# Nondictatorial & Nonmanipulable axioms (was Re: York's version o

Steve Eppley seppley at alumni.caltech.edu
Mon Apr 29 15:07:55 PDT 1996

```Bruce Anderson wrote:
>According to the academically well-known Gibbard-Satterthwaite
>theorem, any voting method essentially either is dictatorial or is
>not strategy proof (i.e., if it is not dictatorial, then it can be
>manipulated).

Isn't this an oversimplification?

The theorem is derived from Arrow's theorem.  The two axioms you
mention above (Nondictatorial and Nonmanipulable) are only two of
five.  The theorem says that all five axioms taken together are
inconsistent.  So it's possible for a voting method to satisfy the
above two as long as it violates at least one of the other three.

I'm sure Bruce has a much better understanding of these axioms than
I do.  As I understand it, the Nondictatorial axiom is another one
of these academic yes/no criteria: "Is there a possible scenario
where some voter's ballot doesn't count?"  The existence of such a
scenario doesn't really say much about how good or bad a voting
method is in normal scenarios.  For example, in a large Condorcet (a
"dictatorial" method) election, who can be said to be dictating?

And the Nonmanipulable axiom doesn't appear to offer much value
either.  If we want voters to matter, it means the method must let
them choose how they will vote--they can't be forced to vote their
sincere preferences since there's no way to know their sincere
preferences.  If someday technology advances to the point where
voters' sincere preferences can be accurately and reliably obtained
(by mind-reading, truth serum, whatever) we'll be able to solve this
conundrum.  In the meantime, a voting method could violate both
axioms and still be the best method.

The Nonmanipulability axiom is closely related to Arrow's axiom that
voter preferences are expressed only as rankings, disallowing the
input of weightings (a.k.a. ratings, a.k.a. point assignments) by
the voters.  If the voting method looks at voters' weightings, it
creates strong incentives for voters to express strategic weightings
instead of "sincere" weightings.  Mike York has claimed without
proof that his Elimination and Renormalization method is much better
than other weight methods at reducing these incentives.

I provided a Nader/Clinton/Dole example showing how the Nader voter
could safely vote {Nader=100, Clinton=1, Dole=0} even if s/he
believes the race will be between Clinton and Dole, since after
Nader's elimination his/her ballot will renormalize to a
full-strength vote for Clinton over Dole.

Also, if the Nader+Clinton voters are a majority and all rate Dole=0,
Dole will lose:

49 Dole=100    Clinton=  0  Nader=  0
26 Dole=  0    Clinton=100  Nader=  1
25 Dole=  0    Clinton=  1  Nader=100

Round One:
Ballots are normalized:
49 Dole=1    Clinton=0     Nader=0
26 Dole=0    Clinton=1     Nader=0.01
25 Dole=0    Clinton=0.01  Nader=1
-----        -----       -----
49           26.25       25.26   Nader eliminated.

Round Two:
Ballots are renormalized:
49 Dole=1    Clinton=0
26 Dole=0    Clinton=1
25 Dole=0    Clinton=1
-----        -----
49           51                  Dole eliminated, Clinton wins.

So York's method may satisfy the Ossipoff Generalized Majority Rule
standard, in that the anti-Dole majority can ensure Dole's defeat.

But to what extent does York's method still have the lesser-of-evils
dilemma?  Will the Nader voters feel a need to protect Clinton from
elimination in normal scenarios, at Nader's expense?

--Steve

```