[EM] Richard's strategy method

[Richard Moore]

MIKE OSSIPOFF wrote:

> Richard, suppose you write an example that includes the
> input information that your method uses, and then use your
> method to show how the voter should vote in that example, in
> Approval.

OK, ZI strategy, four-way race. My utilities are A=100, B=70, C=50,
D = 0.

Assume a vote for a candidate increases that candidate's
likelihood of winning by 3%. Since I have no (reliable) outside
information, I must assume this 3% comes equally from the
other three candidates. Thus my vote decreases their chances
by 1% each. Also, I have no reason to believe the 3% increase
doesn't hold for all four candidates.

A is included regardless of the other utilities.

B is included if the incremental utility of a vote for B is positive.
This incremental utility is

    -.01 * 100 + .03 * 70 - .01 * 50 - .01 * 0 = 0.6

so I include B.

The incremental utility for C is

    -.01 * 100 - .01 * 70 + .03 * 50 - .01 * 0 = -0.2

so I don't include C. Given that C failed the test there is no reason
to crunch the numbers for D.

Non-ZI strategy: The information I use has to take the form
"a vote for i affects candidate j's chances of winning incrementally
by some percentage". This gives a matrix. The diagonal of the
matrix should be all positive numbers and the other elements
all negative (it's a monotonic election, after all). For this example
I'll use a 3-candidate race to keep the matrix small. I'll use
percentages here so I can start with whole numbers.

         2   -1    -1
        -1     3    -2
         0     -1    1

A vote for i is represented by the ith row (each row must add up
to zero). The effect of that vote on candidate j can be found in the
jth column. Assume the candidates are numbered in order of
descending utilities, which are 100, 50, and 0.

A vote for the first candidate is a given, of course.

A vote for the second candidate has the following incremental
utility:

    -.01 * 100 + .03 * 50 - .02 * 0 = 0.5

Since this is positive, I will also vote for B.

No need to run the numbers for C since the lowest-utility
candidate should never be voted for.

It's late; I apologize if this contains any arithmetic errors but
I think it illustrates the procedure.

> That example would show how workable your method actually is.
>
> How would you determine the probabilities that you use--or would
> you estimate them directly?

For the ZI case this is simple -- use +1 for the candidate under
consideration, -1/(N-1) for the other N-1 candidates. Only relative
values are important, so any set of numbers that maintains the same
ratio could be used.

The non-ZI case could only be solved for information of a given nature.
Are you using a poll? If so, do you feel comfortable in assuming that
the only source of error in the poll is random statistical sampling?
Are you using historical data that shows members of party X are 5%
more likely to turn out at the polls than party Y? The information can
take
many forms so there is no one-size-fits-all way of calculating the
probability deltas, and it's quite likely you won't be able to account
for all the error sources. So for non-ZI strategy an old-fashioned
neural
network may still be your best bet. It's been pointed out that this
academic analysis is of more theoretical than practical value. Also, as
I
said previously, in an approval election I would tend to just go with ZI

strategy anyway.

 -- Richard