Transcript Slides.

(More on) characterizing
dominant-strategy implementation
in quasilinear environments
(see, e.g., Nisan’s review: Chapter 9 of Algorithmic Game Theory book)
Tuomas Sandholm
Professor
Computer Science Department
Carnegie Mellon University
Some characterization results (see Nisan’s review chapter)
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Prop. A mechanism is incentive compatible iff
– Agent i’s payment does not depend on his reported vi, but only on the alternative chosen, and
– The mechanism picks an outcome (within its range) that optimizes for each player:
f in argmaxo{ vi(o) – pi(o) }
Can also characterize in the space of social choice functions only:
Def. f satisfies Weak Monotonicity (WMON) if
f(vi ,v-i) = a ≠ b = f(v’i ,v-i)
=>
v’i(b) - vi(b) ≥ v’i(a) - vi(a)
– In words: if social choice changes when a single agent changes his valuation, then it must be
because the agent increased his value of the new choice relative to his value of the old choice.
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Thm. If a mechanism is incentive compatible, then f satisfies WMON. If domains of
preferences Vi are convex sets, then for every f that satisfies (even just local) WMON,
there exists a payment rule such that the mechanism is incentive compatible.
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First part is easy to prove, see page 227 of Algorithmic Game Theory book
Second part holds if
• outcome space is finite [Saks and Yu EC-05], or
• loop integral of f is zero around every sufficiently small triangle [Archer&Kleinberg EC-08]
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They also show that the theorem applies to non-convex Vi by studying functions f that apply to the convex hull
They also show how a truthful f can be stitched together from locally truthful fi’s
Somewhat unsatisfactory: it is not clear exactly what the WMON functions are. WMON
is a local condition. Is there a global condition? Yes for unrestricted or very restricted
Vi. Largely open for practical problems that lie in between.
Unrestricted Vi and affine maximizers
• Affine maximizers are a generalization of Groves
mechanisms
• f in argmaxo{ co + i wi vi(o) }
• Prop. If the payment for agent i is
hi(v-i) - j≠i (wj/wi) vj(o) – co/wi,
then the mechanism is incentive compatible
• Thm (Roberts 1979). If |O| ≥ 3, f is onto O, Vi = RO for
every i, and the mechanism is incentive compatible,
then f is an affine maximizer
Single-parameter domains
• Setting:
– Vi is one-dimensional, i.e., Vi in R
– For each agent, there is a set of equally-preferred
“winning” outcomes and equally preferred “losing”
outcomes
– Assume “normalized”, that is, losing agents pay 0
• Thm. Mechanism is incentive compatible iff
– f is monotone in every vi, and
– every winning agent pays his critical value
(Essentially) uniqueness of prices
• Thm.
– Assume the domains of Vi are connected sets (in
the usual metric in Euclidean space)
– Let (f, p1,…pn) be an incentive compatible
mechanism
– The mechanism (f, p’1,…p’n) is incentive
compatible iff p’i(v1,…vn) = pi (v1,…vn) + hi (v-i)
Network interpretation of incentive
compatibility constraints
• See, e.g., the overview article by Rakesh
Vohra that is posted on the course web page
• Similar approach also available for BayesNash implementation [Weak monotonicity
and Bayes–Nash incentive compatibility
Games and Economic Behavior, Volume 61,
Issue 2, November 2007, Pages 344-358
Rudolf Müller, Andrés Perea, Sascha Wolf]