[EM] round robin tournaments

[fsimmons at pcc.edu]

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