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Unformatted text preview: d by manipulating its inputs to the algorithm to x" the approximation. Truthrevelation is a dominant strategy within a
Groves mechanism if and only if an agent cannot improve on the outcome computed by
the mechanism's algorithm by misrepresenting its own preferences. This observation leads
to useful characterizations of necessary properties for an approximation algorithm to retain
strategyproofness within a Groves mechanism.
Tennenholtz et al. TKDM00 introduce a set of su cient but not necessary axioms
for an approximation algorithm to retain strategyproofness. The most important axiom
essentially introduces the following requirement which the authors also refer to as 1e ciency":
^^
vi ki; ,i; i + X v k ; ^
^
j 6=i j ^
^^ ^
i ,i ; j vi k i ; ,i ; i + X v k^ ; ^
^
j 6=i j ^
i ,i ; j ; ^
^
for all i 6= i ; ,i ; i .
In words, an agent cannot improve the solution with respect to a particular set of inputs
^
i ; ,i by unilaterally misrepresenting its own input i . Strategyproofness follows quite
naturally from this condition, given that a rational agent will only misrepresent its preferences to improve the quality of the solution for reported preferences from other agents and
the agent's true preferences computed by the mechanism. An interesting open question
is the degree to which these axioms restrict the e ciency of an approximation algorithm,
for a particular class of algorithms e.g. constant factor worstcase approximations, etc..
Nisan & Ronen NR00 take a di erent approach and de ne conditions on the range
of an approximation algorithm, and require the algorithm to be optimal in its range
a condition they refer to as maximalinrange for strategyproofness with approximate
winnerdetermination algorithms. The conditions are necessary and su cient.
The maximalinrange condition states that if K0 K is the range of outcomes selected
by the algorithm i.e. k 2 K0 implies there is some set of agent preferences for which the
^
approximation algorithm k = k, then the approximation algorithm must compute the
68 best outcome in this restricted range for all inputs.
^
k = max
k2K
0 X v k;
i2I i i for all 2 , and for some xed K0 K.
Intuitively, strategyproofness follows because the Groves mechanism with this rule
implements a Groves mechanism in the reduced space of outcomes K0 . An agent cannot
improve the outcome of the allocation rule by submitting a corrupted input because there
is no reachable outcome of better quality. Nisan & Ronen partially characterize the necessary ine ciency due to the dual requirements of approximation and strategyproofness,
and claim that all truthful mechanisms with approximate algorithms have unreasonable"
behavior, for an appropriate de nition of unreasonableness.
Lehmann et al. LOS99 consider strategyproof and approximate implementations for
a special case of the combinatorial allocation problem, with singleminded bidders that
care only about one bundle of items. Perhaps surprisi...
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This document was uploaded on 03/19/2014 for the course COMP CS286r at Harvard.
 Fall '13
 DavidParkes

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