Communication requirements of VCG-like mechanisms in convex environments

Ramesh JohariStanford University

Joint work with John N. Tsitsiklis, MIT

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

Outline

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

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

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:

II Achieving efficiency

• In general: utilities are unknown

• Design payments tr to alignefficiency and incentives

• Planner wants to maximize:

• User r wants to maximize:

II VCG mechanisms

• Strategy of user r:declared utility Vr

• Mechanism chooses x(V) s.t.:

• Payment to user r:

II VCG mechanisms

• Strategy of user r:declared utility Vr

• Mechanism chooses x(V) s.t.:

• Payment to user r:

II VCG mechanisms

• Strategy of user r:declared utility Vr

• Mechanism chooses x(V) s.t.:

• User r chooses Vr to maximize:

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

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; ) =

III SSVCG mechanisms

• Mechanism chooses x() s.t.:

• Payment to user r:

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)

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

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

III SSVCG: Efficient NE

But: all NE are not efficient!Example:

Single resource of capacity C = 1User 1 bids huge U’(C ; 1)

All other users:Best response is to “give up”

) User 1 gets everything,regardless of true utilities

III SSVCG: Efficient NE

Given: NE Define: P = { s : xs() > 0 }

J = { j : gj(x()) = 0 }

d(r) = ( gj/xr, 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

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)

IV Networks

J linksCapacity of link j : Cj

User r $ subset of linksX = { 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

V Extensions

• If Ur depends on k-dimensional xr :

Need k-dimensional r

• Designing hr(-r) is similar to VCG:

Budget balance, etc.

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)

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

V Related work

• Semret (1999)• Groves and Ledyard (1979)• Yang and Hajek (2005)

• independent discovery of similar result

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