Algorithmic Mechanism Design

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• Algorithmic Mechanism Design
• Hong Zhu
• Fudan University, Shanghai, China

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Input of Algorithms

• 1? Traditional ?????
• Input ---gt Output
• 2? Internet ???
• Global computing -- ????,?????????????given
  large integer N with 500 bits, ??????????p,????pN
  or not. ???????????,RSA ???????????
• ?????????????????reactions ,??behaviour?????

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My Research

• a
• 5 6
• X 13
  Y
• 7 5
• b

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Mechanism Design

• Concept
• ??????,?????????,??????????
• ????????????,?????????????????????agents?????self-
  interested ????rational,??????????????????????????
  ?????

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Mechanism Design ?????

• Truthfulness---Incentive Compatibility
• ????? That is , a protocol motivates agents to
  tell their private preferences (valuations)
  truthfully

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??????????

• Internet ?????revenue???.
• ?agents?valuations????????????????,?????????,????,
  ???????????

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??????????

• Utilitarian mechanism design
• ????agents?total valuations???,??VCG??,???????

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Nash ??

• n Players
• ??player i, i1,,n.???????S_i, ?????????u_i
  S_1 S_2 S_n R
• ???players?????x_1 ? S_1, x_2 ? S_2,, x_n ?
  S_n,??Nash equlibrium for all I
• u_i(x_1, x_2,,x_i,, x_n )?
• u_i(x_1, x_2,,y_i,, x_n )
• if y_i?x_i

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Single-Minded Auction

• An auctioneer
• sells m heterogeneous commodities ? w1,,wm
• There are n buyers O1,,On .For each buyer Oi,
  there is a unique ?i ? ?, such that, for any
  bundle B ? ?,
• vi the true valuation of each buyer, and
• vi(B) vi(?i), if ?i ? B.
• vi(B) 0, otherwise.
• bi the actual submitted bid of each buyer.
• Pi the payment of each buyer. Note that if a
  buyer does not win the item, Pi 0.

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Single-Minded Auction

• The auctioneer determines
• X (X1,,Xn), the allocation vector of buyers
• Px (Pi,,Pn), the payment vector
• Each buyer aims to maximize the utility value
• ui vi(Xi)-Pi

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Special Case - English auction

• The auctioneer increases the price of the item in
  a continuous manner (or with an increment that is
  insignificantly small) until only one buyer stays
  in.

If you are a buyer, will you bid your valuation
truthfully?
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Special Case - Dutch auction

• The auctioneer starts at a high price and reduces
  it continuously. The auction ends when some buyer
  shouts Mine! to claim the item at the current
  price.

This time, will a buyer bid the truth?
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Incentive compatibility

• We say an auction is incentive compatible (or
  truthful) if the utility of each buyer is
  maximized by bidding his true valuation, i.e., bi
  vi , regardless the bids of other buyers.
• For example,
• English auction is incentive compatible.
• Dutch auction is not incentive compatible.

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Previous Results

• Definition. An allocation algorithm X is monotone
  if for any given bids b-i and a winning
  declaration bi , any higher declaration bi gt bi
  still wins, where b-i is the bids vector of all
  buyers except i.
• Lemma. Let X be a monotone allocation algorithm.
  Then for any b-i, there exists a unique critical
  value ci(X b-i) such that for any bi gt ci(X
  b-i), bi is a winning declaration, and for any bi
  lt ci(X b-i), bi is a losing declaration.

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Previous Results

• Definition. The critical payment PX(b) associated
  with the monotone allocation algorithm X is
  defined by Pi ci(X b-i) if Xi ?i and Pi 0
  otherwise.
• Theorem. Mualem Nisan, AAAI02 A mechanism
  is incentive compatible if and only if its
  allocation algorithm is monotone and its payment
  is critical payment.

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Conclusion

• Theorem. The Greedy Allocation Algorithm based on
  cost ranking associated with the critical payment
  is incentive compatible mechanism, and the
  payment can be interpreted as the price.

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Revenue Consideration

• Definition. (Maximal Revenue Problem) Given
  constant k gt 0 and single-minded auction, does
  there exist regular price vector and trading
  buyers such that the revenue of the auctioneer is
  at least k.
• Theorem. Maximal Revenue Problem is NP-hard.

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Future Breakthrough

• New Tools
• Game theory
• Mathematical Economics - Ecoinformatic
• Computational Model

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STOC01 and ICALP01 by Papadimitriou of UC
Berkeley

• ???? computer logic and combinatorics
• ???? internet has aguable serpassed the von
  Neumann computer as the most complex
  computational artifact Game theory,
  mathematical economics and (??????)
• finite model theory, ????

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• Thank you !