On Jan 25, 2010, at 11:07 PM, Abd ul-Rahman Lomax wrote:
At 03:13 PM 1/25/2010, Juho wrote:
Sure. But equal ranking must be allowed, otherwise noise is
introduced. Borda with equal ranking (and therefore empty ranks,
otherwise equal ranked votes are reduced in strength) is Range. Why
not just use Range, allowing greater precision. One could use a
Range method with N*R resolution, where the "Borda" version has N
equal to 1.
Range style ratings would give more accurate utilities but I used
Borda style to get rid of strategies.
And I pointed out that "strategies" are not to be "gotten rid of"
because they indicate real preference strength, much more than they
distort it. Why would you rate your favorite 100% and everything
else 0%. It must be really important to you, in order for you to
absolutely wipe yourself out of participating in any other "pairwise
contests"!
I might vote A=100 B=100 C=100 D=0 E=0 F=0 G=0 if I believe that
otherwise I might end up in room D or E. If others give more evenly
spread ratings my strategy could be very efficient (and if they
exaggerate my sincere vote might be too weak against their opinions).
Borda style utilities will
distort the true utilities somewhat but the end result may still be
quite fair.
Sure. I either suggested borda ranking and analysis or thought of it
before proposing range, I forget which. Close enough, perhaps. It
certainly is simple to vote! Unless you have a block of rooms that
are equally good for you, in which case ranking is actually harder
if equal ranking isn't allowed.
I didn't include equal rankings since that would not make
any big change in the level of distortion.
It certainly can make a big change. It can strongly overemphasize
some preference strengths and underemphasize others. Suppose that
there are five rooms to be allocated. A>B>C>D>E provides equal
voting power in each adjacent pair. But suppose that the real
situation is A>B=C=D>E. This is fine as to A and E, but lousy as to
B and D, and, further, suppose that in Range, the utilities are,
respectively, 4,3,3,3,0. The vote in the D:E pair is seriously
undervalued. If it comes down to a choice between D and E for a
room, the voter has indicated weak preference with a pure Borda
ballot when, in fact, the preference is strong, D would be almost as
acceptable as A.
(I already gave up the sincere ratings (4,3,3,3,0) when I used the
Borda utilities. (Equal ratings in Borda style could be counted as
4,3,3,3,2 or 4,2,2,2,0.))
(If rooms B, C and D are equal to the voter then giving B the highest
points (and getting room B instead of C or D) is not a problem. In
some cases the voter might lose all of those rooms when voting B>C>D,
or might win B thanks to this, but maybe this is just minor random
noise.)
Forcing the voters to
decide whether A>B or B>A is correct instead of allowing them to vote
A=B is quite ok.
Bad basic concept, too common among voting systems students. Forcing
voters is never a good idea (or so rarely that it should immediately
raise suspicion)! And in this case, what is the problem with
allowing voters who have no preference between A and B to rate or
rank them equally?
As I said, no problem to allow equal voting if it causes no extra
problems. Simple voting is also a nice property and makes voting
easier to the voters. Not having equal rankings is not just forcing
voters to something they don't want.
Picking a random order doesn't distort the outcome
too much even if the voter could not make up his/her mind on which
one
of the two rooms is better.
Robson Rotation introduces some element of fairness here, perhaps,
but this is an election with a quite small number of voters. There
are much simpler solutions that could be even fairer, but a single-
ballot deterministic solution is likely to be suboptimal, or,
alternatively, so complex to vote that nobody would like it.
The method is thus already noisy as it is
and therefore equal rankings might not add very much.
Eh? A high-res range ballot would not be "noisy." What's often
missing in these analyses is that "exaggeration" in a range ballot
conveys useful information, not noise.
Sincere ratings would give noiseless utilities, but I excluded them
for other reasons.
If equal
rankings will not add any complexity to the method they are ok
though.
(DIfferent ways to count the points in case of equal rankings would
have different impact on the method.)
Borda with equal ranking (and, then, correspondingly empty ranks) is
trivial to vote and count. It is, in fact, Range voting. Obviously,
the naive Borda response to equal ranking (start with the bottom
rank and give it 0, then give each increasing rank one more point)
is not usable.
Rather, a Borda ballot with equal ranking would be a ballot with as
many ranks as choices, and voters would then mark the rank for each
choice. Thus overvoting (equal ranking) necessarily results in empty
ranks.
[...] Do you think there are some "iterative methods" that
would achieve more accurate results (or would be necessary for
efficiency or other reasons)?
Sure. If the preference information is collected, analysts could
divide the rooms into blocks of professors seeking them, and,
possibly, blocks of rooms that are equal for a set of professors,
simplifying the problem. Setting aside issues of seniority or other
preference for particular professors, assuming they are peers, then,
the block of professors involved could decide to use, say, random
choice to provide a choice sequence, or some other method. What
would be done, I'd suggest, is that a final choice would be made by
each professor in sequence, the professor chosen by some algorithm,
including chance as a possibility when there is no clear assignment
from the preference information.
So at each iteration, the choice is of one professor, who then
chooses the room from the set indicated by the preference ballots,
reducing the set of professors and rooms by one with each iteration.
Between iterations, any two professors, or set of professors, may
swap rooms before the next choice is made. After the choices are all
made, voluntary swaps remain as a possibility, continuously.
Where professors have equally ranked rooms, proposed assignments
could be made en masse, with individual assignments then occuring by
consent within this block.
Specifying an exact algorithm in advance could be so much more work
than actually doing iterative assignment this way that it would not
be worth it.
However, I suggested an Asset election to create a representative
body that could efficiently "negotiate" the whole process on behalf
of the professorial community, and that includes determining what
specific methods are to be used. An Asset election is very simple:
each eligible member votes for one person to represent them, and
that can be oneself, but it gets more efficient if one chooses
someone else. Choose the person you most trust to do a decent job at
the task, which includes choosing who is most trusted to do a decent
job..... This creates a reduced subset of the original electorate,
which then cooperates to form "seats" on the committee, with 3
seats, the reduced set, I now call electors, can elect a seat
whenever they can agree to assign N/3 votes to it.
If it can't elect at least two seats, quickly, I'd say that the
community needs some professional assistance, it's divided and
contentious....
The missing seat can be filled if the electors can all agree, in
which case representation is complete. But short of complete
representation, two seats can adequately make decisions, and can
consult the remaining electors, at their discretion. Even with the
remaining seat filled, it still requires two votes to make a decision!
The goal of doing this whole process is to produce a result which is
widely perceived as fair, so the committee, I'm sure, will be
motivated to ensure that all concerns are heard, but control over
the process is in their hands for efficiency, so that it isn't
discussed forever, which is, itself, quite undesirable.
Ok, that's one way to do it. If I interpreted you right you didn't
claim that this type of iterative methods would be more accurate or
more efficient than the "exact target definition + approximate
algorithm to find it" approach that I proposed.
If you want to include more choices and parameters to the equation
than what the simple Borda ratings offer then I refer to the other
method that I proposed.
> P.S. There could be also preferences like "I want a room next to my
closest colleagues". If one wants to support also such preferences one
could allow the voters to rank all the possible room allocation
scenarios and then use some Condorcet method to pick the best
allocation. Since the number of different room allocations may often
be too large for manual ranking one would need some mechanism to
derive the rankings from some simpler set of parameters. One could
e.g. use a fixed questionnaire with a list of questions that the
voters could answer and give different weights. These answers could
then be used to rate each room allocation scenario. In theory one
could also allow voters to give their own algorithm (this is however
probably too complex though for most use cases) that takes a room
allocation scenario as input and rates it (or gives directly a ranking
of all the allocations (or why not even pairwise preferences (that
could lead to personal preference cycles))).
Juho
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