[EM] Gini and the Borda count

[Stephen Turner]

There seems to be a connection between the Gini
function discussed here recently and the Borda Count.

On all ballots, rankings are assumed complete and
strict.

Positional methods like the BC correspond bijectively
(apart from a scaling factor) to the lists of n
non-decreasing numbers of the form
w_1=1,w_2,w_3,...,w_(n-1),0=w_n.  

Saari calls this a "w-vector".

Of course, the  correspondence is, for each ballot,
assign 
w_1=1 to the top-ranked candidate
w_2 to the second ranked
...
w_(n-1) to the second-last
w_n=0 to the last.

Then sum over all ballots.  Examples:
for plurality, w_2 = 0
for Borda, the w_i are in arithmetic progression

Now consider the w-vector
w_G obtained by squaring all the entries in the
w-vector for Borda.  This is the one with a connection
to the Gini function.

The positional methods also correspond bijectively to
the convex hull of the n-1 points 

p_i (for 1<=i<n-1) 

in R^n defined by 
p_i=(1,1,...,1,0,0,...0) where there are i ones
followed by n-i zeroes.  

The points p_i even form a simplex of dimension n-2,
within the hyperplane defined by "first coord=1, last
coord=0"  So plurality corresponds to the point p_1
and Borda corresponds to the barycentre/centroid of
the simplex, namely the point obtained by taking the
coordinate-wise mean of the p_i:

(say) p_(BC) = sum_(i=1 to i=n-1) (p_i)/(n-1)

Now this mention of a mean makes one think of other
ways of taking an average, one of which is the Gini
function recently mentioned in this forum.  

For the non-decreasing sequence of reals
0<=b_1<=b_2<=...<=b_r

the value of the Gini function of these is
f_Gini = (1//(r^2))*sum_(j:1<=j<=r) (2r+1-2j)*b_j

The p_i above are not real numbers, but we can decree
an ordering on them by saying p_i > p_j if and only if
i > j.  Then calculating f_Gini for the (n-1) p_i
gives a positional method, and it turns out that
the w-vector for this method is w_G, the one obtained
by squaring all the entries in the Borda vector. 
Curious, no?  Of course, this comes down to the fact
that the sum of the first r positive odd numbers
starting from 1 is r^2.

[Choosing the opposite ordering on the p_i doesn´t
seem to give anything interesting.]

Is there any welfare/utility connection in all this?


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