Suppose that you were voting a range ballot and all you knew was that candidate
X was very likely to
win or be tied for first place. Then you should give max support to each
candidate that you prefer over X,
and zero support to the candidates that you like less than X. But what about X
? How much support
should you give to alternative X ?
If you knew which candidate Y was most likely to be tied with X, then you would
give X full support if you
liked X better than Y, and no support if you liked Y better than X. But we’re
assuming that this
information is unavailable.
So all we know is that X is very likely to either win or be tied for first
place in the range count. Obviously
if X is your favorite, you should give X top rating, and if X is your most
despised option, you should rate X
at minrange. Suppose that X was halfway in between your favorite and worst,
i.e. you would be
indifferent to having X or a coin flip between Favorite and Worst. Then it
seems natural that you would
give X a rating half way between the max and min range values. This line of
reasoning leads one to
conclude that you should just give X and anyone you like the same as X your
sincere rating.
In sum, if your sincere ratings for X, Y, and Z were all equal to r, then you
should rate these alternatives
at level r, all of the alternatives you like better than them with the top
rating, and all of the alternatives
you like less with the bottom rating.
Suppose that when all voters use this strategy it results in X getting the
highest range vote. Then we
could say that X was a stable range winner.
But sometimes it will be the case that no matter which candidate X this
strategy is used on, some other
candidate Y will end up getting the highest range total, i.e. there is no
stable range winner.
It seems to me that we should then seek the alternative that comes nearest to
being a stable range
winner.
How could we measure how close candidate X was to being the stable range winner?
Perhaps we could take the difference in the voted range totals of Y and X as
the measure of distance
from stability, and elect the candidate X that minimizes this difference.
We can automate this strategy by converting each range ballot into a pairwise
matrix as follows:
The (i,j) element of the matrix is the maxrange value if the ballot prefers
alternative i over alternative j. It
is the minrange value if the ballot prefers alternative j over alternative i.
Otherwise, it is just the common
rating for alternatives i and j.
All of these matrices are summed to a matrix M.
The winner is the alternative j that has the smallest value of max{
M(i,j)-M(j,j) | given i not equal to j}.
It seems to me this winner would have an excellent claim as the nearest to
stable range winner.
Comments?
Forest
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