Hi,

Here's a problem for any chess players out there who are also computer
scientists/mathematicians, what is the optimal number of rounds in a
chess tournament with respect to the number of players?

Obviously, the best case would be number of players-1 rounds. But
suppose we had to limit the number of rounds to less than half the
number players, is it always the case that the most number of rounds
possible the optimal number?

(Harialbth) writes:

Hi,

Here's a problem for any chess players out there who are also computer
scientists/mathematicians, what is the optimal number of rounds in a
chess tournament with respect to the number of players?

For what mix of players? The answer is different for 9 players of equal
strength and 1000 players with strengths ranging from beginner to expert.


Obviously, the best case would be number of players-1 rounds.

This is not obvious to me. I just finished running the K-1 section of
the National Elementary. We had nearly 200 players. I doubt we would
have survived 199 rounds. Similarly, I doubt that we would have done a
better job of selecting a winner.

Which raises another question - what do you need as the result of your
chess tournament? Do you want a single clear winner, and nothing else?
do you want to find the best 10 players (in order)? Do you want to rank
order *every* player in the event. The answer to your question depends
on your answer to this question!

But
suppose we had to limit the number of rounds to less than half the
number players, is it always the case that the most number of rounds
possible the optimal number?

No. If you want a clear winner, the ideal method is the one used in the
old Western Open - stop the tournament as soon as there is a clear
leader.

A Swiss event works best when the number of rounds is log2(#players).
If you have fewer rounds, you run the risk of having two, or more,
players tied with perfect scores. If you have more rounds, you run the
risk of seeing the leader stumble and fall back into a many-way tie.

Playing a number of rounds equal to one half the number of players
doesn't strike me as optimal in *any* situation (although I'm sure
someone can invent one!). But...let me try...

If I had n players (and some a priori information, such as ratings), and
wanted to run a total of R rounds, I might run a ratings-guided Swiss
first. After r log2(n) rounds, enter the top
R-r-1 players into a round robin.

Example: 35 players and you want to run only 15 rounds. You might run a
4 round Swiss and then an 11 round RR (with 12 players qualifying from
the Swiss. Note that I skewed this so that you would *not*
(necessarily) have a clear winner from the Swiss.

A possible objection to this scheme is that you will (probably) need to
use tie-breakers to select the bottom players in the RR. The way to
spin this is to view the score group that "just barely qualifies" as
"extra" players - so the ones chosen on tie-break are getting a good
deal (as opposed to the idea that the ones losing out on tiebreak are
getting a rotten deal).

Another scheme in common use is to run qualifying RR's by (randomly or
rating-guided) sections. Again, with 35 players you could divide the
field into 6 fields of 6 players each and play 5 rounds. Then, take the
top 2 players in each section (breaking ties might be a problem here...)
and play an 11 round RR (oops...that's 16 rounds, not 15)

To get fewer than 15 rounds, create 9 sections of 4 players each and
play a 3RR to qualify 1 player from each section plus one "best 2nd
place" to form a field of 10 for a 9RR. 12 rounds total.

(the 3-person section probably has a slight advantage in that it's
easier to win...so it's probably best to declare that the "best 2nd
place" canNOT come from that section)


And so on...


--
Kenneth Sloan
Computer and Information Sciences (205) 934-2213
University of Alabama at Birmingham FAX (205) 934-5473
Birmingham, AL 35294-1170 http://www.cis.uab.edu/sloan/

Harialbth wrote:
Obviously, the best case would be number of players-1 rounds.

It would be better to have twice that many rounds to avoid piece colour
problems.


Dave.

--
David Richerby Cyber-Monk (TM): it's like a man of
www.chiark.greenend.org.uk/~davidr/ God that exists only in your computer!

Obviously, the best case would be number of players-1 rounds.


Leaving aside unscientific factors such as whether you can "survive"
199 rounds, it seems obvious that an all play all tournament is best
because everybody has had the same opposition so there's no element of
chance.

On 9 Apr 2004 23:36:44 -0700, (Harialbth) wrote:

Obviously, the best case would be number of players-1 rounds.

Leaving aside unscientific factors such as whether you can "survive"
199 rounds, it seems obvious that an all play all tournament is best
because everybody has had the same opposition so there's no element of
chance.

All play all may be better but there's still a lot of chance, because
the conditions aren't necessarily equal for all games.

For example, you get a severe one-day stomach upset on a day when
you're playing your main competitor, someone just a few rating points
worse. You lose.

Had you got the same bug on the day when you were playing one of the
competitors whom you outrank by 200 points, you would have beaten him
anyway.

Or, you get a strong opponent in the first round, where he may be more
satisfied with a draw, rather than the last when he needs a win to
place.

Harialbth wrote:

Obviously, the best case would be number of players-1 rounds.


Leaving aside unscientific factors such as whether you can "survive"
199 rounds, it seems obvious that an all play all tournament is best
because everybody has had the same opposition so there's no element of
chance.

There most certainly is an element of chance.

In the 1964 Amsterdam Interzonal there was what was called "The Russian
wall": owing to the way the tournament numbers were set non-USSR players
played "The Russians" (i.e. Soviets, not necessarily Russians) one after
another in successive rounds. It was a distinct disadvantage to run into
this wall early in the tournament, as one was likely to drop many points in
contests with such a formidable group of players; conversely playing them
at the end was very beneficial as FIDE rules prevented all "the Russians"
from qualifying for the Candidates, which made them far more nervous.

If it is obvious that an all play all is superior, then it should be
possible to demonstrate this is a simple easily understood way, preferably
not using the one word proof "obvious".


Regards,

Simon.

If it is obvious that an all play all is superior, then it should be
possible to demonstrate this is a simple easily understood way, preferably
not using the one word proof "obvious".


A stomach upset or nervous opponents can be part of any competition
and not within the parameters of a good tournament algorithm. You can
be the best athlete in the world and have a stomach upset in the
olympic finals, does that make the olympic system flawed??

If A, B and C are in competition, to determine the best player isn't
it "obviously " better if A plays B, B plays C, and C plays A. Than
just let A play the winner of B-C and declare the winner of that game
the champion.

Harialbth wrote:

If A, B and C are in competition, to determine the best player isn't
it "obviously " better if A plays B, B plays C, and C plays A.

At this stage they have played two games each.

Than
just let A play the winner of B-C and declare the winner of that game
the champion.

Now A only has to win one game, whereas B or C must win two.

Why does A get such an advantage?


Consider a competition of twenty players say, where two players are miles
better than the rest: suppose all we are interested in is the winner and we
want him to be the "best player", then an all-play-all would be rather
tiresome, it could lead to the winner being the best "bunny exterminator"
rather than the "best player". Why is this better than a five round swiss,
say, when dropping a half point against a "bunny" may not be too
disastrous. In the all-play-all format the "best player" may take one or
two chances too many and have the nearly impossible task of clawing back
two points vis-a-vis the second ranked player.

Consider also an alternative format of splitting into two groups, with one
favourite in each group, the winners of the groups facing off in the final;
why would this be such a poor way of tackling the problem? The final could
be a mini-match, but overall there could still be fewer games played.

That All-play-all is **always** the optimal format doesn't seem obvious to
me.



Regards,

Simon.

chapman Billy wrote in message
...
Consider a competition of twenty players say, where two players are miles
better than the rest: suppose all we are interested in is the winner and we
want him to be the "best player", then an all-play-all would be rather
tiresome, it could lead to the winner being the best "bunny exterminator"
rather than the "best player". Why is this better than a five round swiss,
say, when dropping a half point against a "bunny" may not be too
disastrous. In the all-play-all format the "best player" may take one or
two chances too many and have the nearly impossible task of clawing back
two points vis-a-vis the second ranked player....

Some persons have contended that Paul Keres 'deserved' to have won the
1959 Candidates Tournament ahead of Mikhail Tal, who had scored only
1-3 against Keres, but who had crushed Fischer (4-0), Benko (3.5-0.5),
Gligoric (3.5-0.5), and Olafsson (3.5-0.5).

Should Tal have been invited to the Watership Down Memorial Tournament? :-)

--Nick

On 11 Apr 2004 10:13:15 -0700, (Harialbth) wrote:

If A, B and C are in competition, to determine the best player isn't
it "obviously " better if A plays B, B plays C, and C plays A. Than
just let A play the winner of B-C and declare the winner of that game
the champion.

Yes. The first is a round robin, the second is a knockout. Knockouts
don't make sense when you have a large number of players in one
location. Knockouts are good when you have a limited number of games
that can be played at once, i.e. a tennis match. But with chess,
everyone can play, so there are better ways than knockouts.

A round robin, double round robin, or quad round robin is good for a
relatively small number of players. For a large number of players, a
Swiss is good. If the number of players is much larger than the
optimum for the number of rounds, you can break it up into sections.
Or you can break it up into small round robins.

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