# [EM] Quantifying manipulability

Forest Simmons fsimmons at pcc.edu
Wed Dec 4 18:44:47 PST 2002

```Here are some preliminary considerations that come to mind in the course
of trying to quantify manipulability.

The kind of manipulability that is most crucial (in my opinion) has to do
with the sensitivity of (near optimal) strategy to variations in
information.  To the degree a method is sensitive to such variations, to
the same degree an election held under that method can be manipulated by
the people that manufacture and disseminate misleading information.

This kind of manipulability is compounded when the method is
non-monotonic.  Imagine a shower in which the water temperature is very
sensitive to the faucet setting, and that to make matters worse, sometimes
a clockwise adjustment corresponds to increasing temperature, and
sometimes to decreasing temperature.

For practical purposes, perhaps we should incorporate the non-monotonicity
into the manipulability definition.

Here's a possible rough outline:

Let V be a set of voter utilities for the candidates.

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

Let E(V,S,M) represent the expected utility of the winner when the voter
utility set is V, the ballot set is S, and the method is M.

Then while holding the method M constant, consider the function

G(V1,V2) = E(V1,F(V2,M),M)

This represents the expected utility of the winner when the true utility
set is V1, while the pundits and pollsters have convinced the electorate
that it is V2.

Note that if V2=V1, then G(V1,V2) is the expected utility of the winner
under perfect information V1, while in the case of V2 not equal to V1,
G(V1,V2) is the expected utility under distorted information V2.

If we generate random V1 and V2 and compare the magnitude of the
difference |G(V1,V1)-G(V1,V2)| with the distance from V1 to V2 using some
appropriate norm |V1-V2|, we get one sample of the sensitivity of output
utility to a variation in information.

Repeated sampling reveals how much damage we can expect on average from
the effects of misinformation.

Specifically, we should average something like

log(|G(V1,V1)-G(V1,V2)|/|V1-V2|)

over sampled values of V1 and V2.

Well this is pretty rough, but I hope it will help get the ball rolling
towards a useful and relevant quantification of manipulability.

Forest

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