Communication requirements of VCG-like mechanisms in convex environments

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Communication requirements of VCG-like mechanisms
in convex environments

• Ramesh Johari
• Stanford University
• Joint work with John N. Tsitsiklis, MIT

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Motivation

• Resource allocation mechanismswith scalar
  strategy spaces
• -single price eff. loss 25
• (J Tsitsiklis)
• -price differentiation no eff. loss
• (Yang Hajek, Maheswaran Basar)
• This talk generalization ofthe price
  differentiation case

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Outline

• Resource allocation model
• VCG mechanisms
• Scalar strategy VCG mechanisms
• Multicommodity flow models
• Extensions and related work

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I Resource allocation model

• N users
• J resources
• Feasible allocations
• X x 2 RN x 0,
• gj(x) 0, j 1, , J
• gj() convex, differentiable
• Assume Slater condition holds

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I Utilities and payoffs

• User r utility Ur(xr) from allocation xr
• concave, strictly increasing, differentiable
• Payment to user r tr
• User rs payoff (in )
• Pr(xr, tr) Ur(xr) tr
• ) Efficient allocation

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II Achieving efficiency

• In general utilities are unknown
• Design payments tr to alignefficiency and
  incentives
• Planner wants to maximize
• User r wants to maximize

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II VCG mechanisms

• Strategy of user rdeclared utility Vr
• Mechanism chooses x(V) s.t.
• Payment to user r

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II VCG mechanisms

• Strategy of user rdeclared utility Vr
• Mechanism chooses x(V) s.t.
• Payment to user r

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II VCG mechanisms

• Strategy of user rdeclared utility Vr
• Mechanism chooses x(V) s.t.
• User r chooses Vr to maximize

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II VCG mechanisms

• Moraltruthful declaration is adominant
  strategy
• Problem
• Strategy spaces are overly complex
• Main insight (for Nash implementation)
• Suffices to elicit only local derivativeof
  utility function

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III SSVCG mechanisms

• VCG-like with scalar strategy spaces.
• Parameterized family U(x ?) s.t.
• x ? U(x ?) is strictly concave
• and strictly increasing, continuous,
  differentiable
• Slope matching
• 8 ? gt 0 and x 0,
• 9 ? gt 0 s.t. U(x ?) ?

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III SSVCG mechanisms

• Mechanism chooses x(?) s.t.
• Payment to user r

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III SSVCG Key lemma

• ? is a Nash equilibriumif and only if for all
  r
• Proof idea If x is optimal, user r can choose
  ?r s.t. Ur(xr) U(xr ?r)

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III SSVCG Key lemma

• ? is a Nash equilibriumif and only if for all
  r
• Proof idea If x is optimal, user r can choose
  ?r s.t. Ur(xr) U(xr ?r)

tr
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III SSVCG Efficient NE

• Corollary
• Efficient Nash equilibrium exists
• Proof idea Given efficient x,each user r
  chooses ?r s.t.
• Ur(xr) U(xr ?r)
• ) Local truthful declaration

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III SSVCG Efficient NE

• But all NE are not efficient!
• Example
• Single resource of capacity C 1
• User 1 bids huge U(C ?1)
• All other users
• Best response is to give up
• ) User 1 gets everything, regardless of true
  utilities

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III SSVCG Efficient NE

• Given NE ?
• Define P s xs(?) gt 0
• J j gj(x(?)) 0
• d(r) ( ?gj/?xr, j 2 J )
• TheoremIf for all r, d(r) is linearly dependent
  on d(s), s ? r, s 2 P,then x(?) is efficient

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III SSVCG Efficient NE

• We know
• x(?) maxx 2 X ?r U(xr ?r)
• For all r
• x(?) 2 max x 2 X Ur(xr) ?s ? r U(xs ?s)
• First order conditions
• linear dependence assumption )
• Ur(xr(?)) U(xr(?) ?r)

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IV Networks

• J links
• Capacity of link j Cj
• User r subset of links
• X x 0 ?r j 2 r xr Cj, for all j
• Assume For all j, two users r1(j), r2(j), s.t.
• Uri (j)(0) 1 and r1(j) r2(j) j
• Then all NE allocations are efficient

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V Extensions

• If Ur depends on k-dimensional xr
• Need k-dimensional ?r
• Designing hr(?-r) is similar to VCG
• Budget balance, etc.

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V Related work

• Yang Hajek (2004),Maheswaran Basar (2004)
• Single resource, capacity 1
• User r chooses bid ?r
• Allocation xr(?) ?r / ?s ?s
• User r pays tr(?) -?r ?(?s ?s)
• Same as
• SSVCG where U(x ?) -? ?(?/x)

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V Related work

• Reichelstein and Reiter (1988)
• more general environments
• not quasilinear,no aggregated goods
• mechanism is asymmetric one user treated
  differently than others
• requires (J - 1) J/(N(N-1)) dimensional
  strategy space per user

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V Related work

• Semret (1999)
• Groves and Ledyard (1979)
• Yang and Hajek (2005)
• independent discovery of similar result