# [EM] Smith Sets with >3 members

Forest Simmons fsimmons at pcc.edu
Thu Feb 28 14:44:57 PST 2002

```On Thu, 28 Feb 2002, Markus Schulze wrote:

> Dear Forest,
>
> every election method that meets the majority criterion is
> vulnerable to "compromising". In so far as a voter will
> usually approve at least that candidate who gets this
> voter's first preference, you cannot circumvent this
> incompability by using "some hybrid method that requires
> information outside of the pairwise comparisons, perhaps
> information not available in preference ballots".
>
> Example:
>
>    40 A > B > C.
>    35 B > C > A.
>    25 C > A > B.
>
>    Suppose that the used election method meets the majority
>    criterion.
>
>    Suppose that A wins the elections. Then the 35 BCA
>    voters can change the winner from A to C by voting CBA
>    (i.e. by "compromising").
>
>    Suppose that B wins the elections. Then the 25 CAB
>    voters can change the winner from B to A by voting ACB
>    (i.e. by "compromising").
>
>    Suppose that C wins the elections. Then the 40 ABC
>    voters can change the winner from C to B by voting BAC
>    (i.e. by "compromising").
>

In other words, when the sincere Smith set is not a singleton, and
Compromise seems to have a better chance than Favorite of getting majority
first place support, then there is a natural strategic incentive to vote
Compromise over Favorite.

So it seems that in your example any method that satisfies the majority
criterion would be as vulnerable to compromising as any other.

So it doesn't matter if we put Approval on the front end (to "seed" single
elimination or bubble sort) or on the back end (to "complete" the election
in the case of no CW) the same compromising incentive would still be
there in full strength.

Still, in the face of usual levels of statistical uncertainty it seems
that some methods satisfying the Condorcet Criterion give more incentive
to compromise than others.

Is this just an illusion?  If not, then why not just apply Borda seeded
single elimination to the Smith set?  Why not take advantage of the extra
expected social utility if it doesn't increase the chance of compromising?

Forest

```