Margins, majority, strategy

[Blake Cretney]

In order to argue in favor of marginal Condorcet, I am going to
suggest a standard that it passes, but Votes-Against fails.

Sincere Expectation Standard
Given that a voter has no knowledge about how others will vote, a
sincere vote must be at least as likely as any insincere vote to
give results that are in some way better in the eyes of the voter.

Or expressed as a more rigid criterion:
-----
Sincere Expectation Criterion (SEC)
Consider a voter with a preference order between the possible
outcomes of the election.  Let us call his sincere ballot, X.  Now,
assuming that every possible legal ballot is equally likely for every
other voter, there must be some justification for the vote X over any
other way to fill out the ballot, which I will call Y.
This justification is given by the following comparisons:

The probability of X electing one of the voter's first choices vs.
the probability of Y electing one of these choices

The probability of X electing one of the voter's first or second
choices vs. the probability of Y electing one of these.

The probability of X electing one of the voter's first, second or
third choices vs. the probability of Y electing one of these.
... And so on through all the voter's choices

X must either do better in one of these comparisons than Y, or equal
in all.  Otherwise the sincere vote can not be justified.
----
In other words, there must be some justification for voting sincerely
even if the voter does not know how any one else is voting.

Votes-Against fails this criterion because if your sincere preference
is A > B=C, it is more likely to your advantage to rate A > B > C or 
A > C > B.  It can back-fire, but the insincere vote is more likely
to get you what you want, so unless you have detailed knowledge about
how everyone else is voting, the insincere vote is better.

So, why is violating SEC a problem?
1.  It divides voters into the naive vs. those who know how to play the
system.  The votes of those who know the trick will be worth more.
2.  It is embarrassing.  Eventually campaign organizers will start to
inform voters about the random preferences strategy.  Voters will feel
that they were deceived in elections where they left candidates unranked
and will be embittered towards the method.  Furthermore, many will dislike
the idea that they are being encouraged to vote randomly, and suspect that 
this will cause random results.
3.  It encourages voters to think strategically.  Once voters are inured
to the idea that a sincere vote is sometimes a bad idea, and that
random preferences should be marked, they will be more likely to
accept other strategies like order-reversal.
4.  Eventually, when everybody knows about the trick, and nobody sincerely
leaves candidates unranked, the method will be equivalent to Margins, with
the exception of the ability to use truncation for some complicated 
strategies.

I have made some comments on your example.
On Mon, 14 Sep 1998 23:05:24   Mike Ositoff wrote:
>
>Hi--
>
>When I posted 2 Margins truncation bad-examples the other day,
>I didn't accompany them with any comment, and so I'd like
>to point out a few things about them.
>
>In both examples, the A voters, by truncating, succeeded in
>gaining the election of A, by defeating a Condorcet winner,
>in violation of the expressed wishes of a majority.
>
>In example 1, Margins, by electing A, elected the only
>candidate with a majority against him.
I notice that when you say X has a majority over Y, you mean a majority
of all voters, not just those expressing a preference between X and Y.
So, it is in effect, a three way race between X, Y, and the abstainers.
Some people might want to use the word majority to mean a majority of
eligible voters, or of the population as a whole, etc.  I do not 
consider any of these uses of "majority" necessarily right or wrong, but
I tend to use it to mean a majority of those expressing a preference
between X and Y.

>
>That sort of thing is predictable for a method that ignores
>majority wishes by not scoring by how many people rank the
>defeater over the defeated.
>
>To defend against that result, the C voters would have to vote
>B equal to C, insincerely voting a less-liked alternative equal
>to a more-liked one. A form of drastic defensive strategy.
>
>***
>
>Example 3:
>
>100 voters.
>
> 44  28  28
>  A   B   C
>  C       B
>
>The A voters are using order-reversal against B. It succeeds,
>using Margins, but is thwarted, in the votes-against versions,
>by the B voters declining to list a 2nd choice.

I do not think your example is very realistic.  It assumes that
B voters have no preference between A and C.  For example, if the
true preferences were

44 A B C
14 B C A
14 B A C
28 C B A

The insincere would be
44 A C B -- order reversal
14 B C A
14 B     -- defensive truncation
28 C B A

That is, I think it more likely that B voters will have preferences
between A and C.  If this is the case, and the B voters who prefer
A to C truncate, this can have the effect of order reversal.  So, the
B voters are not likely quite as helpless as you suggest, even under
Margins.

Furthermore, if they truly have no preference between A and C, then
they will not mind ranking C over A, if A is trying order
reversal.  This is unlikely to even be insincere.  B voters will
be genuinely outraged by A's tactics.

To put it another way, if I vote B > A=C because I hope this will
elect C instead of A, I am voting insincerely.  If I would prefer
to see C elected, even if only to punish A, then the sincere vote
is B > C > A.  This sincere vote is often refered to by Votes-Against
advocates as a drastic strategy.

However, I think we should consider that the same votes could result
44 A C B
28 B
28 C B A
And that they are actually all sincere.  This results in
              Majority of
B > A 56:44   12
A > C 44:28   16
C > B 72:28   44

What we have here is 3 contradictory majority votes.  We have to over-rule
one of them.  It makes sense to over-rule the majority that is least
decisive.  If we use Margins this is B > A, if we use Votes-Against
this is A > C.  So, the question is, if we consider votes to be sincere,
which is more decisive, a higher margin or a higher Votes-Against

Well, Votes-Against says that a majority of 51 to 49 is more decisive
than one of 49 to 1, if there are 100 voters.  Why?  Because "Votes
Against" assumes that the non-participants in the 49 to 1 victory are
abstaining for strategic reasons, and that this throws this victory into
deep suspicion.  However, this has the effect of creating a rather
strange result if the voters are sincere.

So, in conclusion, marginal methods give better results when the voters
are sincere and do not have the problems inherent in violating SEC.



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