Actually, all Paul said is that the analogy is not
perfect.
Condorcet methods are "like" as in "similar to" a
round-robin tournament in sport. The analogy is not identical because in sport
there is a well-determined outcome when team A plays team B, namely either A or
B wins.
Where the analogy breaks down is that in an election
the "team" is an alternative and the "score" that determines whether it wins is
calculated differently depending upon which "condorcet" method is used to
determine which "team" won that "game."
The analogy is an isomorphism if "win" is defined by "A
scores more points than B" in a head-to-head contest between A and B. But for it
to be a perfect analogy, "scores more" needs to be as precisely defined as it is
in sport. This is not the case when voter's prefences for A over B are obtained
from a ballot that includes C, since the voter is not being asked to choose
between A and B on such a ballot.
To be perfectly analogous to the sport metaphor, the
ballot should allow the voter to record a score for one team vs other another
team. Any attempt to infer the voter's preference relative to a third team would
be like adjusting the score between A and B based upon the outcome of the game
played between B and C, and in sport that is not allowed.
The reason that "cycles" can't happen in sport is that
every "game" has a definite outcome, and only involves one pair of contestants
at a time. If a ballot only contained choices between a pair of alternatives,
the mapping from ballot to pairwise-matrix would be just as well-defined, and
irrefutable. But to call any mapping of ranked ballots to the pairwise matrix
"the same as a round roubin sport tournament" is not accurate. It is "similar
to", or "like", but it is nowhere near the "same as."
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of Dave Ketchum
Sent: Sunday, March 13, 2005 8:31 PM
To: 'Alex Small'; [email protected]
Subject: RE: [EM] Round RobinsIf I understand this, Paul is saying that what Condorcet does is not Round Robin BECAUSE Round Robin in sports only has ONE match between each pair of teams,
In sport, there are no "cycles" in a round-robin. In a 3-team round-robin there's only 2-0, 1-1, and 0-2 as possible outcomes for each team, and if one team is 2-0 there's no "cycle". The only possible "cycle" is a 3-team tie with all teams going 1-1 in the tournament.The cases are:2-0 is the winner, the other teams tie 1-1 for second2-0 is the winner, 1-1 is second, 0-2 is third.All teams finish the round-robin 1-1.So the equivalent of a "cycle" is the last case where A beat B but lost to C, B lost to A but beat C, and (if you can't fill in this part you should not read further) C beat A but lost to B.The answer is that in sport the tournament winner in the case of a three-way tie is pre-specified based upon an arbitrary tiebreaker (read: dictator principle)) such as average margin of victory.
Alex Small
Sent: Sunday, March 13, 2005 4:26 PM
To: [email protected]
Subject: [EM] Round Robins
Finally, what rule do people use in sports to break cycles in round robin tournaments? I'd be inclined to use that rule in public proposals for IRR, even if it should turn out that it isn't the optimal rule from a theoretical perspective.
--
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Dave Ketchum 108 Halstead Ave, Owego, NY 13827-1708 607-687-5026
Do to no one what you would not want done to you.
If you want peace, work for justice.
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