Round Robins

[Alex Small]

A few things.

First, I like the point about Instant Round Robin as a name. I think I will start calling it that. Naming something after a theorist is fine in academic circles, but I can't think of too many policies that are widely referred to by the name of some theorist. So I'm cool with the more descriptive name.

The only problem with IRR is when there is no Condorcet winner.
But as far as I know, elections with no CW are totally theoretical.
For them to happen would require voters to seriously confused
about their preferences for different candidates.

Not really. As long as candidates are on a 1D scale from liberal to conservative there won't be any significant number of voters with a preference like left>right>center or right>left>center. However, if there's a second dimension along which to evaluate candidates then it all changes. It could be that we replace the single left-right axis with different axes for different categories of issues, and get canidates who are fiscally conservative and socially liberal, or fiscally liberal and socially conservative. Another way this could happen is if the second axis is some issue like character or experience. A moderate with a sleazy record could wind up being a lot of people's last choice even though he's between the liberal and conservative on the ideological spectrum.

I'm not so sure, Jan, since many Condorcet-efficient methods do not
require all (n-1)*n/2 pairwise comparisons to be carried out. For
example, ROWS is Condorcet-efficient and only requires n-1 comparisons,
which is by far less.

What is ROWS?

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.


Alex Small


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[Paul Kislanko]

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
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.

[Jobst Heitzig]

Dear Alex!

Post by Alex Small
Naming something after a theorist is
fine in academic circles,

There is disagreement about this since it leads too often to the wrong
person getting the credit...

Post by Alex Small

I'm not so sure, Jan, since many Condorcet-efficient methods do not
require all (n-1)*n/2 pairwise comparisons to be carried out. For
example, ROWS is Condorcet-efficient and only requires n-1
comparisons, which is by far less.

What is ROWS?

That was me. ROWS is "Random Order Winner Stays":
1. Sort all candidates into a random order.
2. Compare the first two and drop the defeated one. Compare the winner
of that comparison to the next candidate and again drop the defeated
one. Elect the winner of the last pairwise comparison. This method is
monotonic and Smith-efficient and requires only n-1 pairwise
comparisons. No method can find the Beats-All-Winner with fewer comparisons.

Of course, ROWS is not a good method, since it's not clone-proof, for
example. But like ROACC, we can modify it to meet clone-proofness by
using a more sophisticated order in step 1. For example, this order can
be from bottom to top on a random ballot (as in RBCC), which I would
call RBWS. Or the order could be from least to most approved (as in
TACC), which I would call TAWS. Or by processing X before Y when on the
first randomly drawn ballot on which the approval cutoff divides X and
Y, Y but not X is approved (as in RBACC), which I would call RBAWS. All
these methods are clone-proof, monotonic, and Smith-efficient, and need
only n-1 pairwise comparisons.

Yours, Jobst

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[Dave Ketchum]

Agreed you do not need (n-1)*n/2 pairwise comparisons BUT, seems to me
ROWS went too far:
It will happily and efficiently return the CW if there is one.
It does not know if there is a cycle, though the winner of the n-1
comparisons will, at least, be a cycle member.

Easiest I can think of is another n-1 comparisons to see if the apparent
winner is CW or only a cycle member and, if a member, keep going til you
have the complete cycle.

DWK

Post by Jobst Heitzig
Dear Alex!

Post by Alex Small
Naming something after a theorist is
fine in academic circles,

There is disagreement about this since it leads too often to the wrong
person getting the credit...

Post by Alex Small

I'm not so sure, Jan, since many Condorcet-efficient methods do not
require all (n-1)*n/2 pairwise comparisons to be carried out. For
example, ROWS is Condorcet-efficient and only requires n-1
comparisons, which is by far less.

What is ROWS?

1. Sort all candidates into a random order.
2. Compare the first two and drop the defeated one. Compare the winner
of that comparison to the next candidate and again drop the defeated
one. Elect the winner of the last pairwise comparison. This method is
monotonic and Smith-efficient and requires only n-1 pairwise
comparisons. No method can find the Beats-All-Winner with fewer comparisons.
Of course, ROWS is not a good method, since it's not clone-proof, for
example. But like ROACC, we can modify it to meet clone-proofness by
using a more sophisticated order in step 1. For example, this order can
be from bottom to top on a random ballot (as in RBCC), which I would
call RBWS. Or the order could be from least to most approved (as in
TACC), which I would call TAWS. Or by processing X before Y when on the
first randomly drawn ballot on which the approval cutoff divides X and
Y, Y but not X is approved (as in RBACC), which I would call RBAWS. All
these methods are clone-proof, monotonic, and Smith-efficient, and need
only n-1 pairwise comparisons.
Yours, Jobst

[Jobst Heitzig]

Dear Dave!

Post by Dave Ketchum
Agreed you do not need (n-1)*n/2 pairwise comparisons BUT, seems to me
It will happily and efficiently return the CW if there is one.
It does not know if there is a cycle, though the winner of the n-1
comparisons will, at least, be a cycle member.
Easiest I can think of is another n-1 comparisons to see if the apparent
winner is CW or only a cycle member and, if a member, keep going til you
have the complete cycle.

That's a nice suggestion for the "justification" step of the method but
it doesn't change the winner. Or did you mean to say that a method
should not elect a candidate unless it "knows" in which defeat cycles
s/he is?

Yours, Jobst

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[Dave Ketchum]

Post by Jobst Heitzig
Dear Dave!

Post by Dave Ketchum
Agreed you do not need (n-1)*n/2 pairwise comparisons BUT, seems to me
It will happily and efficiently return the CW if there is one.
It does not know if there is a cycle, though the winner of the n-1
comparisons will, at least, be a cycle member.
Easiest I can think of is another n-1 comparisons to see if the apparent
winner is CW or only a cycle member and, if a member, keep going til you
have the complete cycle.

That's a nice suggestion for the "justification" step of the method but
it doesn't change the winner. Or did you mean to say that a method
should not elect a candidate unless it "knows" in which defeat cycles
s/he is?
Yours, Jobst

I suspect I did not emphasize enough so, assume we are doing only n-1,
A should win, the cycle is A>B>C>A, and C is the first member found (does
not matter if there is a D, for such will be discarded as soon as we have
a cycle member).
PROVIDED B is the next cycle member, it will take over and
recognize A as final winner when found.
BUT IF A is the next cycle member, C will reject it and accept B as
final winner when that is found.

[Dave Ketchum]

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
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.

------------------------------------------------------------------------
--
***@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.

[Paul Kislanko]

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-***@electorama.com
[mailto:election-methods-electorama.com-***@electorama.com] On Behalf Of
Dave Ketchum
Sent: Sunday, March 13, 2005 8:31 PM
To: 'Alex Small'; election-methods-***@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
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.


_____

[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 <***@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-***@electorama.com [mailto:election-methods-electorama.com-***@electorama.com] On Behalf Of Dave Ketchum
Sent: Sunday, March 13, 2005 8:31 PM
To: 'Alex Small'; election-methods-***@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
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.


---------------------------------
--
***@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.




---------------------------------
Do you Yahoo!?
Yahoo! Small Business - Try our new resources site!

[Paul Kislanko]

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...)


_____

From: election-methods-electorama.com-***@electorama.com
[mailto:election-methods-electorama.com-***@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 <***@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-***@electorama.com
[mailto:election-methods-electorama.com-***@electorama.com] On Behalf Of
Dave Ketchum
Sent: Sunday, March 13, 2005 8:31 PM
To: 'Alex Small'; election-methods-***@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
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.


_____
--
***@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.





_____

Do you Yahoo!?
Yahoo! Small Business - Try
<http://us.rd.yahoo.com/evt=31637/*http://smallbusiness.yahoo.com/resources/

Post by Alex Small
our new resources site!

[Dave Ketchum]

Last posts on Round Robins discussed what happens on sports Round Robins
if there is a tie for first - thinking that their solution might be useful
in voting:
Tournament sponsor BETTER have set up a plan ahead of time - of
which I found some via Yahoo!
What follows explains why I see us continuing to develop our
elections plans, rather than copying.

After looking, I LIKE sports Round Robins tournaments as a base BUT, EACH
VOTER ranks candidates and conducts a tournament among all candidates, in
which wins are based on:
Both candidates ranked - ranking selects winner.
One candidate ranked - and wins for loser not playing.
Neither candidate ranked - so no game and no score.

At end of season (election), all tournament results are added together to
determine season winner. It is here that cycles can occur.

The only problem with IRR is when there is no Condorcet winner.
But as far as I know, elections with no CW are totally theoretical.
For them to happen would require voters to seriously confused
about their preferences for different candidates.

I see no confusion in the following:
A and B have serious believers, so that their 65>35 split is expected.
C is much like B, but says SOMETHING in a way that sells a majority
of B backers 33>32.
35 A
32 B
33 C>B

A>C 35>33
C>B 33>32
B>A 65>35

Two reasons for providing for cycles:
They can happen, so be prepared.
If not prepared, you tempt fate too much, and better expect them.

[Gervase Lam]

Date: Sun, 13 Mar 2005 19:17:22 -0800 (PST)
From: Alex Small
Subject: RE: [EM] Round Robins
Say we have a round robin tournament
between soccer teams from USC, UCLA, and UCSB. 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?

For each team, the winning margins versus the other teams are added up,
with losses being denoted by a negative winning margin. The team with
highest net total (known as "goal difference" in soccer) is the winner.

In this case UCLA is the winner because:

UCLA +2 = -1 + 3
UCSB -1 = -3 + 2
USC -1 = +1 - 2

For team sports in general, I think the best analogy is "recursive"
Copeland. For any teams who are given the same Copeland score, Copeland
is then run again on the tied teams.

However, soccer or ice/field hockey don't use Copeland. Instead, a number
of points are given for wins and a lower number of points for ties.
Nevertheless, soccer and hockey fans should easily understand Copeland.

Thanks,
Gervase.

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