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ALLSTAT  1999

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Subject:

SUMMARY: optimal strategy

From:

Tim Cole <[log in to unmask]>

Reply-To:

Tim Cole <[log in to unmask]>

Date:

Wed, 14 Apr 1999 14:28:01 +0100

Content-Type:

text/plain

Parts/Attachments:

Parts/Attachments

text/plain (209 lines)

At 2:34 pm +0100 12/4/99, I wrote:
>Suppose there are N applicants for a job, of which S are to be shortlisted
>(S < N). A panel of P scrutineers decides on the shortlist. Each scrutineer
>can vote for as many candidates as he/she wants, and the candidates
>receiving the most votes are shortlisted.
>
>What is the optimal number of votes for each scrutineer to cast? If they
>vote for all the candidates, or alternatively if they vote for none, they
>make no difference to the process. Obviously some intermediate number of
>votes is the most influential, and I suspect the optimal number is S. Am I
>right?

The answer to my question hinges on the precise definition of "optimal".
Also I need to specify how to handle ties, and what to do if fewer than S
candidates get any votes.

If the question is viewed as a screening test, optimality could mean
maximising the sum of sensitivity (voting for shortlisted candidates) and
specificity (not voting for non-shortlisted candidates). Voting for all and
voting for none then gives a sum (sensitivity+specificity) of 100% in each
case. Some intermediate number of votes ought in general to increase this
sum.

In the absence of knowledge about the other scrutineers, one might assume
as a first step that all candidates have the same expected number of votes
(though this is a fairly unrealistic assumption). In this case the correct
strategy is to vote just for the candidates you want shortlisted, which
will involve no more than S votes, and possibly fewer. As Jim Slattery
points out, any more than S votes are wasted as they cannot all be
shortlisted.

Anyway this analysis to the problem, though imperfect, was sufficient for
my purposes, and I have now used it with fellow governors to select a
shortlist for appointing a new head teacher.

Very many thanks to John Kornak, Eryl Bassett, Dietrich Alte, Miland A I
Joshi, Paul Seed, Jim Slattery, Mike Franklin, MM Peterson and Pat Gaffney
for their comments. Their individual responses appear below.

Tim Cole
------
From: John Kornak <[log in to unmask]>

I suggest that you need to create some kind of cost/loss/quality of
candidate function and some criterion for creating a cut off point. I would
further suggest a Bayesian decision process (but I admit to being biased).

Consider the case where there is one candidate you really want to get the
job that far outshines everyone else. Also, it is difficult to choose
anyone as being the second best i.e. they are all the same standard.
Choosing 5 of these at random to vote for will decrease the chances of your
preferred candidate getting shortlisted.
------
From: Eryl Bassett <[log in to unmask]>

I strongly suspect that you need to specify optimality rather more closely,
if you want to get an "optimum" strategy.

Suppose, for example, you have one strongly favoured candidate, and your
aim is to get that candidate onto the shortlist. I'm pretty sure (though I
haven't got a proof on tap) that the best strategy is just to vote for that
one candidate.

It looks to me as if one might be able to prove that conjecture by some
sort of proof by contradiction.  If A is *your* candidate, and B and C are
other candidates, then (for the case S=2) voting for A and B rather than
just for A could result in  B and C both beating A onto the shortlist.
------
From: dietrich alte <[log in to unmask]>

this is an optimal voting problem. there (at least) are two good books on
the subject which may answer your questions:

* Mathematics and Politics : Strategy, Voting, Power and Proof (Textbooks
inMathematical Sciences)  by Alan D. Taylor; $47.95

* Basic Geometry of Voting, by Donald G. Saari; $39.00.

i read both of them and like them very much. esp. taylors book is easy to
read and gives fascinating insight in the topic. check amazon.com or other
bookstores for more info.
------
From: Miland A I Joshi <[log in to unmask]>

Your problem is very interesting. I would say that the minimum number is S,
in that if it is any less, it is theoretically possible that there will be
less than S candidates mentioned at all, hence eligible for the short list.
However the problem also depends on the size of P, which needs to be large
enough to guarantee that S candidates can be chosen out of the N. Even this
will only work for sure if each panellist is compelled to cast S votes. So
in practice we need to assume that the votes of the panellists will be
correlated such that the short lists will not be very different, and thus
the emergence of a short list is likely. Therefore it seems to me that the
problem is best solved by practical experience rather than by mathematics.
------
From: Paul Seed <[log in to unmask]>

Call the number of votes you cast V. I would have thought V=N/2 (or (N+1)/2
or (N-1)/2 if N is odd). The problem is similar to that of multiple voting
in a multi-member constituency. Usually, the maximum number of votes is S,
but less are permitted.

Your vote is only decisive if it makes or breaks a tie between two
candidates. This occurs if (without your votes) a candidate you voted for
has the same number of votes or one less vote than a candidate that you did
not vote for. Without information about what the other voters will do, you
cannot say what ties or near-ties are more likely. To maximise you chance
of being effective, you need to have as many candidate pairs with one vote
between them as possible.

The number of such pairs is V*(N-V) This is 0 for V=0 or V=N , as you
noted, and maximum for V=N/2 (or as near as possible).

If S<N/2, as it usually is in major elections, and you have only S votes
the best option is V=S.

A problem will arise if V> N/2. Most people think that more votes means
more influence, and try to use their full allocation. This is a mistake.

If you have one candidate to whom you are strongly opposed, the solution is
to vote for every candidate except this one. If votes are restricted, you
should vote for every _marginal_ candidate except this one, assuming that
you can identify them. The strong candidates will get in without your
suport.
------
From: JIM SLATTERY <[log in to unmask]>

S makes a lot of sense. If they vote for more than S they will each vote
for candidates they would not shortlist if left to themselves. If they vote
for less they will fail to support candidates who they would shortlist. It
seems reasonable to choose a number which would make sense no matter how
many scrutineers you have. Again, S is the only possibility since you might
have only one scrutineer.

I suspect that what you really want is a natural definition of
'influential' which makes the intuitive solution the right solution. No
doubt someone will supply one, something like maximising the expected
satisfaction of the scrutineers.
------
From: Michael Franklin <[log in to unmask]>

Not an answer but I think you may need to tighten up on what you mean by
'optimal'. Optimality criteria tend to be very specific and different
criteria lead to different results. (I guessed you mean which scoring will
lead on average to the highest number of the 'top' s candidates being
interviewed - though 'top' is still undefined.)

I'm not sure the answer is S I suspect it may be nearer N/2. Think about
'extreme' case of shortlist of S=1 which in effect is very similar to our
electoral voting system but would it be better for each elector to vote
for, say, 2 candidates as distinguishing between the top two candidates
would be based on roughly twice as many 'successes'?
------
From: MM Peterson <[log in to unmask]>

A difficulty with the problem as posed is that no rule is given for
resolving cases in which after the voting S-x applicants (0<x<=S) receive
more votes than any of the remainder, and a further x+y applicants (0<y)
each receives the SAME number v of votes, where v is less than the number
received by any applicant in the first set but greater than that received
by any other applicant.

For example, suppose there are 10 applicants, A, B, C, D, E, F, G, H, I and
J, the short list is to consist of 3, and that a panel of four scrutineers
casts votes as follows

Votes for 	A  B  C  D  E  F  G  H  I  J

Scrutineer 1	1  1  1
Scrutineer 2	1  1  1
Scrutineer 3	1        1  1
Scrutineer 4	1        1  1

Totals		4  2  2  2  2  0  0  0  0  0

Clearly applicant A with 4 votes is to be on the short list, but which of
B, C, D, E each with 2 votes is to be short-listed?

I shall call applicants in this second category MARGINAL, and those who are
in that category when all votes except those of scrutineer i are counted
i-MARGINAL.  Since the number r of i-marginal applicants can by the rules
proposed be anything from 0 up to N, and since voting by scrutineer i for
any positive number of i-marginal applicants less than r will alter the
number of marginal applicants, the solution to the problem of influence for
the i-th scrutineer is in my view bound to depend on the rule devised for
resolving the problem of the marginal applicants. I don't see how it is
possible to proceed further without some information on this.
------
From: Patrick Joseph Gaffney <[log in to unmask]>

I don't know if statistics can answer the question without asking more
questions (e.g. modelling how people cast their votes -- some interviewers
may be highly correlated and vote similarly, other panels might have more
independent folks).

My suspicion would be to have more than S votes (2S perhaps), if S is
small.  But I'd think of S as the number of candidates required for 2nd
interview in order to get 1 qualified candidate.

Again, nothing statistical about this.  Discupleme.

[log in to unmask]   Phone +44 171 242 9789 x 0740  Fax +44 171 242 2723
Epidemiology & Public Health, Institute of Child Health, London WC1N 1EH, UK




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