Classification Mechanism Design 2, Lecture Notes - Computer Science, Study notes of Interface between Computer Science and Economics

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Classic Mechanism Design (II)

David C. Parkes

Division of Engineering and Applied Science,

Harvard University

CS 286r–Spring 2002

Direct Revelation Mechanisms (DRM)

In a DRM, M = (Θ, g), the strategy space, S = Θ, and an

agent simply reports a type to the mechanism, with outcome

rule, g : Θ → O.

Def. [ incentive-compatible ] A DRM is (Bayes-)Nash

incentive-compatible if truth-revelation is a (Bayes-)Nash

equilibrium, i.e. s

∗

i

(θi) = θi, for all θ ∈ Θ.

Def. [ strategyproof ] A DRM is strategyproof if

truth-revelation is a dominant strategy eq., for all θ ∈ Θ.

Note: The SCF implemented by an incentive-compatible

DRM is precisely the outcome function, g(θ).

“g(θ) is truthfully implementable...”

Proof

Consider mechanism, M = (S, g), that implements SCF,

f (θ), in a dominant strategy equilibrium. In otherwords,

g(s

∗

(θ)) = f (θ), for all θ ∈ Θ, where s

∗

is a dominant

strategy eq.

Construct direct mechanism, M

′

= (Θ, f (θ)). By

contradicton, suppose:

∃θ

′

i

= θi s.t. ui(f (θ

′

i

, θ−i), θi) > ui(f (θi, θ−i), θi)

for some θ

′

i

= θi, some θ−i. But, because

f (θ) = g(s

∗

(θ)), this implies that

u i

(g(s

∗

i

(θ

′

i

), s

∗

−i

(θ −i

)), θ i

) > u i

(g(s

∗

(θ i

), s

∗

−i

(θ −i

)), θ i

which contradicts the strategyproofness of s

∗

in

mechanism, M.

Theoretical Implications

• Focus goals. If M is the only DRM that implements

outcome function k(θ) with properties P then any

mechanism must implement the same transfers as M.

• Impossibility. If no DRM, M, can implement SCF,

f (θ), with properties P, then no mechanism can

implement SCF f (θ).

A modeler can limit the search for an optimal mechanism to

the class of direct, IC mechanisms. Useful, because the

number of mechanisms is huge.

Practical Implications?

• Incentive-compatibility is “free”
  • any outcome implemented by mechanism, M, can be

implemented by incentive-compatible mechanism, M

′ .

• “Fancy” mechanisms are unneccessary
  • any outcome implemented by a mechanism with

complex strategy space, S, can be implemented by a

DRM.

But, few procedures in practical use are direct & IC, perhaps

their are some unmodeled costs, computational problems?

Gibbard-Satterthwaite Impossibility

[Arrow 51, Gibbard & Satterthwaite 73, 75]

Consider SCF, f (θ), and an outcome space O. Let

Rf ⊆ O denote the range of f , i.e.

Rf = {o ∈ O : ∃θ ∈ Θ s.t. o = f (θ)}.

Let o

∗

i

∈ O denote the outcome that maximizes the value,

u

i

(o, θ

i

), over o ∈ R

f

.

Def. [Dictatorial] SCF f (θ) is dictatorial if there is an

agent, h, s.t. f (θ) = o

∗

h

, for all θ.

[Gibbard-Satterthwaite Impossibility] Suppose that the

types include all possible strict orderings over O. A SCF,

f (θ), with |R f

| > 2 , is implementable in dominant

strategies (strategyproof) if and only if it is dictatorial.

Implications: collaborative filtering (Pennock et al.), web

query aggregation (Kumar et al.), voting systems (Cranor).

Introducing Transfers (Side-payments)

Define the outcome space, O = K × R

N

, such that an

outcome rule, o = (k, t 1 ,... , tN ), defines a choice ,

k(s) ∈ K, and a transfer, t i

(s) ∈ R from agent i to the

mechanism, given strategy profile s ∈ S.

Assume quasilinear preferences,

ui(o, θi) = vi(k, θi) − ti

, with valuation function , vi(k, θi) for agent i.

General/No-transfer ⊃ Quasi-linear/Transfer

−→

easier

Budget Balance

Introduce constraints over the total transfers made from

agents to the mechanism. Let s

∗

(θ) denote the equilibrium

strategy of a mechanism.

• weak BB (or feasible )

· ex post:

i

ti(s

∗

(θ)) ≥ 0 , for all θ

· ex ante: Eθ∈Θ

[∑

i

ti(s

∗

(θ))

]

• strong BB

· ex post:

i

ti(s

∗

(θ)) = 0, for all θ

· ex ante: Eθ∈Θ

[∑

i

ti(s

∗

(θ)) = 0

]

ex ante weak ⊃ ex post weak

ex ante strong ⊃ ex post strong

−→

harder

             y

harder

Groves Mechanisms

[Groves 73] Drop: budget-balance.

Def. A Groves mechanism, M = (Θ, k, t 1 ,... , tN ) is

defined with choice rule ,

k

∗

(

θ) = arg max

k∈K

i

vi(k,

θi)

, and transfer rules

ti(

θ) = hi(

θ−i) −

j 6 =i

vj (k

∗

(

θ),

θj )

where hi(·) is an (arbitrary) function that does not depend

on the reported type,

θi, of agent i.

Thm. [Groves 73] Groves mechanisms are strategyproof

and efficient.

Proof. Agent i’s utility for strategy

i

, given

−i

from agents

j 6 = i, is:

ui(

θi) = vi(k

∗

(

θ), θi) − ti(

θ)

= vi(k

∗

(

θ), θi) +

j 6 =i

vj (k

∗

(

θ),

θj ) − hi(

θ−i)

Ignore h i

θ −i

), and notice

k

∗

(

θ) = arg max

k∈K

i

vi(k,

θi)

⇒Strategyproofness, and efficiency immediately follow.

In fact, [Green&Laffont 77], Groves mechanisms are unique ,

in the sense that any mechanism that implements efficient

choice, k

∗

(θ), in truthful dominant strategy must implement

Groves transfers.