[EM] Range Voting in presence of partial information of a certain character

[fsimmons at pcc.edu]

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