Computational Mechanism Design Chapter 3

Truth revelation is a dominant strategy within a

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Unformatted text preview: d by manipulating its inputs to the algorithm to x" the approximation. Truth-revelation 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 strategy-proofness within a Groves mechanism. Tennenholtz et al. TKDM00 introduce a set of su cient but not necessary axioms for an approximation algorithm to retain strategy-proofness. The most important axiom essentially introduces the following requirement which the authors also refer to as 1e ciency": ^^ vi k i; ,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 . Strategy-proofness 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 worst-case 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 maximal-in-range |for strategy-proofness with approximate winner-determination algorithms. The conditions are necessary and su cient. The maximal-in-range 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, strategy-proofness 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 strategy-proofness, and claim that all truthful mechanisms with approximate algorithms have unreasonable" behavior, for an appropriate de nition of unreasonableness. Lehmann et al. LOS99 consider strategy-proof and approximate implementations for a special case of the combinatorial allocation problem, with single-minded 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.

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