSP mechanism is as follows:
1. Sort bidders by si = wibi.
(assume) s1  s2  : : : sn.
2. Allocate slots to bidders 1 : : : k in that order (i.e., bidder i gets the ith slot if i  k).
3. Charge i the mininum bid he needs to retain his slot (i.e., pi =
si+1
wi
).
Intuitively, the weights represent some measure that the search engine assigns to the bidder to show
some notion of the "quality of the bidder". There are dierent ways that search engines assign the
weights:
 Overture model In the Overture model for every i; wi = 1, therefore bidders are ordered
according to the bids only. Furthermore, the payment pi = bi+1 for every i.

 Google model Google assigns weights based on the CTR at the top slot (wi ' i1). Then,
it orders based on wi ' i1. The assumption here is that i1 is static (or slow changing)
information and that the bids are varying more quickly. However, the CTR depends on slot
assignment and so technically there is feedback. Note that a bidder with low CTR needs
to bid higher. This ordering is also called \revenue order" since si = i1bi is the expected
revenue if i is put in slot 1 and there is only one slot.
1.2.1 GSP is not truthful
To understand why GSP is not truthful let us consider a simple example. There are 3 bidders, 2
slots, w1 = w2 = w3 = 1, v1 > v2 > v3. Assume CTR is bidder independent and ij = j and
1 > 2. Therefore, bidder 1 gets slot 1 and bidder 2 gets slot 2 in both VCG and GSP. In VCG
prices for slots 1 and 2 are p1 = (1
2
1
)v2 and p2 = v3, respectively. In GSP prices for slots 1
and 2 are p1 = v2 and p2 = v3, respectively. Since 1 > 2 GSP charges bidder 1 more than VCG.
Therefore, GSP is not truthful since VCG prices are unique truthful prices and allocations are the
same for VCG and GSP.
More concretely, assume that bids of bidders 2 and 3 are xed. Utility of the bidder 1 when
he bids truthfully is 1(v1 p1). However, consider his utility if he bids less than v2 and gets the
second slot: 2(v1 v3). Suppose v3 = 0. Then, for 1 to be truthful we need:
1(v1 p1)  2v1 ) p1  (1
2
1
)v1
But, if v2 > (1
2
1
)v1 and GSP charges bidder 1, v2 then he will bid less than his true value.
Even, more concretely:
Example 1.4 Let w1 = w2 = w3 = 1, v1 = 200; v2 = 180; v3 = 100 and 1 = 0:5; 2 = 0:4.
Suppose the bidders bid truthfully, i.e. bi = vi. Therefore, bidder 1 gets slot 1 and bidder 2 gets
slot 2. The expected utility of the bidder 1 is: 0:5(200 180) = 10. Now, suppose bidder 1 bids less
than his true value (b1 = 110; b2 = v2; b3 = v3). Therefore, bidder 1 gets slot 2 and bidder 2 gets
slot 1. The expected utility of bidder 1 is 0:4(200 100) = 40 which is greater than when he bids
truthfully. In this case, VCG mechanism assigns slot 1 to player 1 and charges her (1 21 )v2 = 36.
Furthermore, it assigns slot 2 to player 2 and charges her v3 = 100.
Therefore, bidding falsely generated more prot, which is not incentive compatible. The intuition
is that CTRs for both slots are close, so the bidder 1 does not care which slot to choose. In the
above setting, as the CTR for slots 1 and 2 are not that dierent, bidder 1 bids less and gets the
2nd slot instead of the rst one. In VCG mechanism payments are taken into account more cleverly.
The payments in VCG are derived by solving the linear program in Equation 1.
We observe that the GSP allocation according to weighted ranking is monotone, i.e. increasing
the bid of the bidder i and keeping the other bids xed results in the same slot or a better slot.
One question is whether the slot allocation according to weighted ranking can be implemented
truthfully or not, i.e. is there a truthful mechanism with GSP allocation rule and a new payment?
From our previous characterization the payment rule is xed for a truthful implementation of
the allocation rule. Recall that,
pi(b; bi) = bxi(b; bi)
Z b
0
xi(z; bj)dz

where, xi(b; bi) is the CTR of the bidder i with bid b and assuming others are at bi. (Recall that
in previous examples of single parameter problems we considered only the case of xi(b; bi) being
0 or 1; here we have to be more general). Therefore, one can explicitly compute the payments by
employing the above calculation.
Denition 1.5 (Laddered Auction) Let w1; : : : ; wn be the weights on bidders which are static
and independent of bids b1; : : : ; bn. The Laddered auction is as follows:
 Allocation is the same as GSP. (Assuming that S1  S2 : : :  Sn where Si = wibi).
 Payment pi = 1i;i
Pk
j=i[(i;j i;j+1)
wj+1
wi
bj+1]
or i;ipi =
Pk
j=i[(i;j i;j+1)
wj+1
wi
bj+1]
Note that i;ipi is the expected payment of the bidder i in slot i.
We expand the above equation as follows:
i;ipi = (i;i i;i+1)
wi+1
wi
bi+1
+(i;i i;i+2)
wi+2
wi
bi+2
+ : : :+ (i;i i;i+k)
wi+k
wi
bi+k
The interpretation of each term in the above equation is as follows: wjwi bj is the minimum bid of
the bidder i to get slot j (for j  i+ 1). (i;j i;j+1) is the extra click that the bidder i is getting
by moving to slot j from slot j + 1 and the price it should pay is the minimum bid required to get
to slot j.
1.3 Equilibrium Issues
The fact that GFP and GSP are not truthful does not mean the revenue is bad. Recall that for
a single item GFP admits a Bayesian Nash equilibrium. We saw that for symmetric bidders the
revenue equivalence theorem implies that revenue from GFP is the same as any other auction that
allocates according to bid order. Note that the equivalence exists under the static model.
In practice the problem is more complicated; In reality, the model is not static, bidders over
multiple days may keep changing their bid on the same keyword to nd the lowest price. Over
time this may result in the loss of revenue for the seller. The Bayesian-Nash equilibrium of GSP
is not known even in the symmetric setting. GSP is neither ecient nor it maximizes the revenue.
However, one can say the following:
Suppose that CTRs are separable, i.e. for every i and j, ij = ij . In other words, CTR
depends on the slot and the quality of bidder in the above product form. Then, one can show that
the allocation produced by GSP is ecient (maximize social welfare) under a \locally-envy-free"
equilibrium.
A locally-envy-free equilibrium is one in which the bidder i has not incentive to switch to i 1
or i + 1 (he may switch to the i + 2 but not with the immediate neighbors; please refer to the
book for more details). Also, please see [1] to see revenue comparison between truthful \laddered"
auction and the GSP auction under equilibrium. There exist equilibria (for a xed valuation) for
which the revenue is the same and for which revenues are dierent.

1.3.1 More issues in sponsored search auctions
Although we looked at static setting, in reality bidders are in a dynamic setting over multiple days
stretching to innity, which makes it quite complicated to analyze. Two phenomenon occurs here:
Bid rotation and vindictive bidding.
 Bid Rotation Bidders take turns occupying the top slot.
 Vindictive Bidding In GSP a bidder can force competitors in a higher slot to pay more by
articially increasing his bid. The advertisers pay based on the click model. One can click on
competitors slots and make them pay more to the search engine.
Budget Constraint Issues Each advertiser has a limited budget per day (or multiple days).
He/She is bidding over many keywords simultaneously. He/She may run out of budget depending
on the search query distribution and clicks.
 Issues for Advertisers One issue is that what the advertisers bid on each keyword given
query distribution information and CTRs. Besides, queries may consist of multiple keywords.
 Issues for Search Enging Dierent bidders are bidding on dierent slots of keywords and
have budget constraints. Issues include allowing which advertiser to bid, or awarding which
advertiser in case of ties to ensure maximum revenue. Consider the following example:
Example 1.6 Assume there are two keywords A and B and two bidders 1 and 2.
Bidder 1 is interested in keywords A and B
Bidder 2 is interested in only A
Suppose query A comes and bidder 1 has little budget and 2 has more budget left. This
scenario may cause the search engine to award keyword A to the bidder 2 even if his bid is
slightly lower. Save bidder 1 for potential arrival of the keyword B in which bidder 2 is not
interested.
References
[1] G. Aggarwal, A. Goel, and R. Motwani. Truthful auctions for pricing search keywords. ACM
conference on Electronic Commerce, 2006.