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Parkes Mechanism Design 1
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Mechanism Design I
David C. ParkesSchool of Engineering and Applied Science,
Harvard University
CS 286r–Spring 2007
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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.
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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?
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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).
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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).
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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
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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
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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.”
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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.
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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.
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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 .
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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!
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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.
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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
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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
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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.
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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)
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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 .
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[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 !
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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)
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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!
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