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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 – PowerPoint PPT presentation

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Title: Communication requirements of VCG-like mechanisms in convex environments


1
Communication requirements of VCG-like mechanisms
in convex environments
  • Ramesh Johari
  • Stanford University
  • Joint work with John N. Tsitsiklis, MIT

2
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

3
Outline
  • Resource allocation model
  • VCG mechanisms
  • Scalar strategy VCG mechanisms
  • Multicommodity flow models
  • Extensions and related work

4
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

5
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

6
II Achieving efficiency
  • In general utilities are unknown
  • Design payments tr to alignefficiency and
    incentives
  • Planner wants to maximize
  • User r wants to maximize

7
II VCG mechanisms
  • Strategy of user rdeclared utility Vr
  • Mechanism chooses x(V) s.t.
  • Payment to user r

8
II VCG mechanisms
  • Strategy of user rdeclared utility Vr
  • Mechanism chooses x(V) s.t.
  • Payment to user r

9
II VCG mechanisms
  • Strategy of user rdeclared utility Vr
  • Mechanism chooses x(V) s.t.
  • User r chooses Vr to maximize

10
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

11
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 ?) ?

12
III SSVCG mechanisms
  • Mechanism chooses x(?) s.t.
  • Payment to user r

13
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)

14
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
15
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

16
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

17
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

18
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)

19
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

20
V Extensions
  • If Ur depends on k-dimensional xr
  • Need k-dimensional ?r
  • Designing hr(?-r) is similar to VCG
  • Budget balance, etc.

21
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)

22
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

23
V Related work
  • Semret (1999)
  • Groves and Ledyard (1979)
  • Yang and Hajek (2005)
  • independent discovery of similar result
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