[EM] round robin tournaments

[fsimmons at pcc.edu]

----- Original Message -----
From: Jameson Quinn 
Date: Friday, June 24, 2011 1:44 am
Subject: Re: [EM] round robin tournaments
To: fsimmons at pcc.edu
Cc: election-methods at lists.electorama.com

> I like this approach. But I think you've multiplied by a factor 
> which should
> be dividing:

Right!

> ...
> 
> > Here's a way to resolve it.
> >
> > Let N=W+L. Let p=W/N. Let q=L/N. Let sigma = sqrt(N*p*q).
> >
> > Measure defeat strength by S=(p-q)*sigma.
> >
> 
> Let's say one team happens to consist of penguins. They score 0. 
> The chances
> of them winning a rematch are 0. Their defeat strength should be...
> infinite. But with your formula, it comes out to 0.
> 

The correct way to avoid dividing by zero is to take the approach of "hypothesis testing" where the null 
hypothesis is that p=q=1/2.  Then sigma=sqrt(Npq)=sqrt(N)/2, so we don't have to divide by zero.

We test whether or not  S=N*(p-.5)/sigma is significantly greater than zero, versus the null hypothesis 
that it is equal to zero.

This S simplifies to (W-L)/sqrt(W+L).

If, instead, we use take sigma as the sample standard deviation based on the sample values of p and q, 
rather than the null hypothesis values of 1/2, we get  

S' = sqrt(W+L)*(W-L)/sqrt(4*W*L) .

Now let 

S'' = (sqrt(W)-sqrt(L))*sqrt(2) .

It turns out that for W>L the following compund inequality always holds:

  S<S"<S'

Since, as you point out, S' is undefined when the sample proportion of failure is zero, and S is on the low 
side, I suggest using S", which is always defined.

Of course, we can disense with the factor of sqrt(2), since we don't care about the exact confidence level 
as much as comparing confidence levels.

So we can safely say that we have more confidence in the repeatability of A's win over B than that of C's 
win over D iff sqrt(A's points)-sqrt(B's points) > sqrt(C's points)-sqrt(D's points).

So far this has been totally in the context of sports tournaments.

Now, Juho suggests looking at the same thing in the context of elections.

It's not going to work there except in zero information elections, where there is no incentive to vote 
insincerely.

But I suggest that on the basis of the above analysis, this sqrt approach is better than the margins 
approach in that context of zero info sincere ballots.

FWS