[EM] extending Myerson's test--more policy positions

[Bart Ingles]

David Catchpole wrote:
> 
> On Fri, 17 Mar 2000, Bart Ingles wrote:
>
> > The proven optimal strategy when no strategy info is available is to
> > vote for candidates for whom the voter gives above-average utilities
> 
> As in above-median ;> ? I take it as a basic rule of thumb that the
> optimal strategy when no strategy info is available, in any game, is
> unrelated to relative magnitudes of utility. If I'm wrong in this case,
> could you direct me to the proof in question?

No, not above-median.  Above average.  Example:  there are four
candidates in the running, whom you rate 10, 9, 8, and 0.  Your average
for these is 6.75, so the optimal strategy is to vote for the first
three.

Example 2:  There are four candidates rated 10, 2, 1, and 0.  Your
average this time is 3.25, so your optimal strategy is to bullet-vote.

This assumes nothing is known about the other voters' preferences, so
that the probabilities of each candidate being in a tie for 1st place
are equal.


The proofs can be found in the Merrill book I mentioned, "Making
Multicandidate Elections More Democratic", Princeton University Press
1988, pp 47-63 and 117-120.

The following were cited in the bibliography of Merrill's book:

Weber, R. J. (1977) "Comparison of Voting Systems" New Haven: Cowles
Foundation discussion paper no. 498A

Merrill, S. (1979) "Approval Voting: A 'Best Buy' Method for
Multicandidate Elections?", Mathematics Magazine 52:98-102


The following discusses approval voting strategy, but I'm not sure if
there are any proofs:  
http://www.kellogg.nwu.edu/faculty/weber/papers/approval.htm



> > I wonder what strategy is optimal under a pairwise system?  Should you
> > refuse to rank candidates whom you rate below a certain number?  If your
> > favorite is one of two known front-runners with a 50-50 chance of
> > defeating the other front-runner, it would seem advantageous to refuse
> > to rank anyone whom you would rate below the middle of the scale.
> 
> The optimal strategy when no strategy info is available, for a Condorcet
> completion system, is sincere expression of preferences... Anyone dare to
> argue? Huh? I would feel that any additional strategy brought about by
> extra information would depend on the completion.

Suppose you are a member of group A below, with vNM utilities as shown
in parentheses:

votes
??        A(1.0)  B(0.1)  C(0.0)
??        B(1.0)  A, C(??)
??        C(1.0)  A, B(??)

If no information is available, you can proceed as though all groups are
equal in size.  Thus in a head-to-head matchup between A and C, you give
A a 50% chance of winning, so the utility of that matchup (ignoring B)
is 0.5.

If B wins, the utility of the outcome is only 0.1.   Why would you ever
want to help B win by ranking him sincerely?  The only way I would do so
is if I knew that A had less than a 10% chance of defeating C.