Mechanism design

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Mechanism design
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Goal of mechanism design

• Implementing a social choice function f(u1, ,
  uA) using a game
• Center auctioneer does not know the agents
  preferences
• Agents may lie
• Goal is to design the rules of the game (aka
  mechanism) so that in equilibrium (s1, , sA),
  the outcome of the game is f(u1, , uA)
• Mechanism designer specifies the strategy sets Si
  and how outcome is determined as a function of
  (s1, , sA) ? (S1, , SA)
• Variants
• Strongest There exists exactly one equilibrium.
  Its outcome is f(u1, , uA)
• Medium In every equilibrium the outcome is f(u1,
  , uA)
• Weakest In at least one equilibrium the outcome
  is f(u1, , uA)

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Revelation principle

• Any outcome that can be supported in Nash
  (dominant strategy) equilibrium via a complex
  indirect mechanism can be supported in Nash
  (dominant strategy) equilibrium via a direct
  mechanism where agents reveal their types
  truthfully in a single step

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Uses of the revelation principle

• Literal Only direct mechanisms needed
• Problems
• Strategy formulator might be complex
• Complex to determine and/or execute best-response
  strategy
• Computational burden is pushed on the center
  (assumed away)
• Thus the revelation principle might not hold in
  practice if these computational problems are hard
• This problem traditionally ignored in game theory
• Even if the indirect mechanism has a unique
  equilibrium, the direct mechanism can have
  additional bad equilibria
• As an analysis tool
• Best direct mechanism gives tight upper bound on
  how well any indirect mechanism can do
• Space of direct mechanisms is smaller than that
  of indirect ones
• One can analyze all direct mechanisms pick best
  one
• Thus one can know when one has designed an
  optimal indirect mechanism (when it is as good as
  the best direct one)

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Implementation in dominant strategies
Strongest form of mechanism design
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Implementation in dominant strategies

• Goal is to design the rules of the game (aka
  mechanism) so that in dominant strategy
  equilibrium (s1, , sA), the outcome of the
  game is f(u1, , uA)
• Nice in that agents cannot benefit from
  counterspeculating each other
• Others preferences
• Others rationality
• Others endowments
• Others capabilities

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Gibbard-Satterthwaite impossibility

• Thrm. If O 3 (and each outcome would be
  the social choice under f for some input profile
  (u1, , uA) ) and f is implementable in
  dominant strategies, then f is dictatorial

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Special case where dominant strategy
implementation is possible Quasilinear
preferences -gt Clarke tax mechanism

• Outcome (x1, x2, ..., xk, m1, m2, ..., mA )
• Quasilinear preferences ui(x, m) mi vi(x1,
  x2, ..., xk)
• Utilitarian setting Social welfare maximizing
  choice
• Outcome s(v1, v2, ..., vA) maxx ?i vi(x1, x2,
  ..., xk)
• Agents payment mi ?j?i vj(s(v)) - ?j?i
  vj(s(v-i)) ? 0 is a tax
• Thrm Every agents dominant strategy is to
  reveal preferences truthfully
• Intuition Agent internalizes the negative
  externality he imposes on others by affecting the
  outcome
• Agent pays nothing if he does not change the
  outcome
• Example k1, x1joint pool built or not,
  mi
• E.g. equal sharing of construction cost -c / A

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Clarke tax mechanism

• Pros
• Social welfare maximizing outcome
• Truth-telling is a dominant strategy
• Feasible in that it does not need a benefactor
  (?i mi ? 0)
• Cons
• Budget balance not maintained (in pool example,
  generally ?i mi lt 0)
• Have to burn the excess money that is collected
• Thrm. Green Laffont 1979. Let the agents
  have arbitrary quasilinear preferences. No
  social choice function that is (ex post) welfare
  maximizing (taking into account money burning as
  a loss) is implementable in dominant strategies
• If there is some party that has no private
  information to reveal and no preferences over x,
  welfare maximization and budget balance can be
  obtained by having that partys payment be m0 -
  ?i1.. mi
• Auctioneer could be called agent 0
• Vulnerable to collusion
• Even by coalitions of just 2 agents

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Another approach for circumventing the
impossibility of dominant-strategy implementation

• Design the game so that (although manipulations
  exist), finding a beneficial manipulation is
  computationally so complex for an agent that the
  agent cannot do that
• E.g. Complexity of Manipulating Elections with
  Few Candidates Conitzer Sandholm AAAI-02,
  TARK-03
• E.g. Universal Voting Protocol Tweaks for Making
  Manipulation Hard Conitzer Sandholm IJCAI-03

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Yet another approach for circumventing the
impossibility of dominant-strategy implementation

• Designing the mechanism automatically to the
  situation at hand Conitzer Sandholm
• Input is the probabilistic information that the
  center has about the agents
• Output is an optimal mechanism where the agents
  are motivated to reveal their preferences
  truthfully, and a social objective is satisfied
  to the optimal extent
• Advantages
• Can be used even without side payments
  quasilinear preferences
• Could achieve better outcomes than Clarke tax
  mechanism
• Circumvents impossibility in many cases
• Complexity of Mechanism Design
• Designing a deterministic mechanism is
  NP-complete
• Designing a randomized mechanism is fast
• No loss in social objective, sometime a gain
• Both results also hold for Bayes-Nash
  implementation
• E.g., metal manufacturers with asymmetric
  production costs