[EM] Round Robins

[Alex Small]

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."
 
 


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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 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.   


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 Dave Ketchum   108 Halstead Ave, Owego, NY  13827-1708   607-687-5026
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