vickery-clark-groves mechanisma general design rule is the vickrey-clarke-groves mechanism. agt-miri...
TRANSCRIPT
![Page 1: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/1.jpg)
Selling one itemVCG mechanism
Vickery-Clark-Groves Mechanism
Maria Serna
Fall 2016
AGT-MIRI VCG mechanism
![Page 2: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/2.jpg)
Selling one itemVCG mechanism
1 Selling one item
2 VCG mechanism
AGT-MIRI VCG mechanism
![Page 3: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/3.jpg)
Selling one itemVCG mechanism
1- item: Buyer’s utility
Bidders have private values vi for the item
A winning bidder gets a utility of ui = vi − p − i
A losing bidder pays nothing and gets ui = 0
AGT-MIRI VCG mechanism
![Page 4: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/4.jpg)
Selling one itemVCG mechanism
1- item: Buyer’s utility
Bidders have private values vi for the item
A winning bidder gets a utility of ui = vi − p − i
A losing bidder pays nothing and gets ui = 0
AGT-MIRI VCG mechanism
![Page 5: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/5.jpg)
Selling one itemVCG mechanism
1- item: Buyer’s utility
Bidders have private values vi for the item
A winning bidder gets a utility of ui = vi − p − i
A losing bidder pays nothing and gets ui = 0
AGT-MIRI VCG mechanism
![Page 6: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/6.jpg)
Selling one itemVCG mechanism
1- item: Buyer’s utility
Bidders have private values vi for the item
A winning bidder gets a utility of ui = vi − p − i
A losing bidder pays nothing and gets ui = 0
AGT-MIRI VCG mechanism
![Page 7: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/7.jpg)
Selling one itemVCG mechanism
1-item: Seller’s incentive
Maximize social welfare: SP auctions
Maximize revenue: SP auctions with reserve price
For example, reserve-pr [0,100]Reserve price is independent of the number of playersOptimality assumes a technical assumption on thedistributions.
Revenue equivalence results
AGT-MIRI VCG mechanism
![Page 8: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/8.jpg)
Selling one itemVCG mechanism
1-item: Seller’s incentive
Maximize social welfare: SP auctions
Maximize revenue: SP auctions with reserve price
For example, reserve-pr [0,100]Reserve price is independent of the number of playersOptimality assumes a technical assumption on thedistributions.
Revenue equivalence results
AGT-MIRI VCG mechanism
![Page 9: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/9.jpg)
Selling one itemVCG mechanism
1-item: Seller’s incentive
Maximize social welfare: SP auctions
Maximize revenue: SP auctions with reserve price
For example, reserve-pr [0,100]Reserve price is independent of the number of playersOptimality assumes a technical assumption on thedistributions.
Revenue equivalence results
AGT-MIRI VCG mechanism
![Page 10: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/10.jpg)
Selling one itemVCG mechanism
1-item: Seller’s incentive
Maximize social welfare: SP auctions
Maximize revenue: SP auctions with reserve price
For example, reserve-pr [0,100]Reserve price is independent of the number of playersOptimality assumes a technical assumption on thedistributions.
Revenue equivalence results
AGT-MIRI VCG mechanism
![Page 11: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/11.jpg)
Selling one itemVCG mechanism
1-item: Seller’s incentive
Maximize social welfare: SP auctions
Maximize revenue: SP auctions with reserve price
For example, reserve-pr [0,100]Reserve price is independent of the number of playersOptimality assumes a technical assumption on thedistributions.
Revenue equivalence results
AGT-MIRI VCG mechanism
![Page 12: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/12.jpg)
Selling one itemVCG mechanism
Efficiency
We saw that in 1-item auctions SP achieve efficiency with truthfuldominant strategies.
Can this be achieved in other settings?
Moving from a specific example (1-item auctions) to a moregeneral mechanism design setting.
Objective: Design the right incentives such that the efficientoutcome will be chosen.
AGT-MIRI VCG mechanism
![Page 13: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/13.jpg)
Selling one itemVCG mechanism
Efficiency
We saw that in 1-item auctions SP achieve efficiency with truthfuldominant strategies.
Can this be achieved in other settings?
Moving from a specific example (1-item auctions) to a moregeneral mechanism design setting.
Objective: Design the right incentives such that the efficientoutcome will be chosen.
AGT-MIRI VCG mechanism
![Page 14: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/14.jpg)
Selling one itemVCG mechanism
Efficiency
We saw that in 1-item auctions SP achieve efficiency with truthfuldominant strategies.
Can this be achieved in other settings?
Moving from a specific example (1-item auctions) to a moregeneral mechanism design setting.
Objective: Design the right incentives such that the efficientoutcome will be chosen.
AGT-MIRI VCG mechanism
![Page 15: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/15.jpg)
Selling one itemVCG mechanism
Efficiency
We saw that in 1-item auctions SP achieve efficiency with truthfuldominant strategies.
Can this be achieved in other settings?
Moving from a specific example (1-item auctions) to a moregeneral mechanism design setting.
Objective: Design the right incentives such that the efficientoutcome will be chosen.
AGT-MIRI VCG mechanism
![Page 16: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/16.jpg)
Selling one itemVCG mechanism
Example: The roommates problem
Consider two roommates who would like to buy a TV for theirapartment.
TV costs $100
They should decide:
Do they want to buy a TV together?If so, how should they share the costs?
Individual preferences have to be taken into consideration indeciding the efficient outcome.
AGT-MIRI VCG mechanism
![Page 17: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/17.jpg)
Selling one itemVCG mechanism
Example: The roommates problem
Consider two roommates who would like to buy a TV for theirapartment.
TV costs $100
They should decide:
Do they want to buy a TV together?If so, how should they share the costs?
Individual preferences have to be taken into consideration indeciding the efficient outcome.
AGT-MIRI VCG mechanism
![Page 18: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/18.jpg)
Selling one itemVCG mechanism
Example: The roommates problem
Consider two roommates who would like to buy a TV for theirapartment.
TV costs $100
They should decide:
Do they want to buy a TV together?If so, how should they share the costs?
Individual preferences have to be taken into consideration indeciding the efficient outcome.
AGT-MIRI VCG mechanism
![Page 19: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/19.jpg)
Selling one itemVCG mechanism
Example: The roommates problem
Consider two roommates who would like to buy a TV for theirapartment.
TV costs $100
They should decide:
Do they want to buy a TV together?If so, how should they share the costs?
Individual preferences have to be taken into consideration indeciding the efficient outcome.
AGT-MIRI VCG mechanism
![Page 20: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/20.jpg)
Selling one itemVCG mechanism
Example: The roommates problem
Consider two roommates who would like to buy a TV for theirapartment.
TV costs $100
They should decide:
Do they want to buy a TV together?If so, how should they share the costs?
Individual preferences have to be taken into consideration indeciding the efficient outcome.
AGT-MIRI VCG mechanism
![Page 21: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/21.jpg)
Selling one itemVCG mechanism
Example: The buying of multiple items
Each bidder has a value of vi for an item.
But now we have 5 items!
Each bidder wants only one item.
What is an efficient outcome?
valuations $70 $30 $27 $25 $12 $5 $2
sell the items to the 5 bidders with the highest values
How to design the auction?A general design rule is the Vickrey-Clarke-Groves mechanism.
AGT-MIRI VCG mechanism
![Page 22: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/22.jpg)
Selling one itemVCG mechanism
Example: The buying of multiple items
Each bidder has a value of vi for an item.
But now we have 5 items!
Each bidder wants only one item.
What is an efficient outcome?
valuations $70 $30 $27 $25 $12 $5 $2
sell the items to the 5 bidders with the highest values
How to design the auction?A general design rule is the Vickrey-Clarke-Groves mechanism.
AGT-MIRI VCG mechanism
![Page 23: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/23.jpg)
Selling one itemVCG mechanism
Example: The buying of multiple items
Each bidder has a value of vi for an item.
But now we have 5 items!
Each bidder wants only one item.
What is an efficient outcome?
valuations $70 $30 $27 $25 $12 $5 $2
sell the items to the 5 bidders with the highest values
How to design the auction?A general design rule is the Vickrey-Clarke-Groves mechanism.
AGT-MIRI VCG mechanism
![Page 24: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/24.jpg)
Selling one itemVCG mechanism
Example: The buying of multiple items
Each bidder has a value of vi for an item.
But now we have 5 items!
Each bidder wants only one item.
What is an efficient outcome?
valuations $70 $30 $27 $25 $12 $5 $2
sell the items to the 5 bidders with the highest values
How to design the auction?A general design rule is the Vickrey-Clarke-Groves mechanism.
AGT-MIRI VCG mechanism
![Page 25: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/25.jpg)
Selling one itemVCG mechanism
Example: The buying of multiple items
Each bidder has a value of vi for an item.
But now we have 5 items!
Each bidder wants only one item.
What is an efficient outcome?
valuations $70 $30 $27 $25 $12 $5 $2
sell the items to the 5 bidders with the highest values
How to design the auction?A general design rule is the Vickrey-Clarke-Groves mechanism.
AGT-MIRI VCG mechanism
![Page 26: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/26.jpg)
Selling one itemVCG mechanism
Example: The buying of multiple items
Each bidder has a value of vi for an item.
But now we have 5 items!
Each bidder wants only one item.
What is an efficient outcome?
valuations $70 $30 $27 $25 $12 $5 $2
sell the items to the 5 bidders with the highest values
How to design the auction?A general design rule is the Vickrey-Clarke-Groves mechanism.
AGT-MIRI VCG mechanism
![Page 27: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/27.jpg)
Selling one itemVCG mechanism
Example: The buying of multiple items
Each bidder has a value of vi for an item.
But now we have 5 items!
Each bidder wants only one item.
What is an efficient outcome?
valuations $70 $30 $27 $25 $12 $5 $2
sell the items to the 5 bidders with the highest values
How to design the auction?A general design rule is the Vickrey-Clarke-Groves mechanism.
AGT-MIRI VCG mechanism
![Page 28: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/28.jpg)
Selling one itemVCG mechanism
Example: The buying of multiple items
Each bidder has a value of vi for an item.
But now we have 5 items!
Each bidder wants only one item.
What is an efficient outcome?
valuations $70 $30 $27 $25 $12 $5 $2
sell the items to the 5 bidders with the highest values
How to design the auction?
A general design rule is the Vickrey-Clarke-Groves mechanism.
AGT-MIRI VCG mechanism
![Page 29: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/29.jpg)
Selling one itemVCG mechanism
Example: The buying of multiple items
Each bidder has a value of vi for an item.
But now we have 5 items!
Each bidder wants only one item.
What is an efficient outcome?
valuations $70 $30 $27 $25 $12 $5 $2
sell the items to the 5 bidders with the highest values
How to design the auction?A general design rule is the Vickrey-Clarke-Groves mechanism.
AGT-MIRI VCG mechanism
![Page 30: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/30.jpg)
Selling one itemVCG mechanism
1 Selling one item
2 VCG mechanism
AGT-MIRI VCG mechanism
![Page 31: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/31.jpg)
Selling one itemVCG mechanism
Vickrey-Clarke-Groves (VCG) mechanisms
Goal: implement the efficient outcome in dominant strategies.
VCG is a general method generalizing SP auctions.
Solution: players should pay the damage they impose onsociety.
You can maximize efficiency by:
Choosing the efficient outcome (given the bids) as allocation.Each player pays his social cost (welfare).
Payment pi , for bidder i , is obtained as
Optimal welfare (for the other players) if player i was notparticipating.minus welfare of the other players from the chosen outcomeIn a single item auctionwhen i wins the object this payment is 2nd highest bid minus0, otherwise this payment is highest bid minus highest bid.
AGT-MIRI VCG mechanism
![Page 32: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/32.jpg)
Selling one itemVCG mechanism
Vickrey-Clarke-Groves (VCG) mechanisms
Goal: implement the efficient outcome in dominant strategies.
VCG is a general method generalizing SP auctions.
Solution: players should pay the damage they impose onsociety.
You can maximize efficiency by:
Choosing the efficient outcome (given the bids) as allocation.Each player pays his social cost (welfare).
Payment pi , for bidder i , is obtained as
Optimal welfare (for the other players) if player i was notparticipating.minus welfare of the other players from the chosen outcomeIn a single item auctionwhen i wins the object this payment is 2nd highest bid minus0, otherwise this payment is highest bid minus highest bid.
AGT-MIRI VCG mechanism
![Page 33: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/33.jpg)
Selling one itemVCG mechanism
Vickrey-Clarke-Groves (VCG) mechanisms
Goal: implement the efficient outcome in dominant strategies.
VCG is a general method generalizing SP auctions.
Solution: players should pay the damage they impose onsociety.
You can maximize efficiency by:
Choosing the efficient outcome (given the bids) as allocation.Each player pays his social cost (welfare).
Payment pi , for bidder i , is obtained as
Optimal welfare (for the other players) if player i was notparticipating.minus welfare of the other players from the chosen outcomeIn a single item auctionwhen i wins the object this payment is 2nd highest bid minus0, otherwise this payment is highest bid minus highest bid.
AGT-MIRI VCG mechanism
![Page 34: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/34.jpg)
Selling one itemVCG mechanism
Vickrey-Clarke-Groves (VCG) mechanisms
Goal: implement the efficient outcome in dominant strategies.
VCG is a general method generalizing SP auctions.
Solution: players should pay the damage they impose onsociety.
You can maximize efficiency by:
Choosing the efficient outcome (given the bids) as allocation.Each player pays his social cost (welfare).
Payment pi , for bidder i , is obtained as
Optimal welfare (for the other players) if player i was notparticipating.minus welfare of the other players from the chosen outcomeIn a single item auctionwhen i wins the object this payment is 2nd highest bid minus0, otherwise this payment is highest bid minus highest bid.
AGT-MIRI VCG mechanism
![Page 35: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/35.jpg)
Selling one itemVCG mechanism
Vickrey-Clarke-Groves (VCG) mechanisms
Goal: implement the efficient outcome in dominant strategies.
VCG is a general method generalizing SP auctions.
Solution: players should pay the damage they impose onsociety.
You can maximize efficiency by:
Choosing the efficient outcome (given the bids) as allocation.Each player pays his social cost (welfare).
Payment pi , for bidder i , is obtained as
Optimal welfare (for the other players) if player i was notparticipating.minus welfare of the other players from the chosen outcomeIn a single item auctionwhen i wins the object this payment is 2nd highest bid minus0, otherwise this payment is highest bid minus highest bid.
AGT-MIRI VCG mechanism
![Page 36: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/36.jpg)
Selling one itemVCG mechanism
Vickrey-Clarke-Groves (VCG) mechanisms
Goal: implement the efficient outcome in dominant strategies.
VCG is a general method generalizing SP auctions.
Solution: players should pay the damage they impose onsociety.
You can maximize efficiency by:
Choosing the efficient outcome (given the bids) as allocation.Each player pays his social cost (welfare).
Payment pi , for bidder i , is obtained as
Optimal welfare (for the other players) if player i was notparticipating.minus welfare of the other players from the chosen outcomeIn a single item auctionwhen i wins the object this payment is 2nd highest bid minus0, otherwise this payment is highest bid minus highest bid.
AGT-MIRI VCG mechanism
![Page 37: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/37.jpg)
Selling one itemVCG mechanism
Vickrey-Clarke-Groves (VCG) mechanisms
Goal: implement the efficient outcome in dominant strategies.
VCG is a general method generalizing SP auctions.
Solution: players should pay the damage they impose onsociety.
You can maximize efficiency by:
Choosing the efficient outcome (given the bids) as allocation.Each player pays his social cost (welfare).
Payment pi , for bidder i , is obtained as
Optimal welfare (for the other players) if player i was notparticipating.
minus welfare of the other players from the chosen outcomeIn a single item auctionwhen i wins the object this payment is 2nd highest bid minus0, otherwise this payment is highest bid minus highest bid.
AGT-MIRI VCG mechanism
![Page 38: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/38.jpg)
Selling one itemVCG mechanism
Vickrey-Clarke-Groves (VCG) mechanisms
Goal: implement the efficient outcome in dominant strategies.
VCG is a general method generalizing SP auctions.
Solution: players should pay the damage they impose onsociety.
You can maximize efficiency by:
Choosing the efficient outcome (given the bids) as allocation.Each player pays his social cost (welfare).
Payment pi , for bidder i , is obtained as
Optimal welfare (for the other players) if player i was notparticipating.minus welfare of the other players from the chosen outcome
In a single item auctionwhen i wins the object this payment is 2nd highest bid minus0, otherwise this payment is highest bid minus highest bid.
AGT-MIRI VCG mechanism
![Page 39: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/39.jpg)
Selling one itemVCG mechanism
Vickrey-Clarke-Groves (VCG) mechanisms
Goal: implement the efficient outcome in dominant strategies.
VCG is a general method generalizing SP auctions.
Solution: players should pay the damage they impose onsociety.
You can maximize efficiency by:
Choosing the efficient outcome (given the bids) as allocation.Each player pays his social cost (welfare).
Payment pi , for bidder i , is obtained as
Optimal welfare (for the other players) if player i was notparticipating.minus welfare of the other players from the chosen outcomeIn a single item auction
when i wins the object this payment is 2nd highest bid minus0, otherwise this payment is highest bid minus highest bid.
AGT-MIRI VCG mechanism
![Page 40: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/40.jpg)
Selling one itemVCG mechanism
Vickrey-Clarke-Groves (VCG) mechanisms
Goal: implement the efficient outcome in dominant strategies.
VCG is a general method generalizing SP auctions.
Solution: players should pay the damage they impose onsociety.
You can maximize efficiency by:
Choosing the efficient outcome (given the bids) as allocation.Each player pays his social cost (welfare).
Payment pi , for bidder i , is obtained as
Optimal welfare (for the other players) if player i was notparticipating.minus welfare of the other players from the chosen outcomeIn a single item auctionwhen i wins the object this payment is 2nd highest bid minus0,
otherwise this payment is highest bid minus highest bid.
AGT-MIRI VCG mechanism
![Page 41: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/41.jpg)
Selling one itemVCG mechanism
Vickrey-Clarke-Groves (VCG) mechanisms
Goal: implement the efficient outcome in dominant strategies.
VCG is a general method generalizing SP auctions.
Solution: players should pay the damage they impose onsociety.
You can maximize efficiency by:
Choosing the efficient outcome (given the bids) as allocation.Each player pays his social cost (welfare).
Payment pi , for bidder i , is obtained as
Optimal welfare (for the other players) if player i was notparticipating.minus welfare of the other players from the chosen outcomeIn a single item auctionwhen i wins the object this payment is 2nd highest bid minus0, otherwise this payment is highest bid minus highest bid.
AGT-MIRI VCG mechanism
![Page 42: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/42.jpg)
Selling one itemVCG mechanism
VCG: payments in a 5-item auction
Assume that bids = valuations$70 $30 $27 $25 $12 $5 $2
Optimal welfare if player i was not participating.$99 $139 $142 $144 $157 $164 $164
Welfare of the other players from the chosen outcome$94 $ 134 $137 $139 $157 $164 $164
This gives payments$5 $ 5 $ 5 $ 5 $5 $0 $0
VCG rules for k-item auctions:
Highest k bids win an object.The winners pay the (k + 1)st bid.
Here, again, truthfulness is a dominant strategy.
AGT-MIRI VCG mechanism
![Page 43: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/43.jpg)
Selling one itemVCG mechanism
VCG: payments in a 5-item auction
Assume that bids = valuations$70 $30 $27 $25 $12 $5 $2
Optimal welfare if player i was not participating.$99 $139 $142 $144 $157 $164 $164
Welfare of the other players from the chosen outcome$94 $ 134 $137 $139 $157 $164 $164
This gives payments$5 $ 5 $ 5 $ 5 $5 $0 $0
VCG rules for k-item auctions:
Highest k bids win an object.The winners pay the (k + 1)st bid.
Here, again, truthfulness is a dominant strategy.
AGT-MIRI VCG mechanism
![Page 44: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/44.jpg)
Selling one itemVCG mechanism
VCG: payments in a 5-item auction
Assume that bids = valuations$70 $30 $27 $25 $12 $5 $2
Optimal welfare if player i was not participating.
$99 $139 $142 $144 $157 $164 $164
Welfare of the other players from the chosen outcome$94 $ 134 $137 $139 $157 $164 $164
This gives payments$5 $ 5 $ 5 $ 5 $5 $0 $0
VCG rules for k-item auctions:
Highest k bids win an object.The winners pay the (k + 1)st bid.
Here, again, truthfulness is a dominant strategy.
AGT-MIRI VCG mechanism
![Page 45: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/45.jpg)
Selling one itemVCG mechanism
VCG: payments in a 5-item auction
Assume that bids = valuations$70 $30 $27 $25 $12 $5 $2
Optimal welfare if player i was not participating.$99 $139 $142 $144 $157 $164 $164
Welfare of the other players from the chosen outcome$94 $ 134 $137 $139 $157 $164 $164
This gives payments$5 $ 5 $ 5 $ 5 $5 $0 $0
VCG rules for k-item auctions:
Highest k bids win an object.The winners pay the (k + 1)st bid.
Here, again, truthfulness is a dominant strategy.
AGT-MIRI VCG mechanism
![Page 46: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/46.jpg)
Selling one itemVCG mechanism
VCG: payments in a 5-item auction
Assume that bids = valuations$70 $30 $27 $25 $12 $5 $2
Optimal welfare if player i was not participating.$99 $139 $142 $144 $157 $164 $164
Welfare of the other players from the chosen outcome
$94 $ 134 $137 $139 $157 $164 $164
This gives payments$5 $ 5 $ 5 $ 5 $5 $0 $0
VCG rules for k-item auctions:
Highest k bids win an object.The winners pay the (k + 1)st bid.
Here, again, truthfulness is a dominant strategy.
AGT-MIRI VCG mechanism
![Page 47: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/47.jpg)
Selling one itemVCG mechanism
VCG: payments in a 5-item auction
Assume that bids = valuations$70 $30 $27 $25 $12 $5 $2
Optimal welfare if player i was not participating.$99 $139 $142 $144 $157 $164 $164
Welfare of the other players from the chosen outcome$94 $ 134 $137 $139 $157 $164 $164
This gives payments$5 $ 5 $ 5 $ 5 $5 $0 $0
VCG rules for k-item auctions:
Highest k bids win an object.The winners pay the (k + 1)st bid.
Here, again, truthfulness is a dominant strategy.
AGT-MIRI VCG mechanism
![Page 48: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/48.jpg)
Selling one itemVCG mechanism
VCG: payments in a 5-item auction
Assume that bids = valuations$70 $30 $27 $25 $12 $5 $2
Optimal welfare if player i was not participating.$99 $139 $142 $144 $157 $164 $164
Welfare of the other players from the chosen outcome$94 $ 134 $137 $139 $157 $164 $164
This gives payments
$5 $ 5 $ 5 $ 5 $5 $0 $0
VCG rules for k-item auctions:
Highest k bids win an object.The winners pay the (k + 1)st bid.
Here, again, truthfulness is a dominant strategy.
AGT-MIRI VCG mechanism
![Page 49: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/49.jpg)
Selling one itemVCG mechanism
VCG: payments in a 5-item auction
Assume that bids = valuations$70 $30 $27 $25 $12 $5 $2
Optimal welfare if player i was not participating.$99 $139 $142 $144 $157 $164 $164
Welfare of the other players from the chosen outcome$94 $ 134 $137 $139 $157 $164 $164
This gives payments$5 $ 5 $ 5 $ 5 $5 $0 $0
VCG rules for k-item auctions:
Highest k bids win an object.The winners pay the (k + 1)st bid.
Here, again, truthfulness is a dominant strategy.
AGT-MIRI VCG mechanism
![Page 50: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/50.jpg)
Selling one itemVCG mechanism
VCG: payments in a 5-item auction
Assume that bids = valuations$70 $30 $27 $25 $12 $5 $2
Optimal welfare if player i was not participating.$99 $139 $142 $144 $157 $164 $164
Welfare of the other players from the chosen outcome$94 $ 134 $137 $139 $157 $164 $164
This gives payments$5 $ 5 $ 5 $ 5 $5 $0 $0
VCG rules for k-item auctions:
Highest k bids win an object.The winners pay the (k + 1)st bid.
Here, again, truthfulness is a dominant strategy.
AGT-MIRI VCG mechanism
![Page 51: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/51.jpg)
Selling one itemVCG mechanism
VCG: payments in a 5-item auction
Assume that bids = valuations$70 $30 $27 $25 $12 $5 $2
Optimal welfare if player i was not participating.$99 $139 $142 $144 $157 $164 $164
Welfare of the other players from the chosen outcome$94 $ 134 $137 $139 $157 $164 $164
This gives payments$5 $ 5 $ 5 $ 5 $5 $0 $0
VCG rules for k-item auctions:
Highest k bids win an object.The winners pay the (k + 1)st bid.
Here, again, truthfulness is a dominant strategy.
AGT-MIRI VCG mechanism
![Page 52: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/52.jpg)
Selling one itemVCG mechanism
VCG: payments in a 5-item auction
Assume that bids = valuations$70 $30 $27 $25 $12 $5 $2
Optimal welfare if player i was not participating.$99 $139 $142 $144 $157 $164 $164
Welfare of the other players from the chosen outcome$94 $ 134 $137 $139 $157 $164 $164
This gives payments$5 $ 5 $ 5 $ 5 $5 $0 $0
VCG rules for k-item auctions:
Highest k bids win an object.The winners pay the (k + 1)st bid.
Here, again, truthfulness is a dominant strategy.
AGT-MIRI VCG mechanism
![Page 53: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/53.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 54: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/54.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 55: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/55.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 56: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/56.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome:
buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 57: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/57.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 58: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/58.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 59: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/59.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 60: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/60.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 61: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/61.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 62: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/62.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.
Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 63: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/63.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.
p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 64: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/64.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80.
Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 65: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/65.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70
But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 66: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/66.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!
Cost is not covered!
AGT-MIRI VCG mechanism
![Page 67: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/67.jpg)
Selling one itemVCG mechanism
VCG: Roommates
TV cost $100
Bidders are willing to pay v1 and v2 this is private information.
Efficient outcome: buy if v1 + v2 > 100
VCG ensures:
Efficient outcome.Truthful revelation.
What are the VCG payments?
Consider values v1 = 70, v2 = 80.
With player 1: value for the others is 80.Without player 1: welfare for the others is 100.p1 = 100 − 80. Similarly for player 2, p2 = 100 − 70But, total payment is 20 + 30 < 100!Cost is not covered!
AGT-MIRI VCG mechanism
![Page 68: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/68.jpg)
Selling one itemVCG mechanism
VCG: budget balanced?
In general, p1 = 100 − v2 and p2 = 100 − v1.
p1 + p2 = 100 − v1 + 100 − v2 = 100 − (v1 + v2 − 100) < 100
Whenever we can buy, the cost is not covered!
In some cases, the VCG mechanism is not budget-balanced:spends more than it collects from the players!
This is a real problem!
There isn’t much we can do: It can be shown that there is nomechanism that is both efficient and budget balanced.
AGT-MIRI VCG mechanism
![Page 69: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/69.jpg)
Selling one itemVCG mechanism
VCG: budget balanced?
In general, p1 = 100 − v2 and p2 = 100 − v1.
p1 + p2 = 100 − v1 + 100 − v2 = 100 − (v1 + v2 − 100) < 100
Whenever we can buy, the cost is not covered!
In some cases, the VCG mechanism is not budget-balanced:spends more than it collects from the players!
This is a real problem!
There isn’t much we can do: It can be shown that there is nomechanism that is both efficient and budget balanced.
AGT-MIRI VCG mechanism
![Page 70: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/70.jpg)
Selling one itemVCG mechanism
VCG: budget balanced?
In general, p1 = 100 − v2 and p2 = 100 − v1.
p1 + p2 = 100 − v1 + 100 − v2 = 100 − (v1 + v2 − 100) < 100
Whenever we can buy, the cost is not covered!
In some cases, the VCG mechanism is not budget-balanced:spends more than it collects from the players!
This is a real problem!
There isn’t much we can do: It can be shown that there is nomechanism that is both efficient and budget balanced.
AGT-MIRI VCG mechanism
![Page 71: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/71.jpg)
Selling one itemVCG mechanism
VCG: budget balanced?
In general, p1 = 100 − v2 and p2 = 100 − v1.
p1 + p2 = 100 − v1 + 100 − v2 = 100 − (v1 + v2 − 100) < 100
Whenever we can buy, the cost is not covered!
In some cases, the VCG mechanism is not budget-balanced:spends more than it collects from the players!
This is a real problem!
There isn’t much we can do: It can be shown that there is nomechanism that is both efficient and budget balanced.
AGT-MIRI VCG mechanism
![Page 72: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/72.jpg)
Selling one itemVCG mechanism
VCG: budget balanced?
In general, p1 = 100 − v2 and p2 = 100 − v1.
p1 + p2 = 100 − v1 + 100 − v2 = 100 − (v1 + v2 − 100) < 100
Whenever we can buy, the cost is not covered!
In some cases, the VCG mechanism is not budget-balanced:spends more than it collects from the players!
This is a real problem!
There isn’t much we can do: It can be shown that there is nomechanism that is both efficient and budget balanced.
AGT-MIRI VCG mechanism
![Page 73: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/73.jpg)
Selling one itemVCG mechanism
VCG: budget balanced?
In general, p1 = 100 − v2 and p2 = 100 − v1.
p1 + p2 = 100 − v1 + 100 − v2 = 100 − (v1 + v2 − 100) < 100
Whenever we can buy, the cost is not covered!
In some cases, the VCG mechanism is not budget-balanced:spends more than it collects from the players!
This is a real problem!
There isn’t much we can do: It can be shown that there is nomechanism that is both efficient and budget balanced.
AGT-MIRI VCG mechanism
![Page 74: Vickery-Clark-Groves MechanismA general design rule is the Vickrey-Clarke-Groves mechanism. AGT-MIRI VCG mechanism. Selling one item VCG mechanism Example: The buying of multiple items](https://reader033.vdocuments.us/reader033/viewer/2022053123/60ac201ff3c48e75f525d2a8/html5/thumbnails/74.jpg)
Selling one itemVCG mechanism
VCG: budget balanced?
In general, p1 = 100 − v2 and p2 = 100 − v1.
p1 + p2 = 100 − v1 + 100 − v2 = 100 − (v1 + v2 − 100) < 100
Whenever we can buy, the cost is not covered!
In some cases, the VCG mechanism is not budget-balanced:spends more than it collects from the players!
This is a real problem!
There isn’t much we can do: It can be shown that there is nomechanism that is both efficient and budget balanced.
AGT-MIRI VCG mechanism