[email protected] 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).
----
Election-Methods mailing list - see http://electorama.com/em for list info
