[EM] Quantifying manipulability

[Alex Small]

Forest Simmons said:
> Let F(V,M) represent the set of voter ballots that are optimal for the
> voters with utility set V under method M.

For whatever it's worth, I don't think F(V,M) is in general unique.

F(V,M) must be a Nash equilibrium.  If any voter can get a better outcome
by voting a different way then it is not optimal for him.

Now, depending on V there may be more than one Nash equilibrium.  There
must be some V for which there is more than one Nash equilibrium. 
Otherwise I could build the machine that I talked about several months
ago, assign every voter his single optimal strategy given everybody else's
assigned strategies, and pick the winner that way.  The voters would have
no incentive to falsify their preferences, but that would violate the
Gibbard-Satterthwaite Theorem.

So, there must be some V for which there is more than one Nash
equilibrium.  The machine is then impossible to build because we must now
find a procedure for selecting among the equilibrium outcomes, which gets
us back to our original problem.

Anyway, I don't know how important this is, but the non-uniqueness of
(V,M) may be relevant.



Alex


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