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 mechanisms with scalar strategy spaces: -single price: eff. loss · 25% (J & Tsitsiklis) -price differentiation: no eff. loss (Yang & Hajek, Maheswaran & Basar) This talk: generalization of the price differentiation case

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

4. IResource allocation model N users J resources Feasible allocations: X = { x 2 R N : x ¸ 0, g j ( x ) · 0, j = 1, …, J } g j ( ¢ ) : convex, differentiable Assume: Slater condition holds

5. IUtilities and payoffs User r : utility U r ( x r ) from allocation x r concave, strictly increasing, differentiable Payment to user r : t r User r s payoff (in $$$): P r ( x r, t r ) = U r ( x r ) + t r ) Efficient allocation:

6. IIAchieving efficiency In general: utilities are unknown Design payments t r to align efficiency and incentives Planner wants to maximize: User r wants to maximize:

7. IIVCG mechanisms Strategy of user r : declared utility V r Mechanism chooses x ( V ) s.t.: Payment to user r :

8. IIVCG mechanisms Strategy of user r : declared utility V r Mechanism chooses x ( V ) s.t.: Payment to user r :

9. IIVCG mechanisms Strategy of user r : declared utility V r Mechanism chooses x ( V ) s.t.: User r chooses V r to maximize:

10. IIVCG mechanisms Moral: truthful declaration is a dominant strategy Problem: Strategy spaces are overly complex Main insight (for Nash implementation): Suffices to elicit only local derivative of utility function

11. IIISSVCG 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. IIISSVCG mechanisms Mechanism chooses x ( ) s.t.: Payment to user r :

13. IIISSVCG: Key lemma is a Nash equilibrium if and only if for all r : Proof idea: If x * is optimal, user r can choose r s.t. U r ( x r * ) = U ( x r * ; r )

14. IIISSVCG: Key lemma is a Nash equilibrium if and only if for all r : Proof idea: If x * is optimal, user r can choose r s.t. U r ( x r * ) = U ( x r * ; r ) trtr

15. IIISSVCG: Efficient NE Corollary: Efficient Nash equilibrium exists Proof idea: Given efficient x *, each user r chooses r s.t. U r ( x r * ) = U ( x r * ; r ) ) Local truthful declaration

16. IIISSVCG: 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. IIISSVCG: Efficient NE Given: NE Define: P = { s : x s ( ) > 0 } J = { j : g j ( x ( )) = 0 } d ( r ) = ( g j / x r, j 2 J ) Theorem: If for all r, d ( r ) is linearly dependent on d ( s ), s r, s 2 P, then x ( ) is efficient

18. IIISSVCG: Efficient NE We know: x ( ) = max x 2 X r U ( x r ; r ) For all r : x ( ) 2 max x 2 X U r ( x r ) + s r U ( x s ; s ) First order conditions + linear dependence assumption ) U r ( x r ( )) = U ( x r ( ) ; r )

19. IVNetworks J links Capacity of link j : C j User r $ subset of links X = { x ¸ 0 : r : j 2 r x r · C j, for all j } Assume: For all j, two users r 1 ( j ), r 2 ( j ), s.t. U r i ( j ) (0) = 1 and r 1 ( j ) = r 2 ( j ) = { j } Then all NE allocations are efficient

20. VExtensions If U r depends on k -dimensional x r : Need k -dimensional r Designing h r ( - r ) is similar to VCG: Budget balance, etc.

21. VRelated work Yang & Hajek (2004), Maheswaran & Basar (2004): Single resource, capacity = 1 User r chooses bid r Allocation: x r ( ) = r / s s User r pays: t r ( ) = - r ( s s ) Same as: SSVCG where U ( x ; ) = - ( / x )

22. VRelated 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. VRelated work Semret (1999) Groves and Ledyard (1979) Yang and Hajek (2005) independent discovery of similar result

24. IIISSVCG: Efficient NE But: all NE are not efficient! Example: Two users, U 1 ( 1 ) = 2, U 2 ( 1 ) = 3 X = { ( x 1, x 2 ) : 5 x 1 + x 2 · 6, x 1 + x 2 · 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