# [EM] Some unfortunately too strong Defensive Strategy Criterion

Jobst Heitzig heitzig-j at web.de
Fri Apr 30 15:37:03 PDT 2004

```In the last days, I thought about some form of strategy-proofness like
the following:

Criterion:
Suppose that, with all voters voting sincerely, the method elects A, but
some voter prefers B to A and can get B elected by voting insincerely.
Then those voters not preferring B to A must have a way of voting which
ensures that A or some option C gets elected which the first voter ranks
*below* A, so that either the sincere result can be guaranteed or the
incentive to vote insincerely is removed.

However, I then came up with the following, very simple 3-by-3-example
which seems to render those thoughts ridiculous...

Problematic Example:

Sincere preferences
Voter 1: A>B>C
Voter 2: B>C=A
Voter 3: C>A>B

If voter 2 would have B>C>A instead, all three options would necessarily
get a winning probability of 1/3 if only the method satisfies
neutrality and anonymity. Hence in the sincere case where voter 2 has
C=A instead of C>A, A must get a winning probability larger than 1/3,
while C must get a winning probability below 1/3, if only the method
satisfies some weak version of monotonicity. Almost all methods would
elect A with probability 1 here, I guess. This means that voter 2 can
improve the chances of its favourite B by voting insincerely C>A instead
of sincerely C=A, a strategy which has been called "burying" here. Now
the only voter who does not profit from this strategy is voter 1 since
also C's (voter 3's favourite) chances have been improved. The only
B>C, thus producing a situation where C gets elected. However, this does
not remove voter 2's incentive to vote insincerely because C is no worse
to him than the original winner A...

What I think is noteworthy here is that this example seems to affect
*all* acceptable methods whatsoever (more precisely, those which satisfy
neutrality, anonymity, and some small amount of monotonicity and
decisiveness)!

I sincerely hope I missed some essential point in that example... Can
anyone tell what this would look like with Approval (I mean, what is a
sincere Approval vote in the first place?)?

Jobst

```