Implementation in Bayes-Nash equilibrium

1
Implementation in Bayes-Nash equilibrium

• Tuomas Sandholm
• Computer Science Department
• Carnegie Mellon University

2
Implementation in Bayes-Nash equilibrium

• Goal is to design the rules of the game (aka
  mechanism) so that in Bayes-Nash equilibrium (s1,
  , sA), the outcome of the game is f(u1, ,
  uA)
• Weaker requirement than dominant strategy
  implementation
• An agents best response strategy may depend on
  others strategies
• Agents may benefit from counterspeculating each
  others
• preferences
• rationality
• endowments
• capabilities
• Can accomplish more than under dominant strategy
  implementation
• E.g., budget balance Pareto efficiency (social
  welfare maximization) under quasilinear
  preferences

3
Expected externality mechanism dAspremont
Gerard-Varet 79 Arrow 79

• Like Groves mechanism, but sidepayment is
  computed based on agents revelation vi ,
  averaging over possible true types of the others
  v-i
• 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)
• Others expected welfare when agent i announces
  vi is ?(vi) ?v-i p(v-i) ?j?i vj(s(vi , v-i))
• Measures change in expected externality as agent
  i changes her revelation
• Thrm. Assume quasilinear preferences and
  statistically independent valuation functions vi.
  A utilitarian social choice function f v -gt
  (s(v), m(v)) can be implemented in Bayes-Nash
  equilibrium if mi(vi) ?(vi) hi(v-i) for
  arbitrary function h
• Unlike in dominant strategy implementation,
  budget balance achievable
• Intuitively, have each agent contribute an equal
  share of others payments
• Formally, set hi(v-i) - 1 / (A-1) ?j?i
  ?(vj)
• Does not satisfy participation constraints (aka
  individual rationality constraints) in general
• Agent might get higher expected utility by not
  participating

4
Myerson-Satterthwaite impossibility

• Avrim is selling a car to Tuomas, both are risk
  neutral, quasilinear
• Each party knows his own valuation, but not the
  others valuation
• The probability distributions are common
  knowledge
• Want a mechanism that is
• Ex post budget balanced
• Ex post Pareto efficient Car changes hands iff
  vbuyer gt vseller
• (Interim) individually rational Both Avrim and
  Tuomas get higher expected utility by
  participating than not
• Thrm. Such a mechanism does not exist (even if
  randomized mechanisms are allowed)
• This impossibility is at the heart of more
  general exchange settings (NYSE, NASDAQ,
  combinatorial exchanges, ) !

5
Proof

• Sellers valuation is sL w.p. a and sH w.p. (1-a)
• Buyers valuation is bL w.p. b and bH w.p. (1-b).
  Say bH gt sH gt bL gt sL
• By revelation principle, can focus on truthful
  direct revelation mechanisms
• p(b,s) probability that car changes hands given
  revelations b and s
• Ex post efficiency requires p(b,s) 0 if (b
  bL and s sH), otherwise p(b,s) 1
• Thus, EpbbH 1 and Epb bL a
• Eps sH 1-b and Eps sL 1
• m(b,s) expected price buyer pays to seller
  given revelations b and s
• Since parties are risk neutral, equivalently
  m(b,s) actual price buyer pays to seller
• Buyer pays what seller gets paid ? ex post budget
  balance
• Emb (1-a) m(b, sH) a m(b, sL)
• Ems (1-b) m(bH, s) b m(bL, s)
• Individual rationality (IR) requires
• b Epb Emb ? 0 for b bL, bH
• Ems s Eps ? 0 for s sL, sH
• Bayes-Nash incentive compatibility (IC) requires
• b Epb Emb ? b Epb Emb for all
  b, b
• Ems s Eps ? Ems s Eps for all
  s, s
• Suppose ab ½, sL0, sHy, bLx, bHxy, where 0
  lt 3x lt y. Now,

6
Myerson-Satterthwaite impossibility

• Actually, the impossibility applies to any
  priors, as long as
• the priors supports overlap, and
• the priors dont have gaps
• The inefficiency is caused by two-sided private
  information

7
Full implementation

• Here we dont necessarily assume that payments
  are possible
• Want to implement so that the right social choice
  function is achieved in all equilibria
• Virtual implementation relax this by allowing a
  social choice within ? to be implemented
• Thm Serrano Vohra GEB 2005. Consider pure
  strategies only, and assume no-total-indifference.
  A social choice function in such environments is
  virtually Bayesian implementable iff it satisfies
  Bayesian incentive compatibility and a condition
  called virtual monotonicity.
• Virtual monotonicity is weak in the sense that it
  is generically satisfied in environments with at
  least three alternatives
• This implies that in most environments virtual
  Bayesian implementation is as successful as it
  can be (incentive compatibility is the only
  condition needed)