Transcript Mechanism design Goal of mechanism design Implementing a social choice function f(u1, …, u|A|) using a game Center = “auctioneer” does not.

Mechanism design
Goal of mechanism design
Implementing a social choice function f(u1, …, u|A|) using a game
Center = “auctioneer” does not know the agents’ preferences
Agents may lie
Goal is to design the rules of the game (aka mechanism) so that in
equilibrium (s1, …, s|A|), the outcome of the game is f(u1, …, u|A|)
• Mechanism designer specifies the strategy sets Si and how outcome is
determined as a function of (s1, …, s|A|)  (S1, …, S|A|)
• Variants
•
•
•
•
– Strongest: There exists exactly one equilibrium. Its outcome is f(u1, …, u|A|)
– Medium: In every equilibrium the outcome is f(u1, …, u|A|)
– Weakest: In at least one equilibrium the outcome is f(u1, …, u|A|)
Revelation principle
• Any outcome that can be supported in Nash (dominant
strategy) equilibrium via a complex “indirect” mechanism
can be supported in Nash (dominant strategy) equilibrium
via a “direct” mechanism where agents reveal their types
truthfully in a single step
Constructed “direct revelation” mechanism
Agent 1’s
Strategy
preferences
formulator
..
.
Agent |A|’s
Strategy
preferences
formulator
Strategy
Strategy
Original
“complex”
“indirect”
mechanism
Outcome
Uses of the revelation principle
• Literal: “Only direct mechanisms needed”
– Problems:
• Strategy formulator might be complex
– Complex to determine and/or execute best-response strategy
– Computational burden is pushed on the center (assumed away)
– Thus the revelation principle might not hold in practice if these
computational problems are hard
– This problem traditionally ignored in game theory
• Even if the indirect mechanism has a unique equilibrium, the
direct mechanism can have additional bad equilibria
• As an analysis tool
– Best direct mechanism gives tight upper bound on how
well any indirect mechanism can do
• Space of direct mechanisms is smaller than that of indirect ones
• One can analyze all direct mechanisms & pick best one
• Thus one can know when one has designed an optimal indirect
mechanism (when it is as good as the best direct one)
Implementation in dominant
strategies
Strongest form of mechanism design
Implementation in dominant strategies
• Goal is to design the rules of the game (aka mechanism)
so that in dominant strategy equilibrium (s1, …, s|A|),
the outcome of the game is f(u1, …, u|A|)
• Nice in that agents cannot benefit from
counterspeculating each other
–
–
–
–
Others’ preferences
Others’ rationality
Other’s endowments
Other’s capabilities …
Gibbard-Satterthwaite impossibility
•
Thrm. If |O | ≥ 3 (and each outcome would be the social choice under f for some
input profile (u1, …, u|A|) ) and f is implementable in dominant strategies, then f is
dictatorial
General preferences
Quasilinear preferences
Special case where dominant strategy implementation is
possible: Quasilinear preferences -> Clarke tax mechanism
• Outcome (x1, x2, ..., xk, m1, m2, ..., m|A| )
• Quasilinear preferences: ui(x, m) = mi + vi(x1, x2, ..., xk)
• Utilitarian setting: Social welfare maximizing choice
– Outcome s(v1, v2, ..., v|A|) = maxx i vi(x1, x2, ..., xk)
• Agent’s payment mi = ji vj(s(v)) - ji vj(s(v-i))  0 is a “tax”
• Thrm: Every agent’s dominant strategy is to reveal preferences truthfully
– Intuition: Agent internalizes the negative externality he imposes on others by
affecting the outcome
• Agent pays nothing if he does not change the outcome
• Example: k=1, x1=”joint pool built” or “not”, mi = $
– E.g. equal sharing of construction cost: -c / |A|
General preferences
Pool
Quasilinear preferences
Pool
ui =10
No pool
$0
ui =5
u i =10
No pool
$0
ui =5
Clarke tax mechanism…
• Pros
– Social welfare maximizing outcome
– Truth-telling is a dominant strategy
– Feasible in that it does not need a benefactor (i mi  0)
• Cons
– Budget balance not maintained (in pool example, generally i mi < 0)
• Have to burn the excess money that is collected
• Thrm. [Green & Laffont 1979]. Let the agents have arbitrary quasilinear
preferences. No social choice function that is (ex post) welfare maximizing
(taking into account money burning as a loss) is implementable in
dominant strategies
• If there is some party that has no private information to reveal and no
preferences over x, welfare maximization and budget balance can be
obtained by having that party’s payment be m0 = - i=1.. mi
– Auctioneer could be called “agent 0”
– Vulnerable to collusion
• Even by coalitions of just 2 agents
Another approach for circumventing
the impossibility of dominantstrategy implementation
• Design the game so that (although manipulations
exist), finding a beneficial manipulation is
computationally so complex for an agent that the
agent cannot do that
– E.g. “Complexity of Manipulating Elections with Few
Candidates” [Conitzer & Sandholm AAAI-02, TARK-03]
– E.g. “Universal Voting Protocol Tweaks for Making
Manipulation Hard” [Conitzer & Sandholm IJCAI-03]
Yet another approach for
circumventing the impossibility of
dominant-strategy implementation
• Designing the mechanism automatically to the situation at hand [Conitzer & Sandholm]
– Input is the probabilistic information that the center has about the agents
– Output is an optimal mechanism where the agents are motivated to reveal their
preferences truthfully, and a social objective is satisfied to the optimal extent
– Advantages:
• Can be used even without side payments & quasilinear preferences
• Could achieve better outcomes than Clarke tax mechanism
General preferences
• Circumvents impossibility in many cases
– “Complexity of Mechanism Design”
• Designing a deterministic mechanism is NP-complete
• Designing a randomized mechanism is fast
– No loss in social objective, sometime a gain
• Both results also hold for Bayes-Nash implementation
– E.g., metal manufacturers with asymmetric production costs
Quasilinear prefs