New paper: average-case manipulation of elections

The classic Gibbard-Satterthwaite theorem states that every nontrivial voting method can be manipulated.   In the late 1980’s an approach to circumvent this impossibility was suggested  by Bartholdi, Tovey, and Trick: maybe a voting method can be found where the manipulation is computationally hard — and thus effective manipulation would be practically impossible?  Indeed voting methods whose manipulation problem was NP-complete were found by them as well as later works.  However, this result is not really satisfying: the NP-completeness only ensures that the manipulation cannot be solved in the worst case; it is quite possible that for most instances manipulation is easy, and thus the voting method can still be effectively manipulated.   What would be desired is a voting method where manipulation would be computationally hard everywhere, except for a negligible fraction of inputs.  In the last few years several researchers have indeed attempted “amplifying” the NP-hardness of manipulation in a way that may get closer to this goal.

The new paper of Marcus Isaksson, Guy Kindler, and Elchanan Mossel titled The Geometry of Manipulation – a Quantitative Proof of the Gibbard-Satterthwaite Theorem shatters these hopes.  Solving the main open problem in a paper by Ehud Freidgut, Gil Kalai, and myself, they show that every nontrivial neutral voting method can be manipulated on a non-negligible fraction of inputs.  Moreover, a random flip of the order of 4 (four) alternatives will be such a manipulation, again with non-negligible probability.  This provides a quantitative version of the Gibbard-Satterthwaite theorem,  just like Gil Kalai previously obtained a quantitative version of Arrow’s theorem.

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