[EM] More 0-info pairwise strategy

[Markus Schulze]

Dear Blake,

you wrote (30 Mar 2000):
> Markus Schulze wrote (29 Mar 2000):
> > This is an example where it is advantageous to vote
> > insincerely in a zero information situation:
> > 
> > Suppose that MinMax(margins) is used. Suppose that there
> > are four candidates. Suppose that your sincere opinion
> > is A > B > C > D.
> > 
> > Where is the problem? The problem is: It is possible that
> > the worst defeat of candidate B is against candidate A
> > and that the worst defeat of candidate C is against
> > candidate D and that -by going to the polls and voting
> > sincerely- you simultaneously increase the worst
> > defeat of candidate B and decrease the worst defeat
> > of candidate C and therefore change the winner from
> > candidate B to candidate C.
> > 
> > Now suppose that you have zero information. One possible
> > strategy is that you presume that the other voters vote
> > randomly. Of course, this is certainly not the best
> > strategy. But it is a plausible one.
> > 
> > The unique advantage of voting A > B > C > D sincerely
> > instead of A = B > C > D insincerely is that you could
> > change the winner from candidate B to candidate A. 
>
> This is where I disagree.  It is also possible that you
> could change the winner from C to A.  For example, using
> my base rule for constructing examples:
>
> B>A b+12 (this is a margin, of course)
> C>B b+20
> A>C b+11
> A,B,C>D b+21
> 
> One vote of the form A>B>C>D will decrease A's largest
> victory, and cause A to win.  A=B>C>D only takes you half
> way there, and in this case causes a tie.
>
> The important point, which distinguishes margins, is that
> a vote of A>B is just as likely to decrease the largest
> loss of A as it is to increase the largest loss of B,
> unless we know which is already winning. So, p(C,A)=p(B,C).
>
> As a result, the change in expected utility by voting
> insincerely is negative, independent of candidate
> utilities.

I don't agree with your statement that "a vote of A>B is
just as likely to decrease the largest loss of A as it is
to increase the largest loss of B, unless we know which is
already winning." Even if a given voter has no information
about the other voters, he still has information about
his own opinion and about his own strategy. If he votes A>B,
then he has to conclude that -if the other voters vote
randomly- the probability that the worst defeat of
candidate A is against B is slightly smaller than the
probability that the worst defeat of candidate B is against
candidate A.

CASE 1: If this voter votes A>B, then he has to conclude
that the probability that the worst defeat of candidate A
is against B is smaller than the probability that the worst
defeat of candidate B is against candidate A. Therefore he
has to conclude that it would have been better if he had
voted B>A.

CASE 2: If this voter votes B>A, then he has to conclude
that the probability that the worst defeat of candidate B
is against A is smaller than the probability that the worst
defeat of candidate A is against candidate B. Therefore he
has to conclude that it would have been better if he had
voted A>B.

Therefore -if the difference of the von Neumann-Morgenstern
utilities of candidate A and candidate B is sufficiently
small- it makes neither sense to vote A>B nor to vote
B>A.

Markus Schulze
schulze at sol.physik.tu-berlin.de
schulze at math.tu-berlin.de
markusschulze at planet-interkom.de