Introduction to Mechanism Design

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Introduction to Mechanism Design

• Patchrawat Uthaisombut
• University of Pittsburgh
• September 5th, 2002

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Mechanism design

• Mechanism Design Problems
• Auction of single item
• Construction of a Public Property
• Auction of digital goods

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Mechanism Design

• Game theory
• Given a game and rational players who want to
  maximize their payoffs, what will happen?
• Mechanism Design
• Inverse of game theory
• Given a goal, design a game so that the players,
  driven by their self-interests, end up achieving
  the goal of the game designer.

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valuation vi
322
200
643
150
307
750
400
bid bi
Want to give to the player with maximum vi
charge ci
Not seeking to maximize profit.
utility ui vi ci
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Designing a Game
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First price auction

• Each player
• The players simultaneously submit bids bis.
• Relabel players so that b1gtb2gtgtbn.
• Player 1 gets the item and pays b1. The others
  pay nothing.
• Utility of player 1 v1-b1.
• The others utilities 0

v3
v2
v1
0
b3
b2
b1
0
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Second price auction

• Same as First price auction except that player 1
  pays b2.
• Utility of player 1 v1-b2.
• The others utilities 0

v3
v2
v1
0
b3
b2
b1
0
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Privacy in Auction

• How much information can be withheld yet
  everybody is certain that there is no
  discrepancy?
• Bid of each bidder
• Identity of bidders

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Privacy in Auction

• Conceal everything.
• The item goes to nobody.
• Reveal only the value of the highest bid.
• The auctioneer could lie about the 2nd highest
  bid.
• Reveal identity of 2 highest bidders their bids
• Ok?
• Reveal identity of 2 highest bidders 2nd
  highest bid.
• Ok?
• Other combinations?

v3
v2
v1
0
b3
b2
b1
0
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valuation vi
322
200
643
150
307
750
400
cost C
reported value ri
Want to construct if Sum vi gt C
charge ci
Recovering the cost C is not required.
utility ui vi ci
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General Form of a Mechanism

• Citizens report their valuation ri
• ri? vi possible if citizen i lies.
• Based on ris
• Decide whether to build P.
• If so, charge each citizen an amount ci.
• Citizen is utility is ui vi ci if the
  fountain is constructed. Otherwise, ui 0.

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Some Attempts

• Each citizen pays C/n
• A citizen with vi gt C/n could lie.
• Charge the amount they declare
• Lie and report smaller value.
• Charge max(0, C- Sumj? i rj)
• rj amount reported by i.
• This encourages the citizens to report their
  valuations truthfully.

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Analysis

• Want to construct if Sum vj gt C.
• Will construct if Sum rj gt C.
• ci max(0, C- Sumj? i rj) and ui vi ci
• We will show that telling the truth is a players
  best strategy. (that is, player i should set rj
  vi)
• If player is lie doesnt change the outcome,
  there is no gain or lost to i.
• Case 1 is lie tips the sum above C
• Case 2 is lie tips the sum below C

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Mechanism design

• Mechanism Design Problems
• Auction of single item
• Construction of a Public Property
• Auction of digital goods

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Auction of unlimited-supply items

• Unlimited supply of identical items
• More items than consumers
• Cost of reproducing is negligible
• Pay-per-view movies, downloadable audio files
• Consumer is valuation vi private info
• The maximum amount willing to pay
• Design auction procedure to maximize the profit.
• Will focus on single round, sealed bid auctions.

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An upper bound

• Upper bound
• Sum vi
• Can this be achieved?
• Everybody must get an item.
• A consumer always gets an item and pays her
  reported valuation.
• Consumer could lie and report lower valuation.

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Single pricing

• Fix a price p.
• Consumer i gets an item if vi gt p and pays p.
• If vi is known, we can compute optimal p.
• Setting p too high
• Not enough items sold.
• Setting p too low
• Not enough profit per item.
• Problem We dont know vi.

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Some solutions

• Partially solved by Goldberg et.al. 2000
• Competitive
• Close to maximum profit
• Single round, sealed bid
• Multi-price
• Different consumers pay different prices.
• Randomized
• Number of winning bid is not fixed in advance
• Truthful
• Consumers do best when truthfully declare their
  valuation
• Simple strategy for consumers.
• Independent of strategies of other consumers

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Open Problems

• Design a game to maximize the throughput of a
  network.
• Auctions of multiple items
• Sharing in Peer-to-peer networks
• Packets forwarding in ad hoc networks
• Cooperation among agents