Bidding to the Top: Position-based Auctions
Gagan Aggarwal
Joint work with Jon Feldman and
S. Muthukrishnan
Position 1
Position 2
Position 4
Generalized Second Price (GSP) Auction
Advertisers enter bids for keyword “car wash”.
Query comes, ads ranked by bid.
Price of ad = bid of next advertiser in ranking.
Bid
Price
ACarWash.com
$0.32
$0.24
BrightAndClean.com
$0.24
$0.17
CleanCars.com
$0.17
$0.14
DoVisitUs.com
$0.14
-
Advertiser pays only if user clicks on ad.
Google: [effective bid] = [bid] £ [quality]
Maturing Business of Sponsored Search
Sponsored Search is big business.
Significant portion of revenue of Google, Yahoo!, etc.
Increasing percentage of advertising budget of many large
advertisers.
Huge tail of small advertisers.
Advertisers are demanding more features, more options,
more control.
• Analytics and Conversion Tracking
• Ad Diagnostics
• Content Ads
• Negative Keywords
• Position Preference
Outline of Talk
Motivation for position control.
Generalization of GSP auction (GGSP?) to handle
position constraints.
Characterization of equilibria and relationship with
“truthful” VCG auction.
Parallels some of the results of [Edelman, Ostrovsky and
Schwarz, 2006] for GSP.
The Importance of Being Placed High
Position among ads has many effects.
Higher ads get more clicks.
Placing above a particular competitor is often important.
Higher ads have positive branding effect, even if they do not get
clicked. [IAB study]
Is there a natural generalization of GSP that takes
position constraints as input?
We study prefix constraints:
Advertiser i gives bid bi and position cutoff i
Ad i is never placed below position i
Top-down Auction Mechanism (GGSP)
Advertiser i gives bid bi and position cutoff i
For each position j from 1 … k :
Run a second-price auction among advertisers whose i ¸ j.
Place the winner at position j and remove her from the advertiser
pool.
Example: GGSP
Suppose number of positions available k = 3.
bid
Cutoff i
price
A.com
$6.10
2
$5.90
B.com
$5.90
3
$3.00
C.Com
$5.30
1
-
D.com
$3.00
3
$2.30
E.com
$2.30
3
Top-down Auction Mechanism (GGSP)
Advertiser i gives bid bi and position cutoff i
For each position i from 1 … k :
Run a second-price auction among advertisers whose i ¸ j.
Place the winner at position i and remove her from the advertiser
pool.
Is this a “good” mechanism?
… what even makes a mechanism good?
GSP is Not Truth-Revealing
GSP is not truth-revealing, i.e. it may not be optimal to
bid true value-per-click.
Example:
Existing bidders
Bid
Clicks
Alice.com
$3.00
100
Bob.com
$2.00
80
Carol.com has value $3.10 per click.
Utility of Position 1 = ($3.10 - $3.00) £100 = $10.
Utility of Position 2 = ($3.10 - $2.00) £ 80 = $88.
Locally stable.
Vickrey-Clarke-Groves (VCG) Auction
VCG is a generic truthful mechansim:
Allocation = the one that maximizes social welfare or total value
(assuming value = bid)
Price (i ) = cost imposed by i on others
= total increase in others’ value if i were to disappear.
To run VCG for prefix auctions:
Find max-value cutoff-respecting matching of ads to positions.
For each advertiser i ,
• Remove i from the pool of advertisers.
• Recalculate max matching.
• Set price(i ) =  value of all other advertisers.
Ordering Property
Definition: An ad shown higher has a higher bid.
GSP and GGSP have ordering property.
With position constraints, VCG does not have ordering
property.
Example: VCG Outcome
Suppose number of positions available k = 3.
Bid
Cutoff i
Position
A.com
$6.10
2
2
B.com
$5.90
3
3
C.Com
$5.30
1
1
D.com
$3.00
3
-
E.com
$2.30
3
Ordering Property
Definition: An ad shown higher has a higher bid.
GSP and GGSP have ordering property.
With position constraints, VCG does not have ordering
property.
Without position constraints, VCG has ordering property as long
as the click-through rates are “separable”.
Truthful
Locally
Stable
Ordering
Property
VCG
Yes
Yes
No
GGSP
No
Yes
Yes
Our Main Theorem
Theorem:
For a fixed set of values, the top-down auction
mechanism (GGSP) has a Nash equilibrium whose
(allocation, pricing) is the same as VCG.
Furthermore,
This equilibrium is envy-free (symmetric).
Among all envy-free equilibria, this one is bidderoptimal.
Key Points in Analysis
(No Position Constraints)
A
B
C
D
..
.
Z
To show that advertisers don’t envy each
other in the VCG outcome, we need a
handle on VCG prices of different positions
relative to each other.
Without position constraints, simple
relationship:
VCG price (B) =  value(i)
i = C…Z
VCG price (C) =

i = D…Z
value(i)
price(B) = price(C) + value(C)
Now VCG Prices are More Unruly
A
When an advertiser is removed, things can
change dramatically …
B
We need to better understand the structure
of the change in allocation.
C
D
..
.
Z
A Nugget
A
B
C
D
..
.
Z
Chain: sequence of one-step changes in
optimal allocation when an advertiser
removed.
Claim: The smallest chain formed on removal
of an advertiser cannot contain a down link
followed by an up link.
A Nugget
A
B
C
D
..
.
Z
Chain: sequence of one-step changes in
optimal allocation when an advertiser
removed.
Claim: The smallest chain formed on removal
of an advertiser cannot contain a down link
followed by an up link.
A Nugget
Claim: A shortest chain cannot contain a down
link followed by an up link.
A
B
…
i
…
j
ci = clicks at pos i [ci > … > cn ]
vi = value of bidder in VCG pos i
Proof: Use shortcut.
v = ci (vj – vk) + ck (vk- vj)
= (ci – ck) (vj – vk)
…
k
…
ci ¸ ck ?
… yes
vj ¸vk ?
… yes, otherwise original
solution would switch j and k.
Z
v ¸ 0, i.e. new allocation is no worse and has a shorter chain.
General Position-Based Bids
Arbitrary Ranges – Top and bottom cutoffs. For this,
GGSP is
Locally stable.
No ordering property.
May not have equilibrium matching the VCG outcome.
What if an advertiser could submit different bids for
different positions?
Since no natural ordering of positions left, ordering property
doesn’t make sense any more.
Suppose allocation using maximum matching.
Theorem: Maximum matching + local stability  VCG.
Open Questions
What makes a mechanism “good”?
Is truthfulness a goal in itself or a means to a goal?
If it is not the goal, what is? Stability? Ease of
understanding? Simple bidding strategies? Something
else …?
Thanks for your attention! 