We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem...
Transcript of We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem...
![Page 1: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/1.jpg)
![Page 2: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/2.jpg)
We give a simple allocation and pricing mechanism
whose Nash equilibrium solves a very large optimization problem
This Talk
![Page 3: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/3.jpg)
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
![Page 4: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/4.jpg)
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.
![Page 5: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/5.jpg)
Introduction to Sponsored Search
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Sponsored Search
![Page 7: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/7.jpg)
Sponsored Search
![Page 8: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/8.jpg)
Sponsored Search
Ads
Ads
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Sponsored Search
![Page 10: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/10.jpg)
Sponsored Search
![Page 11: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/11.jpg)
Sponsored Search
![Page 12: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/12.jpg)
Sponsored Search
![Page 13: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/13.jpg)
Sponsored Search
Bid
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Sponsored Search
Bid
Impressions
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Sponsored Search
Bid
Impressions
Price per click
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Sponsored Search
✓
![Page 17: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/17.jpg)
Sponsored Search
✓Click
![Page 18: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/18.jpg)
Sponsored Search
✓
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Variability in Sponsored SearchA one-shot or repeated auction:
eg. “Dominos Pizza” (exact match)
CTR
BID
0.10
0.05
0
6420
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Variability in Sponsored SearchA mixture of auctions:
eg. “ … Dominos … Pizza …” (phrase match)
CTR
BID
0.10
0.05
0
6420
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Variability in Sponsored SearchA bigger mixture of auctions:
eg. “ … Delivery … Pizza …” (broad match)
CTR
BID
0.10
0.05
0
6420
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Variability in Sponsored SearchA continuum of auctions:
eg. “ … Delivery … Pizza …” + location + time
(broad match) + Searcher
CTR
BID
0.10
0.05
0
6420
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Information and Temporal Asymmetry
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Information and Temporal Asymmetry
The Searcher
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Information and Temporal Asymmetry
The Searcher The Platform
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Information and Temporal Asymmetry
The Searcher The Platform The Advertiser
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Information and Temporal Asymmetry
The Searcher The Platform The Advertiser
Search Distribution
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Information and Temporal Asymmetry
The Searcher The Platform The Advertiser
Search Distribution When a search occurs
Click-Through
Assignment
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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
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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
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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
![Page 32: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/32.jpg)
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
![Page 33: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/33.jpg)
Auction or Optimize?
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Two Auctions
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
![Page 35: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/35.jpg)
Two Auctions
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
Auction 1:
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Two Auctions
Assign
– Bid of ad i
– Search Type
– CTR given bids
– Click-Through ad i slot l
in order
Auction 1:
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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:
![Page 38: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/38.jpg)
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:
![Page 39: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/39.jpg)
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:
![Page 40: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/40.jpg)
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:
![Page 41: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/41.jpg)
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
![Page 42: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/42.jpg)
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
![Page 43: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/43.jpg)
Immediate Advantages
![Page 44: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/44.jpg)
Immediate Advantages
![Page 45: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/45.jpg)
Immediate Advantages
There is not ordering of Ads!Generalized 2nd Price Breaks down. Our mechanism and results hold true.
![Page 46: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/46.jpg)
Is it GSP or VCG?
![Page 47: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/47.jpg)
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.”
![Page 48: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/48.jpg)
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?
![Page 49: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/49.jpg)
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.
![Page 50: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/50.jpg)
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.
![Page 51: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/51.jpg)
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
![Page 52: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/52.jpg)
The VCG Mechanism
![Page 53: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/53.jpg)
The VCG Mechanism● Advertiser's utilities
![Page 54: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/54.jpg)
The VCG Mechanism● Advertiser's utilities● bid utilities
![Page 55: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/55.jpg)
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
![Page 56: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/56.jpg)
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
![Page 57: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/57.jpg)
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Maximize Value
![Page 58: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/58.jpg)
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
![Page 59: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/59.jpg)
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Platform Prices:
![Page 60: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/60.jpg)
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Platform Prices:
Everyone else'svalue
![Page 61: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/61.jpg)
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Platform Prices:
Everyone else'svalue
Value without you there
![Page 62: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/62.jpg)
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Platform Prices:
![Page 63: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/63.jpg)
The VCG Mechanism● Advertiser's utilities● bid utilities● Assignment constraints
Platform Assigns:
Platform Prices:
Equilibrium Advertizer:
![Page 64: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/64.jpg)
The VCG Mechanism
TheoremThe VCG mechanism has a dominate strategies equilibrium that is:
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The VCG Mechanism
TheoremThe VCG mechanism has a dominate strategies equilibrium that is:
– Incentive compatible
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The VCG Mechanism
TheoremThe VCG mechanism has a dominate strategies equilibrium that is:
– Incentive compatible
– Efficient
![Page 67: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/67.jpg)
Directly Applying VCG
![Page 68: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/68.jpg)
Directly Applying VCGPros
1. Result applies in very general settings
2. Allocation of Adverts is provably optimal
![Page 69: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/69.jpg)
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
![Page 70: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/70.jpg)
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
![Page 71: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/71.jpg)
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
![Page 72: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/72.jpg)
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.
![Page 73: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/73.jpg)
A Massive Optimization
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A Massive Optimization
Max Utility
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A Massive Optimization
Max Utility
Mean click-rate
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A Massive Optimization
Max Utility
Mean click-rate
Per impressionAssignment Constraints
![Page 77: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/77.jpg)
A Massive Optimization
Max Utility
Mean click-rate
Per impressionAssignment Constraints
small
![Page 78: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/78.jpg)
A Massive Optimization
Max Utility
Mean click-rate
Per impressionAssignment Constraints
small Large!
![Page 79: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/79.jpg)
A Massive Optimization
Max Utility
Mean click-rate
Per impressionAssignment Constraints
small Large! Uncountably infinite!!
![Page 80: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/80.jpg)
A Massive Optimization
● even if we knew all the parameters, it's impossible to solve this optimization off-line
![Page 81: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/81.jpg)
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...
![Page 82: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/82.jpg)
A Small Optimization
When a search occurs, solve:
![Page 83: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/83.jpg)
A Small Optimization
When a search occurs, solve: Assignment Problem
![Page 84: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/84.jpg)
A Small Optimization
When a search occurs, solve: Assignment Problem
![Page 85: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/85.jpg)
A Small Optimization
When a search occurs, solve:
Lots of polynomial time algorithms:
Assignment Problem
![Page 86: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/86.jpg)
A Small Optimization
When a search occurs, solve:
Lots of polynomial time algorithms: Hungarian ; Hopcroft-Karp ; Bertsekas' Auction ...
Assignment Problem
![Page 87: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/87.jpg)
Optimization DecompositionSolve the big optimization with many little optimizations
![Page 88: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/88.jpg)
Optimization Decomposition
1. Substitution:
Solve the big optimization with many little optimizations
![Page 89: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/89.jpg)
Optimization Decomposition
1. Substitution: 2. Separability:
Solve the big optimization with many little optimizations
![Page 90: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/90.jpg)
Optimization Decomposition
1. Substitution: 2. Separability:
MAIN IDEA: Substitute utility for Separate out the resulting optimization
Solve the big optimization with many little optimizations
![Page 91: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/91.jpg)
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
![Page 92: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/92.jpg)
Our Results
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A Preliminary Calculation
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A Preliminary Calculation
![Page 95: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/95.jpg)
A Preliminary Calculation
![Page 96: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/96.jpg)
A Preliminary Calculation
LF-TransformAssignment
Problem
![Page 97: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/97.jpg)
A Preliminary Optimization Result
![Page 98: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/98.jpg)
A Preliminary Optimization Result
To solve the Massive Optimization
![Page 99: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/99.jpg)
A Preliminary Optimization Result
To solve the Massive Optimization
Advertiser's must signal average prices
![Page 100: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/100.jpg)
A Preliminary Optimization Result
To solve the Massive Optimization
Advertiser's must signal average prices
Platform solves Assignment when each search occurs
![Page 101: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/101.jpg)
A Preliminary Optimization Result
Advertiser's must signal average prices
Platform solves Assignment when each search occurs
![Page 102: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/102.jpg)
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.
![Page 103: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/103.jpg)
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.
![Page 104: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/104.jpg)
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.
![Page 105: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/105.jpg)
Main Theorem and Mechanism Design
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Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
![Page 107: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/107.jpg)
Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
A Nash Equilibrium is then:
![Page 108: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/108.jpg)
Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
A Nash Equilibrium is then:
![Page 109: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/109.jpg)
Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
A Nash Equilibrium is then:
These Prices
![Page 110: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/110.jpg)
Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
A Nash Equilibrium is then:
These Prices
at Nash Equilibrium
![Page 111: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/111.jpg)
Main Theorem and Mechanism DesignAdvertizers maximizes rewards:
A Nash Equilibrium is then:
These Prices
at Nash Equilibrium solve the Massive Optimization
![Page 112: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/112.jpg)
Proof of Main Theorem
Optimality condition for the dual:
envelope thrm
Fenchel-Moreau thrm
Substitute
integrate
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Proof of Main Theorem
Optimality condition for the dual:
envelope thrm
Fenchel-Moreau thrm
Substitute
The correct price!!
integrate
![Page 114: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/114.jpg)
How to Implement the Prices
![Page 115: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/115.jpg)
How to Implement the PricesTwo Price Implementations:
![Page 116: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/116.jpg)
How to Implement the Prices
1. Let
and price
Two Price Implementations:
![Page 117: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/117.jpg)
How to Implement the Prices
1. Let
and price
2. A discounted-VCG price
Two Price Implementations:
![Page 118: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/118.jpg)
How to Implement the Prices
1. Let
and price
2. A discounted-VCG price
Two Price Implementations:
the same average price
![Page 119: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/119.jpg)
How to Implement the Prices
1. Let
and price
2. A discounted-VCG price
Two Price Implementations:
the same average price
![Page 120: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/120.jpg)
A massively distributed VCG mechanism
![Page 121: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/121.jpg)
A very simple pay-per click mechanism:
A massively distributed VCG mechanism
![Page 122: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/122.jpg)
A very simple pay-per click mechanism:
Assignment Pricing
A massively distributed VCG mechanism
![Page 123: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/123.jpg)
A very simple pay-per click mechanism:
Assignment Pricing
at Nash equilibrium solves the Massive Optimization
A massively distributed VCG mechanism
![Page 124: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/124.jpg)
Further Results and Extensions
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Dynamics and Convergence
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Dynamics and Convergence
A natural dynamic:
![Page 127: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/127.jpg)
Dynamics and Convergence
A natural dynamic:
Lyapunov function:
![Page 128: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/128.jpg)
Dynamics and Convergence
A natural dynamic:
Lyapunov function:
![Page 129: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/129.jpg)
Dynamics and Convergence
A natural dynamic:
Lyapunov function:
![Page 130: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/130.jpg)
Further Extensions
Controlling number of slots:
![Page 131: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/131.jpg)
Further Extensions
Multivariate utilities:
![Page 132: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/132.jpg)
Further Extensions
Budget constraints:
![Page 133: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/133.jpg)
Summary of the talk
![Page 134: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/134.jpg)
Summary of the talk
● Massively decomposed VCG implementation – Simple– Flexible
![Page 135: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/135.jpg)
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
![Page 136: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/136.jpg)
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.
![Page 137: We give a simple allocation and pricing mechanismblaszczy/FB60/slides/Kelly.pdfoptimization problem This Talk. We give a simple allocation and pricing mechanism whose Nash equilibrium](https://reader034.fdocuments.in/reader034/viewer/2022042323/5f0d39257e708231d43947af/html5/thumbnails/137.jpg)
Thank you for listening!