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Communication requirements of VCG-like mechanisms in convex environments

Communication requirements of VCG-like mechanisms in convex environments. Ramesh Johari Stanford University Joint work with John N. Tsitsiklis, MIT. Motivation. Resource allocation mechanisms with scalar strategy spaces: -single price: eff. loss · 25% (J & Tsitsiklis)

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Communication requirements of VCG-like mechanisms in convex environments

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  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 • Nusers • Jresources • Feasible allocations: X= { x2RN : 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 r’s 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 r:declared utilityVr • Mechanism chooses x(V) s.t.: • Payment to user r:

  8. II VCG mechanisms • Strategy of user r:declared utilityVr • Mechanism chooses x(V) s.t.: • Payment to user r:

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

  10. II VCG mechanisms • Moral:truthful 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 > 0 and x¸ 0, 9 > 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 hugeU’(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() > 0 } J = { j : gj(x()) = 0 } d(r) = ( gj/xr, j2J ) Theorem:If for all r, d(r) is linearly dependent on d(s), sr, s2P,then x() is efficient

  18. III SSVCG: Efficient NE We know: x() = maxx2XrU(xr ; r) For all r: x() 2 maxx2XUr(xr) + srU(xs ; s) First order conditions + linear dependence assumption ) Ur’(xr()) = U(xr() ; r)

  19. IV Networks Jlinks Capacity of link j : Cj User r$ subset of links X = { x¸ 0 : r : j2rxr·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

  24. III SSVCG: Efficient NE But: all NE are not efficient! Example: Two users, U1’(1) = 2, U2’(1) = 3 X = { (x1, x2) : 5x1+x2·6, x1+x2·1 } Then: • (1, 1) is not an efficient allocation • (1, 1) is an NE allocation:  s.t. U’(1 ; 1) = 4 ; U’(1 ; 2) = 1

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