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We give a simple allocation and pricing mechanism
whose Nash equilibrium solves a very large optimization problem
This Talk
We give a simple allocation and pricing mechanism
whose Nash equilibrium solves a very large optimization problem
Very Large = over the infinite results of a search engine.
This Talk
Outline● A Introduction to Sponsored Search
– Bids, Impressions, Click-Through Rate, Advertizers, Platform
● Auction or Optimize?
– Our Mechanism, Generalized 2nd Price, VCG Mechanism, Decomposition.
● Our Results
– Main Theorem, Implementations
● Further Results and Extensions
– Dynamics, Multivariate Utilities, General Page Layouts, Budgets.
Introduction to Sponsored Search
Sponsored Search
Sponsored Search
Sponsored Search
Ads
Ads
Sponsored Search
Sponsored Search
Sponsored Search
Sponsored Search
Sponsored Search
Bid
Sponsored Search
Bid
Impressions
Sponsored Search
Bid
Impressions
Price per click
Sponsored Search
✓
Sponsored Search
✓Click
Sponsored Search
✓
Variability in Sponsored SearchA one-shot or repeated auction:
eg. “Dominos Pizza” (exact match)
CTR
BID
0.10
0.05
0
6420
Variability in Sponsored SearchA mixture of auctions:
eg. “ … Dominos … Pizza …” (phrase match)
CTR
BID
0.10
0.05
0
6420
Variability in Sponsored SearchA bigger mixture of auctions:
eg. “ … Delivery … Pizza …” (broad match)
CTR
BID
0.10
0.05
0
6420
Variability in Sponsored SearchA continuum of auctions:
eg. “ … Delivery … Pizza …” + location + time
(broad match) + Searcher
CTR
BID
0.10
0.05
0
6420
Information and Temporal Asymmetry
Information and Temporal Asymmetry
The Searcher
Information and Temporal Asymmetry
The Searcher The Platform
Information and Temporal Asymmetry
The Searcher The Platform The Advertiser
Information and Temporal Asymmetry
The Searcher The Platform The Advertiser
Search Distribution
Information and Temporal Asymmetry
The Searcher The Platform The Advertiser
Search Distribution When a search occurs
Click-Through
Assignment
Information and Temporal Asymmetry
The Searcher The Platform The Advertiser
Search Distribution When a search occurs
Click-Through
Assignment
Receives average information
Click-Through
Assignment
Information and Temporal Asymmetry
The Searcher The Platform The Advertiser
Search Distribution
UnknownTo everyone
When a search occurs
Click-Through
Assignment
Receives average information
Click-Through
Assignment
Information and Temporal Asymmetry
The Searcher The Platform The Advertiser
Search Distribution
UnknownTo everyone
When a search occurs
Click-Through
Assignment
Platform knowsAdvertiser doesn't
Receives average information
Click-Through
Assignment
Information and Temporal Asymmetry
The Searcher The Platform The Advertiser
Search Distribution
UnknownTo everyone
When a search occurs
Click-Through
Assignment
Platform knowsAdvertiser doesn't
Receives average information
Click-Through
Assignment
Platform knowsAdvertiser knows
Auction or Optimize?
Two Auctions
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
Two Auctions
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
Auction 1:
Two Auctions
Assign
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
in order
Auction 1:
Two Auctions
Assign
Pay, per-click
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
in order
Auction 1:
Two Auctions
Assign
Pay, per-click
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
in order
Auction 1: Auction 2:
Two Auctions
Assign
Pay, per-click
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
Assign max matching
in order
Auction 1: Auction 2:
Two Auctions
Assign
Pay, per-click
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
Assign max matching
Pay, per-click
in order
Auction 1: Auction 2:
Two Auctions
Assign
Pay, per-click
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
Assign max matching
Pay, per-click
in order
Auction 1: Auction 2:
Generalized 2nd Price
Two Auctions
Assign
Pay, per-click
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
Assign max matching
Pay, per-click
in order
Auction 1: Auction 2:
Generalized 2nd Price A VCG Auction
Immediate Advantages
Immediate Advantages
Immediate Advantages
There is not ordering of Ads!Generalized 2nd Price Breaks down. Our mechanism and results hold true.
Is it GSP or VCG?
Is it GSP or VCG?
Google (2006) said:
“Google’s unique auction model uses Nobel Prize winning economic theory … the AdWords™ Discounter makes sure that they only pay what they need in order to stay ahead of their nearest competitor.”
Is it GSP or VCG?
Google (2006) said:
“Google’s unique auction model uses Nobel Prize winning economic theory … the AdWords™ Discounter makes sure that they only pay what they need in order to stay ahead of their nearest competitor.”
Q. Is this really true?
Is it GSP or VCG?
Google (2006) said:
“Google’s unique auction model uses Nobel Prize winning economic theory … the AdWords™ Discounter makes sure that they only pay what they need in order to stay ahead of their nearest competitor.”
Q. Is this really true?A. Not really.
Is it GSP or VCG?
Google (2006) said:
“Google’s unique auction model uses Nobel Prize winning economic theory … the AdWords™ Discounter makes sure that they only pay what they need in order to stay ahead of their nearest competitor.”
Q. What did they really mean?Q. Is this really true?
A. Not really.
Is it GSP or VCG?
Google (2006) said:
“Google’s unique auction model uses Nobel Prize winning economic theory … the AdWords™ Discounter makes sure that they only pay what they need in order to stay ahead of their nearest competitor.”
Q. What did they really mean?A. The VCG Mechanism...
Q. Is this really true?A. Not really.
Vickrey Clark Groves
The VCG Mechanism
The VCG Mechanism● Advertiser's utilities
The VCG Mechanism● Advertiser's utilities● bid utilities
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Maximize Value
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Platform Prices:
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Platform Prices:
Everyone else'svalue
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Platform Prices:
Everyone else'svalue
Value without you there
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Platform Prices:
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Platform Prices:
Equilibrium Advertizer:
The VCG Mechanism
TheoremThe VCG mechanism has a dominate strategies equilibrium that is:
The VCG Mechanism
TheoremThe VCG mechanism has a dominate strategies equilibrium that is:
– Incentive compatible
The VCG Mechanism
TheoremThe VCG mechanism has a dominate strategies equilibrium that is:
– Incentive compatible
– Efficient
Directly Applying VCG
Directly Applying VCGPros
1. Result applies in very general settings
2. Allocation of Adverts is provably optimal
Directly Applying VCGPros
1. Result applies in very general settings
2. Allocation of Adverts is provably optimal
Cons
1. Advertisers submit their entire utility function
Directly Applying VCGPros
1. Result applies in very general settings
2. Allocation of Adverts is provably optimal
Cons
1. Advertisers submit their entire utility function
2. Utility isn't for a single adauction but for all adauctions
Directly Applying VCGPros
1. Result applies in very general settings
2. Allocation of Adverts is provably optimal
Cons
1. Advertisers submit their entire utility function
2. Utility isn't for a single adauction but for all adauctions
3. Platform needs to solve a massive optimization
Directly Applying VCGPros
1. Result applies in very general settings
2. Allocation of Adverts is provably optimal
Cons
1. Advertisers submit their entire utility function
2. Utility isn't for a single adauction but for all adauctions
3. Platform needs to solve a massive optimization
This talk: We deal with these issue by appropriately decomposing this massive optimization.
A Massive Optimization
A Massive Optimization
Max Utility
A Massive Optimization
Max Utility
Mean click-rate
A Massive Optimization
Max Utility
Mean click-rate
Per impressionAssignment Constraints
A Massive Optimization
Max Utility
Mean click-rate
Per impressionAssignment Constraints
small
A Massive Optimization
Max Utility
Mean click-rate
Per impressionAssignment Constraints
small Large!
A Massive Optimization
Max Utility
Mean click-rate
Per impressionAssignment Constraints
small Large! Uncountably infinite!!
A Massive Optimization
● even if we knew all the parameters, it's impossible to solve this optimization off-line
A Massive Optimization
● even if we knew all the parameters, it's impossible to solve this optimization off-line
● Still … maybe we can solve a lot of small optimizations...
A Small Optimization
When a search occurs, solve:
A Small Optimization
When a search occurs, solve: Assignment Problem
A Small Optimization
When a search occurs, solve: Assignment Problem
A Small Optimization
When a search occurs, solve:
Lots of polynomial time algorithms:
Assignment Problem
A Small Optimization
When a search occurs, solve:
Lots of polynomial time algorithms: Hungarian ; Hopcroft-Karp ; Bertsekas' Auction ...
Assignment Problem
Optimization DecompositionSolve the big optimization with many little optimizations
Optimization Decomposition
1. Substitution:
Solve the big optimization with many little optimizations
Optimization Decomposition
1. Substitution: 2. Separability:
Solve the big optimization with many little optimizations
Optimization Decomposition
1. Substitution: 2. Separability:
MAIN IDEA: Substitute utility for Separate out the resulting optimization
Solve the big optimization with many little optimizations
Optimization Decomposition
1. Substitution: 2. Separability:
MAIN IDEA: Substitute utility for Separate out the resulting optimization
THE RESULT: A massively distributed VCG Mechanism
Solve the big optimization with many little optimizations
Our Results
A Preliminary Calculation
A Preliminary Calculation
A Preliminary Calculation
A Preliminary Calculation
LF-TransformAssignment
Problem
A Preliminary Optimization Result
A Preliminary Optimization Result
To solve the Massive Optimization
A Preliminary Optimization Result
To solve the Massive Optimization
Advertiser's must signal average prices
A Preliminary Optimization Result
To solve the Massive Optimization
Advertiser's must signal average prices
Platform solves Assignment when each search occurs
A Preliminary Optimization Result
Advertiser's must signal average prices
Platform solves Assignment when each search occurs
A Preliminary Optimization Result
Advertiser's must signal average prices
Platform solves Assignment when each search occurs
● Decomposed on the timescales of Platform and Advertisers.
A Preliminary Optimization Result
Advertiser's must signal average prices
Platform solves Assignment when each search occurs
● Decomposed on the timescales of Platform and Advertisers. ● Search distribution is not required.
A Preliminary Optimization Result
Advertiser's must signal average prices
Platform solves Assignment when each search occurs
● Decomposed on the timescales of Platform and Advertisers. ● Search distribution is not required.● But it's an optimization result, we must incentivize this behaviour.
Main Theorem and Mechanism Design
Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
A Nash Equilibrium is then:
Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
A Nash Equilibrium is then:
Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
A Nash Equilibrium is then:
These Prices
Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
A Nash Equilibrium is then:
These Prices
at Nash Equilibrium
Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
A Nash Equilibrium is then:
These Prices
at Nash Equilibrium solve the Massive Optimization
Proof of Main Theorem
Optimality condition for the dual:
envelope thrm
Fenchel-Moreau thrm
Substitute
integrate
Proof of Main Theorem
Optimality condition for the dual:
envelope thrm
Fenchel-Moreau thrm
Substitute
The correct price!!
integrate
How to Implement the Prices
How to Implement the PricesTwo Price Implementations:
How to Implement the Prices
1. Let
and price
Two Price Implementations:
How to Implement the Prices
1. Let
and price
2. A discounted-VCG price
Two Price Implementations:
How to Implement the Prices
1. Let
and price
2. A discounted-VCG price
Two Price Implementations:
the same average price
How to Implement the Prices
1. Let
and price
2. A discounted-VCG price
Two Price Implementations:
the same average price
A massively distributed VCG mechanism
A very simple pay-per click mechanism:
A massively distributed VCG mechanism
A very simple pay-per click mechanism:
Assignment Pricing
A massively distributed VCG mechanism
A very simple pay-per click mechanism:
Assignment Pricing
at Nash equilibrium solves the Massive Optimization
A massively distributed VCG mechanism
Further Results and Extensions
Dynamics and Convergence
Dynamics and Convergence
A natural dynamic:
Dynamics and Convergence
A natural dynamic:
Lyapunov function:
Dynamics and Convergence
A natural dynamic:
Lyapunov function:
Dynamics and Convergence
A natural dynamic:
Lyapunov function:
Further Extensions
Controlling number of slots:
Further Extensions
Multivariate utilities:
Further Extensions
Budget constraints:
Summary of the talk
Summary of the talk
● Massively decomposed VCG implementation – Simple– Flexible
Summary of the talk
● Massively decomposed VCG implementation – Simple– Flexible
● Can be Implemented– Relevant timescale– Relevant information asymmetry– Low Computational Overhead– Applies to different page layouts
Summary of the talk
● Massively decomposed VCG implementation – Simple– Flexible
● Can be Implemented– Relevant timescale– Relevant information asymmetry– Low Computational Overhead– Applies to different page layouts
● Provably solves an Infinitely Large Optimization.
Thank you for listening!
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