[EM] Round robin tournament statistics

[Kristofer Munsterhjelm]

fsimmons at pcc.edu wrote:
> 
> ----- Original Message -----
> From: Kristofer Munsterhjelm 
> Date: Saturday, June 25, 2011 2:26 pm
> Subject: Round robin tournament statistics
> To: EM 
> Cc: Forest W Simmons 
> 

>> It gets more difficult when one takes ties into account, though. 
>> For most games, no pair is exactly tied in the long run, but one 
>> could imagine a game where if both players cooperate, there's
>> always a tie (such as two players in chess agreeing to always do a 
>> grandmaster draw, based on tit-for-tat reasoning). Then a long run
>> of ties would in itself be significant: it means that neither
>> player is (or chooses to be) any better than the other. Just
>> eliminating ties from consideration, as you did in the winner
>> calculation, wouldn't work because it could take a really long time
>> before a non-tie result is granted.
>>
> 
> That's where the "Independent Identically Distributed" proviso comes in.  If there is any kind of mutual 
> strategy, this condition cannot hold.

It doesn't have to be strategy. I've been considering this problem in 
the setting of coevolutionary algorithms. Say, for instance, you're 
trying to make a game AI in a shooter. Then, very early "random" 
programs might not know to shoot at each other at all, which means 
nobody ever dies, and so it's always a tie.

> What I was more concerned with, ultimately, was how equal rankings would affect the significance of the 
> defeat.  In other words, suppose there are 100 ballots, and W=40 support the winner, L=10 support the 
> loser, and the other fifty rank them equally or truncate them both.  Does this 40 to 10 defeat have the 
> same significance as a 40 to 10 defeat in which there were only fifty ballots total?
> 
> According to the above model (with the independent identical distribution condition) the answer is yes.
> 
> That makes things nice for comparing pairwise defeat strengths in the case of sincere rankings.
> 
> As I mentioned before, these sincere rankings are most likely in the case of informal polls before the 
> actual election.

I also imagine it would be useful in places where it's hard to 
strategize or the context means there won't be any strategy. Such 
examples might be computers in a redundant system voting about an 
observation under uncertainty (the "strategy" will be a random 
distortion) or actual round robin tournaments (where engineering a 
Condorcet cycle based on just one's own matchups would be quite hard).