[EM] round robin tournaments

[robert bristow-johnson]

On Jun 24, 2011, at 3:13 AM, Juho Laatu wrote:

> On 24.6.2011, at 1.56, fsimmons at pcc.edu wrote:
>
>> In sports round robin tournaments where burial strategy is  
>> meaningless, what is the best measure of
>> defeat strength?  In other words, if you were going to use, say,  
>> beatpath to find a total order of the teams
>> from best to worst after the conclusion of a tournament, would you  
>> use wv, margins, or something else
>> to measure defeat strength?
>>
>> Let's think about basketball for example.  Let W be the winning  
>> points and L be the losing points from a
>> match between two teams.  The margins defeat strength would be W- 
>> L.  The wv defeat strength would
>> be just W.  Another possibility would be the point ratio W/L or  
>> equivalently log(W) - log(L).  Between
>> margins and the point ratio would be  sqrt(W) - sqrt(L), which  
>> seems better to me from an intuitive point
>> of view.
>>
>> What rational basis could we use to decide between these measures?
>
> We should understand how the results of the winner selection  
> algorithm relate to our real world needs. A good explanation and  
> justification is what we need.
>
> The basic explanation behind margins is "number of required extra  
> votes to win everyone else pairwise". For the proportion / ratio  
> approach the corresponding explanation would be "proportion/ratio of  
> required extra votes to win everyone else pairwise". If people think  
> that margins is too much on one side and ratio is too much on the  
> other side, then they might be happy with the sqrt approach. I don't  
> have a good natural explanation available for it. Maybe there is one  
> (a bit more complex than for the other two) (maybe below in your  
> mail). The explanation for winning votes is also a bit more complex  
> and a bit one-sided (since it does not count the defending votes). I  
> think winning votes have some strategic defence flavour and implicit  
> approval flavour in it (but others may have different viewpoints on  
> what is the best explanation for winning votes).

my spin on why Margins makes the most sense is:

    1.  the margins of logs is sorta the same as the percentage defeat  
strength.  in an election with 100 voters, a 55/45 defeat is precisely  
as strong (as indicative of the voter sentiment) as a 550/450 defeat  
in an election of 1000 voters.  both represent the same degree of  
rejection by voters of the 45% candidate.

    2.  the measure of importance of an election is proportional the  
number of voters participating in it.  if very few people weigh in on  
an election, it must not be very important.

    3.  so, just like in the beginning of the Dead Poets Society, we  
multiply these two measures of import together to get a composite  
measure of import.  if in a Condorcet cycle you have one pairwise  
election with a 10% margin and 100 voters (a margin of 10 votes) and  
another that is 5% of 400 voters (a margin of 20 votes), while the  
later displays a smaller degree of rejection of voters it involves so  
many more voters that it speaks more loudly than the result of the  
former.

    4.  simplicity has its attraction.

    5.  Winning Votes communicates something regarding the number of  
voters participating, but says nothing about how close the election  
was.  an election with a razor-thin result, even with a lot of people  
voting, does not measure well the will of the people.  if it's 99,999  
voters and 50,000 said Candidate A and 49,999 said Candidate B, that  
does *not* say that Candidate A has such a great mandate to lead.   
Margins says his/her mandate to lead is 1 and Winning Votes says it's  
50,000.  how can that make any sense?

if there are N candidates, the number of pairs is N*(N-1)/2, there are  
two vote totals per pair, so the set of vote totals is N*(N-1).  the  
entire election is decided solely on those N*(N-1) numbers.  if there  
is a CW, the entire election is decided on the set of margins (again,  
back to half: N*(N-1)/2), in fact with a CW it's based on the signum  
function of those margins: N*(N-1)/2 bits of information (ignoring  
ties).

again, y'all heard me bitch about the stupid rectangular "Defeat  
Matrix".  i think the layout should be a triangle arranged in such a  
way that the UR half of the defeat matrix is folded down onto the LL  
half.  the winning candidate of a pair is listed above the loser for  
each pair in one cell of the "Defeat Triangle".  the election can be  
survey with a glance (whereas with the stupid defeat matrix, you have  
to look across the matrix to see the other salient number and then you  
have to remember which number goes with which candidate).  if margins  
are used, each pair of numbers can be replaced with a single number,  
the margin, and the entire election is decided simply by those N*(N-1)/ 
2 numbers and the pair ordering that comes with them.

i think the stupid rectangular Defeat Matrix should be ditched and  
replaced with a Defeat Triangle with N*(N-1)/2 cells and three numbers  
in each cell: Winner total voters, Loser total votes, and the W-L  
margin.

whether it's Schulze, Ranked Pairs, or MinMax (using margins), the  
determination of the election result is based on the same N*(N-1)/2  
numbers.

--

r b-j                  rbj at audioimagination.com

"Imagination is more important than knowledge."