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VCG Protocol

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									             Comp325 Algorithmic and Game Theoretic Foundation for
             Internet Economics/Xiaotie Deng




 Lecture 12 VCG Protocol
http://www.csc.liv.ac.uk/~deng/COMP325.html
                         Comp325 Algorithmic and Game Theoretic Foundation for


  Generalization of Vickrey Auction
                         Internet Economics/Xiaotie Deng




– Recall that Vickrey Auction is the second price auction. 

– K-Vickrey takes the (k+1) price for all top k bidders, when all items 
  are the same.
– GSP is also a generalization but not truthful.

– VCG Generalizes the Vickrey Auction for multiple different items
                      Comp325 Algorithmic and Game Theoretic Foundation for


                 VCG Mechanism
                      Internet Economics/Xiaotie Deng




• Allocation: 
      • Optimization of Social Welfare, i.e., the total value of 
        the allocated items to the buyers.
• Pricing: 
      • Each player in the auction pays the opportunity 
        cost that their presence introduces to all the 
        other players. 
         Comp325 Algorithmic and Game Theoretic Foundation for


      Optimal Allocation
         Internet Economics/Xiaotie Deng




•  
      Comp325 Algorithmic and Game Theoretic Foundation for


      VCG Pricing
      Internet Economics/Xiaotie Deng




•  
      Comp325 Algorithmic and Game Theoretic Foundation for


        Example
      Internet Economics/Xiaotie Deng




•  
                 Comp325 Algorithmic and Game Theoretic Foundation for
                 Internet Economics/Xiaotie Deng
                      Exercise

• buyers a, b, c
• Items: football, CD
• Buyer a likes football only for 3 pounds
• Buyer b likes football for 4 pounds and CD for 
  two pound
• Buyer c likes to have both football and CD for a 
  total of 5 bounds.
            Comp325 Algorithmic and Game Theoretic Foundation for
            Internet Economics/Xiaotie Deng
      THEOREM: VCG is truthful

•  
                  Comp325 Algorithmic and Game Theoretic Foundation for
                  Internet Economics/Xiaotie Deng
      Difference of Utilities: True vs. Lying

•  
          Comp325 Algorithmic and Game Theoretic Foundation for
          Internet Economics/Xiaotie Deng
      Lying cannot be better

•  
    VCG Allocation for Sponsored 
            Comp325 Algorithmic and Game Theoretic Foundation for
            Internet Economics/Xiaotie Deng

               Search
 
    VCG Pricing for Sponsored Search
              Comp325 Algorithmic and Game Theoretic Foundation for
              Internet Economics/Xiaotie Deng




 
    VCG Revenue for Sponsored 
           Comp325 Algorithmic and Game Theoretic Foundation for
           Internet Economics/Xiaotie Deng

             Search
 
    VCG Revenue vs Social Optimum
             Comp325 Algorithmic and Game Theoretic Foundation for
             Internet Economics/Xiaotie Deng



 
                 Comp325 Algorithmic and Game Theoretic Foundation for
                 Internet Economics/Xiaotie Deng
         Why Truthful is Important?

• Players do not need to engage in the guess gain 
  of other players’ strategies so that it saves the 
  social cost. 
                 Comp325 Algorithmic and Game Theoretic Foundation for
                 Internet Economics/Xiaotie Deng
    Why Social Optimum is Important ?

• Exercise: Find the answers in a list with simple 
  examples.
                    Comp325 Algorithmic and Game Theoretic Foundation for

        Revenue Equivalence Theorem
                    Internet Economics/Xiaotie Deng




• Any bidder's payment under the forward-looking 
  equilibrium is equal to her payment under VCG 
  mechanism for the auction.
• For sponsored search auction, the auctioneer's 
  revenue in forward-looking equilibrium is equal 
  to her revenue under VCG mechanism for the 
  auction.
  – It is an RET though the conditions for it to hold were 
    not known previously.
                    Comp325 Algorithmic and Game Theoretic Foundation for
                    Internet Economics/Xiaotie Deng
            Exercise (do one of them)

• Find out the VCG solution when n=m for the 
  sponsored search auction.
• Find the revenue for the forward looking Nash 
  equilibrium and show it is the same as VCG.
• For a set of advertisement slots with # of clicks 
  being: 10, 40, 60, participants with values 1, 3, 4, 7, 
  8, 9. Is it a Nash equilibrium when all bid truthfully 
  under GSP?
• What is the forward looking Nash equilibrium bids 
  and prices
• What are the bids for VCG, allocation and prices?

								
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