mechanism design i - harvard university design i david c. parkes school of engineering and applied...

Parkes Mechanism Design 1

'

&

$

%

Mechanism Design I

David C. ParkesSchool of Engineering and Applied Science,

Harvard University

CS 286r–Spring 2007


'

&

$

%

Mechanism Design

Central question: what social choice functions can be

implemented in distributed systems with private information

and rational agents?

– impose incentive based constraints

Note: Mechanism design assumes unlimited computation

and communication. Key concept is that of a “rational” agent.


'

&

$

%

Central Ideas

• Design criteria

– participation constraints

– incentive-compatibility constraints

– budget-balance constraints

• Revelation principle

• Impossibility and possibility results

– what are the limits that the “information problem” and

rationality place on implementation?


'

&

$

%

Basic Definitions

• Set of possible outcomes O

• Agents i ∈ N , with preference types θi ∈ Θi.

• Utility, ui(o, θi), over outcome o ∈ O

• Mechanism M = (S, g) defines a strategy space

S = S1 × . . . × Sn, and an outcome function

g : SN → O.

• Agent i plays strategy si(θi) ∈ Si, given type θi, and

the mechanism implements outcome

g(s1(θ1), . . . , sn(θn)).

• Defines a game: Utility to agent i from strategy profile s,

is ui(g(s(θ)), θi), which we write as shorthand

ui(s, θi).


'

&

$

%

Classic Implementation Concept

Mechanism M = (S, g) implements social choice function

f : Θ → O if:

g(s∗1(θ1), . . . , s∗n(θn)) = f(θ), ∀θ ∈ Θ

for an equilibrium strategy (s∗1, . . . , s∗n).


'

&

$

%

Equilibrium Concepts

Mechanism M = (S, g) implements f in eq. if, for all

θ ∈ Θ, g(s∗(θ)) = f(θ), where s∗ is a eq.

• Bayesian-Nash: (need a common prior F (θ))

Eθ−i

[ui(s∗i (θi), s

∗−i(θ−i), θi)] ≥ Eθ

−i[ui(s

′i(θi), s

∗−i(θ−i), θi)],

∀i, ∀θi, ∀s′i 6= s

∗i

• Dominant strategy:

ui(s∗i (θi), s−i(θ−i), θi) ≥ ui(s

′i(θi),s−i(θ−i), θi),

∀i,∀θ, ∀s′i 6= s

∗i , ∀s−i

Bayes-Nash ⊃ Dominant

−→

harder


'

&

$

%

Mechanism Desiderata

• Allocative Efficiency

– select the outcome that maximizes total utility

• Fairness

– select the outcome that minimizes the variance in

utility

• Revenue maximization

– select the outcome that maximizes revenue to a seller

(or more generally, utility to one of the agents)

• Budget-balanced

– implement outcomes that have balanced transfers

across agents

• Pareto Optimal


'

&

$

%

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 BNE, i.e. for all

agent i and θi ∈ Θi,

Eθ−i

[ui(g(θi, θ−i), θi)] ≥ Eθ−i

[ui(g(θ̂i, θ−i), θi)], ∀θ̂i 6= θi

Def. [strategyproof ] A DRM is strategyproof (or truthful) if

truth-revelation is a DSE, i.e. for all agent i and θi ∈ Θi,

ui(g(θi, θ−i), θi) ≥ ui(g(θ̂i, θ−i), θi), ∀θ−i ∈ Θ−i, ∀θ̂i 6= θi

We say that (social choice function) g is “truthfully

implementable.”


'

&

$

%

The Revelation Principle

[Gibbard 73;Green & Laffont 77]

Thm. For any mechanism, M, there is a direct and

incentive-compatible mechanism with the same outcome.

“the computations that go on within the mind of any bidder in the

nondirect mechanism are shifted to become part of the mechanism

in the direct mechanism.” [McAfee&McMillan, 87]

Consider:

–the IC direct-revelation implementation of a first-price

sealed-bid auction

–the IC direct-revelation implementation of an English

(ascending-price) auction.


'

&

$

%

Proof

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

in a dominant strategy equilibrium. In otherwords,

g(s∗(θ)) = f(θ) for all θ ∈ Θ, where s∗ is a DSE.

Construct direct mechanism, M′ = (Θ, f). By

contradicton, suppose:

ui(f(θ′i, θ−i), θi) > ui(f(θi, θ−i), θi)

for some θ′i 6= θi, some θ−i. But, because

f(θ) = g(s∗(θ)), this implies that

ui(g(s∗i (θ′i), s

∗−i(θ−i)), θi) > ui(g(s∗(θi), s

∗−i(θ−i)), θi)

which contradicts the strategyproofness of s∗ in

mechanism, M.


'

&

$

%

Theoretical Implications

Putting computational constraints to one side, we can limit

the search for a useful mechanism to the class of direct and

incentive-compatible mechanisms.

• Focus goals. Formulate as an explicit optimization

problem; e.g. maximize expected payoff to the seller

across all DRM’s that satisfy IC and IR constraints

(Myerson’81).

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

then no (arbitrarily hairy) mechanism can implement

SCF f .


'

&

$

%

Practical Implications?

• Incentive-compatibility is “free” from an implementation

perspective (but NOT strategyproofness)

– any outcome implemented by mechanism, M, can be

implemented by incentive-compatible mechanism, M′.

• BUT BUT BUT , what about computation?

– few procedures in practical use are direct and

incentive-compatible, perhaps their are some

unmodeled costs, computational problems?

• Problems include : Shifts computation to center; Pref.

elicitation costs; Not transparent; Requires a center.

Return to these issues later in the course!


'

&

$

%

Single-Peaked Preferences

[Moulin 80]

Suppose O ⊆ R (i.e. choosing a point on the real line.)

Preferences are “single-peaked” if for every i there exists a

peak, p(θi) ∈ O, s.t. for any d, d′ ∈ O, s.t. d′ < d ≤ p(θi)

or p(θi) ≤ d < d′ then ui(d, θi) > ui(d′, θi).

Def. [median rule] Ask agents to declare their peaks, and

select the median peak.

Thm. This “median rule” mechanism is Pareto efficient and

strategy proof.


'

&

$

%

Introducing Payments

Define the outcome space, O = K × Rn, such that an

outcome rule, o = (k, t1, . . . , tn), defines a choice,

k(s) ∈ K, and a transfer, ti(s) ∈ R from agent i to the

mechanism, given strategy profile s ∈ S.

Assume quasilinear preferences,

ui(o, θi) = vi(k, θi) − ti

where vi(k, θi) is the valuation function of agent i.

General/No-transfer ⊃ Quasi-linear/Transfer

−→

easier


'

&

$

%

Individual Rationality Constraints

Suppose the mechanism is incentive compatible and the

agent has no “outside option” value. Roughly: should

choose to participate.

• ex ante individual-rationality

– agents choose to participate before they know their

own types; Eθ∈Θ [ui(f(θ), θi)] ≥ 0

• interim individual-rationality

– agents can withdraw once they know their own type;

Eθ−i∈Θ

−i[ui(f(θi, θ−i), θi)] ≥ 0

• ex post individual-rationality

– agents can withdraw from the mechanism at the end;

ui(f(θ), θi) ≥ 0 .

ex ante ⊃ interim ⊃ ex post

−→

harder


'

&

$

%

Groves Mechanisms

[Groves 73]

Def. A Groves mechanism, M = (Θ, (k, t)) is defined with

choice rule,

k∗(θ̂) = arg max

k∈K

X

i

vi(k, θ̂i)

and transfer rules

ti(θ̂) = hi(θ̂−i) −X

j 6=i

vj(k∗(θ̂), θ̂j)

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

the reported type, θ̂i, of agent i.

Thm. [Groves 73] Groves mechanisms are strategyproof

and efficient.


'

&

$

%

Proof. Agent i’s utility for strategy θ̂i, given θ̂−i from agents

j 6= i, is:

ui(θ̂i) = vi(k∗(θ̂), θi) − ti(θ̂)

= vi(k∗(θ̂), θi) +

X

j 6=i

vj(k∗(θ̂), θ̂j) − hi(θ̂−i)

Ignore hi(θ̂−i), and notice that choice

k∗(θ̂) = arg max

k∈K

X

i

vi(k, θ̂i)

i.e. it maximizes the sum of reported values, and therefore

the agent should announce θ̂i = θi to maximize its own

payoff.

Thm. Groves mechanisms are unique in the sense that any

mechanism that implements efficient choice in truthful

dominant strategy must collect Groves transfers. (Green &

Laffont’77)


'

&

$

%

Vickrey-Clarke-Groves Mechanism

[Vickrey61, Clarke71, Groves73]

Def. [VCG mechanism] Implement efficient outcome,

k∗ = maxk

P

j vj(k, θ̂j), and compute transfers

ti(θ̂) =X

j 6=i

vj(k−i

, θ̂j) −X

j 6=i

vj(k∗, θ̂j)

where k−i = maxk

P

j 6=i vj(k, θ̂j).

Thm. The VCG mechanism is strategyproof, efficient, and

ex post IR.

Alternative description:

pvick,i(θ) = vi(k∗, θi) − [V (N) − V (N \ i)]

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

subproblem restricted to agents L ⊆ N .


'

&

$

%

[Note 1:] given strategies θ̂−i, each agent’s adjusted

payment, vi(k∗, θ̂i) − [V (N) − V (N \ i)], sets

vi(k∗, θ̂i) − [V (N) − V (N \ i)] +

X

j 6=i

vj(k∗, θ̂j)

=V (N \ i)

i.e., this is the least value agent i could have bid and

maintained outcome k∗.

[Note 2:] Each agent’s equilibrium utility is:

πvick,i = vi(k∗, θi) − [vi(k

∗, θi) − V (N) + V (N \ i)]

= V (N) − V (N \ i)

i.e., equal to its marginal contribution to the welfare of the

system.

* potential for a budget-balance problem here !


'

&

$

%

Example: Shortest Path.

[Nisan 99]

40

S T

30

60 10

2050

2040

30

Biconnected graph, G = (N, E), cost cl ≥ 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 −ˆ

(V − dG) − (V − dG/l)˜

= −cl − (dG/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. VCG & reserve prices also maximizes revenue across

all mechanisms when there is a perfectly efficient

aftermarket. (Ausubel& Cramton’99)

This unification provides quite direct results of a number of

central negative results in the mechanism design literature.

Next time!

mechanism design i - harvard university design i david c. parkes school of engineering and applied...

Documents