[EM] Round Robins

Sun Mar 13 19:34:30 PST 2005

Actually, I addressed this in my original post. In a sporting tournament
involving a 3-team round-robin, the only possible tie for first is if every
team goes 1-1 in the match.

Which team is the winner is based upon an arbirtrary criterion declared
ahead of time by the tournament sponsor. 

There is no "next level of competition" for a team to advance to. It is a
round-robin tournament that ends when the teams get back on their buses. As
to which gets the champoionship trophy, it depends upon the tiebreakers
defined by the tournament sponsor. (My preference would be total wins vs the
home team, to make sure that a visiting team got it, but they didn't ask
me...)

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From: election-methods-electorama.com-bounces at electorama.com
[mailto:election-methods-electorama.com-bounces at electorama.com] On Behalf Of
Alex Small
Sent: Sunday, March 13, 2005 9:17 PM
To: Election Methods List
Subject: RE: [EM] Round Robins

OK, maybe Condorcet elections aren't exactly analogous to round robin sports
tournaments, but I still want somebody, anybody, to tell me how the winner
is determined in a round robin if each of the 3 teams wins one game and
loses one game.  I've been told that the method of resolution has something
to do with margins of victory, but I'm wondering if anybody can offer a more
precise explanation.

I'll make my question concrete.  Say we have a round robin tournament
between soccer teams from USC, UCLA, and UCSB.  (And if it turns out that
these 3 schools don't compete in the same league, I'm hoping somebody will
be kind enough to answer the hypothetical question anyway.)  Say that USC
beats UCLA 2-1, UCLA beats UCSB 4-1, and UCSB beats USC 2-0.

Who would be declared the winner of that round robin and advance to the next
level of competition?  If a version of that "cycle resolution" method can be
formulated for public elections and it doesn't have any egregiously awful
flaws (no method is perfect, after all), I'd be just as happy to offer that
as a public proposal.  It would have the virtue of bein gsomething that
people already know.

Paul Kislanko <kislanko at airmail.net> wrote:

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: election-methods-electorama.com-bounces at electorama.com
[mailto:election-methods-electorama.com-bounces at electorama.com] On Behalf Of
Dave Ketchum
Sent: Sunday, March 13, 2005 8:31 PM
To: 'Alex Small'; election-methods-electorama.com at electorama.com
Subject: RE: [EM] Round Robins

If 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 second
 2-0 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: election-methods-electorama.com at electorama.com
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.   

  _____  

-- 
 davek at clarityconnect.com    people.clarityconnect.com/webpages3/davek
 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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