[EM] Quantifying manipulability

Alex Small asmall at physics.ucsb.edu
Wed Dec 4 13:27:01 PST 2002

```barnes99 said:
> The Borda Count will  punish an insincere vote in some cases, and that is
> actually an incentive to  vote sincerely. If too many people sincerely
> put Nader between Bush and Gore,  and voted either Bush>Nader>Gore or
> some information to successfully manipulate the  BC,

True.  It would be interesting to find out if there are any general
results on Nash equilibria and the Borda Count.  In that 3-candidate
example, if we assume that most people rank Nader last, there is a
disincentive for too many voters to falsely vote Nader in the middle.

Here's the problem as I see it:  In most ranked methods, reversing the
rankings of #2 and #3 only hurts #2 (presumably to help #1 at #2's
expense).  In Borda, reversing the ranking of #2 and #3 hurts #2 but also
hurts #1 to some extent.  Granted, this isn't a quantitative statement.

> As a start, maybe we could quantify the number of voters who could
> successfully manipulate the outcome in a particular example. Here's a
> simple  example to kick off the game:
>
> 1 ABC
> 1 ACB
> 1 CAB
> 1 CBA
> 1 BCA
> 1 BAC

This case isn't really susceptible to analysis because ANY reasonable
ranked method will give a 3-way tie, leaving the outcome in the hands of
Justice Scalia ;)  Even if we perturb the numbers a little to break the
tie, so that each category has between 9 and 11 voters, say, any
reasonable method will be very unstable:  It will be very easy to change
the outcome with a strategic adjustment because you're close to a 3-way
tie.

It's usually better to analyze strategic voting with specific criteria
(e.g. montonicity, FBC, etc.).  Individual criteria can usually be
analyzed in cases that aren't near-ties.  Maybe we need to invent more
criteria to understand the manipulability of Borda, or maybe those
commonly discussed on the list are sufficient.  In any case, we need
criteria to move this discussion forward.

Alex

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