I did the whole thing over with a trinomial model that took ties into account
as follows:
Let X1, X2, ... be independent identically distributed random variables with
values in
the set {-1, 0, 1}. Let Prob(1)=p, Prob(0)=q, and Prob(-1)=r.
Let XSum be the sum X1+X2+...
Then XSum = W-L, where W is the number of one's and L is the number of -1's.
Under the null hypothesis that p=r, the expected value of XSum is zero.
Under the null hypothesis the variance of each X in the sum is 2*p=p+r, so the
variance of XSum is
N*(p+r)=N*p+N*r, which is estimated by W+L. So the standard deviation can be
resonably estimated
by
sqrt(W+L).
If the alternative hypothesis is p>r, the appropriate statistic to test is
S=(Xsum-(Expected value of XSum))/(Variance of XSum),
which (as we have seen under the null hypothesis) is estimated by
(W-L-0)/sqrt(W+L),
or S=(W-L)/sqrt(W+L),
exactly the same result that we got by ignoring the ties!
Since S is non-negative, squaring is a monotone transformation, and one half of
S squared is an equally
valid measure of defeat strength.
It turns out that this latter quantity is equal to the difference in the
arithmetic and harmonic means of W
and L:
(W+L)/2 - 2W*L/(W+L)
So the greater the difference in the arithmetic and harmonic means of W and L,
the greater the likelihood
that p is really greater than r, i.e. the greater the statistical significance
of the defeat.
Of course, as I have emphasized before, this measure of defeat strength is
designed for sincere votes.
If pre-election polls can be considered sincere, then this method can be used
to determine the relative
strengths of pairwise wins in the polls, i.e. to deternmine the relative
likelihoods of the pairwise contests
in the polls being borne out in the election itself.
And of course, in the sports context, the method can be used to determine the
relative strengths of the
defeats in a round robin tournament.
So if River, Ranked Pairs, or Beatpath is used to find the tournament winner,
then this method should be
used to determine the defeat strengths.
FWS
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