[EM] Gini and the Borda count
Stephen Turner
smturner0 at yahoo.es
Tue Jun 13 09:54:19 PDT 2006
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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