Classification Mechanism Design 3, Lecture Notes - Computer Science, Study notes of Interface between Computer Science and Economics

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Classic Mechanism Design (III)

David C. Parkes

Division of Engineering and Applied Science,

Harvard University

CS 286r–Spring 2002

Vickrey-Clarke-Groves Mechanism

[Vickrey61, Clarke71, Groves73] (VCG or “Pivotal” mechanism.)

Def. [VCG mechanism] Implement efficient outcome,

k

∗

= maxk

j

vj (k,

θj ), and compute transfers

ti(

θ) =

j 6 =i

vj (k

−i

,

θj ) −

j 6 =i

vj (k

∗

,

θj )

where k

−i

= maxk

j 6 =i

vj (k,

θj ).

Thm. The VCG mechanism is strategyproof, efficient, and

interim IR.

Alternative description:

pvick,i(θ) = vi(k

∗

, θi) − [V (I) − V (I \ i)]

where V (K) is the value of the efficient allocation in the

subproblem restricted to agents K ⊆ I.

Example: Shortest Path.

[Nisan 99]

S

T

Biconnected graph, G = (N, E), cost c

l

≥ 0 per edge

l ∈ E, edges srategic. Assume large value V to send

message.

Goal : route packets along the lowest-cost path from S to T.

VCG Payment edge e:

pvick,l = −cl −

[

(V − dG) − (V − d G/l

]

= −c l

− (d G/l

− dG)

The Centrality of VCG

[Krishna & Perry 98]

Thm. Among all efficient and interim IR mechanisms, the

VCG maximizes the expected transfers from agents.

Note: this is interesting, shows that the best mechanism

amongst all (Bayes-Nash), etc. is dominant strategy.

Thm. Given preferences, Θ, there exists a (weak) BB and

efficient mechanism, with interim IR, if and only if the VCG

has positive expected surplus.

... leads to quite direct proofs of Myerson-Satterthwaite,

other negative results.

Expected Externality Mechanism

[Arrow79,d’Aspremont&Gerard-Varet79] Retain Bayes-Nash, and

relax interim IR to ex ante IR, and try to achieve BB.

The d’AGVA mechanism (or expected-Groves mechanism),

uses the same allocation as the Groves, but computes an

transfer term averaged across all possible types of agents.

[P.55, Parkes-Diss]

Thm. The d’AGVA mechanism is efficient, ex post

budget-balanced, but only ex ante IR.

Demonstrates: (wrt Eff. mech. des.):

(a) ex ante IR really does make mechanism design “easier” than

interim IR (compare Myerson-Satterthwaite with d’AGVA)

(b) Bayes-Nash implementation really does make mechanism

design “easier” than dominant-strategy equilibrium (compare

Green-Laffont impossibility with d’AGVA).

Cost-Sharing Problems

Choice set K, N buyers, 1 seller. Transfers t 1 ,... , tN and

t

s

. Values vi(k) ≥ 0 for buyers, and cost cs(k) ≥ 0 for

seller. Quasi-linear utility functions, ui(k, ti) = vi(k) − ti,

and us(k, ti) = −cs(k) − ti.

Example: Multi-cast cost sharing.

Welfare=40+15-(20+10)=

20 30 10 15 40

20 20

5

10 5

5

10

15

Desirable Properties

[Assume the seller is truthful.] Compute outcome k

∗

(θ) and

transfers ti(θ), ts(θ).

Use revelation principle , focus on IC mechanisms.

• EFF. Select maxk

i

vi(k

∗

(θ), θi) − cs(k

∗

(θ)), for all

θ ∈ Θ.

• BB. Transfers

i

ti(θ) + ts(θ) = 0, for all θ ∈ Θ.

• No-profit. Transfers −cs(k

∗

(θ)) − ts(θ) = 0, for all

θ ∈ Θ.

• Buyer-SP. Satisfy: vi(k

∗

i

(θi, θ−i), θi) − ti(θi, θ−i) ≥

vi(k

∗

i

θi, θ−i), θi) − ti(

θi, θ−i) for all

θi 6 = θi, θi, and

θ−i.

• IR. Satisfy: vi(k

∗

i

(θi, θ−i), θi) − ti(θi, θ−i) ≥ 0 , for all

θ ∈ Θ.

Group Strategyproofness

[this comes for “free” when we worry about BB]

• Buyer-GSP. For all coalitions S ⊆ I,

θS 6 = θS , θS , θ−S ,

either ui(

θS , θ−S , θi) ≤ ui(θS , θ−S , θi), ∀i ∈ S, or

∃i ∈ S s.t. ui(

θS , θ−S , θi) < ui(θS , θ−S , θi).

No coalition of agents can manipulate the outcome of the

mechanism without making one of the agents in the coalition

worse off.

Simplifying: A Binary Choice Model

[Moulin&Shenker 99]

Suppose I agents, either receive the service or not (binary

choice). Let R ⊆ I denote the receiver set. Define C(R) as

the cost of providing service to R agents.

Eff: R(θ) = arg maxR

i∈R

vi − C(R), ∀θ

BB,No-Profit:

i∈R(θ)

ti(θ) = C(R(θ)), ∀θ.

GSP, IP.

Def. Mechanism M = (Θ, R, ti) satisfies the core property

if and only if

i∈Q

t i

(θ) ≤ C(Q), ∀Q ⊆ I

i.e., there is no incentive for a subset of agents to break from

the grand coalition.

Cross-monotonic Cost Sharing

Def. Cost sharing method, ξ(Q, i) is cross-monotonic if and

only if

ξ(Q, i) ≥ ξ(R, i), ∀Q ⊆ R

Note: this is also weak cross-monotonic, satisfying:

i∈Q

ξ(Q, i) ≥

i∈Q

ξ(R, i), ∀Q ⊆ R

Prop. A mechanism that implements transfers,

ti = ξ(R

∗

, i), for some weak cross-mononotic, ξ(·),

implements a core allocation for each subset of users.

Coalitional StrategyProof Cost-Sharing

Mechanisms

[Moulin& Shenker, 99]

Given cross-monotonic cost-sharing method, ξ(Q, i),

mechanism M(ξ) computes the receiver set R

∗

and

transfers ti = ξ(R

∗

, i) as follows:

Def. Mechanism M(ξ):

Agents report values, ˆv; initialize R

∗

← I.

Select an agent i ∈ R

∗

at random, if vˆi < ξ(R

∗

, i) then

drop i from R

∗

.

Continue until vˆi ≥ ξ(R

∗

, i) for all i ∈ R

∗

.

Implement R

∗

and transfers ξ(R

∗

, i).

Thm. Given cross-monotonic, ξ(Q, i), then M(ξ) is BB and

GSP.

Example: Shapley Mechanism

Jain & Vazirani : assume a general biconnected network, propose a

centralized approximation mechanism; not submodular and

Shapely does not apply.

Additional Implementation Concepts

• Repeated implementation : can begin to implement

more [Kalai 97]

• if the planner learns and is more patient than the

agents, and agents in a multi-round game, then can

achieve dom. strategy implementation (in limit if center

has no time discounting)

• reduce to a one-shot revelation game
• Large societies :
• can get approx. EFF and approx. balance in large

double auctions [McAfee92, Satterthwaite&Williams89, Rustichini

et al.95]