# Re: Traviesties in the methods Europeans Use for Pairing Algorithms

> jazzerciser@xxxxxxxxxxx (-) wrote:
>> By pairing a player with one loss -THREE- ranks apart while
>> pairing another player with one loss -TWO- ranks apart or -ONE-
>> rank apart, you have not achieved "essentially equal rights" among
>> all players in the top group.

Robert Jasiek <jasiek@xxxxxxxx> wrote:
> Ok, I have to be more specific about the axiom of equal rights: It
> refers to the assumption of equal tournament-playing-strength at
> the beginning of the tournament and thus the assumption of equal
> probabilities to win the tournament. Within the tournament, the top
> players are not distinguished by their
> outside-of-the-tournament-playing-strengths. There is no equal right to
> play against opponents so as to get equal sums of rank (or rating) differences.

However awful this may sound, the reality of tournaments is
that not all players have an equal probability to win the tournament.
Without prejudicing the "dark horse players" (long-shot come from
behind candidates) tournament directors can take into account the
unbalanced probabilities when setting up pairing schemes that more
rapidly converge upon the actual maximum likelihood. Whatever the
means, every tournament goal is to match up with reality precisely,
and as soon as possible. Use of initial ranks in the effort to match
up reality is not unfair if initial ranks are not too wildly inaccurate.
The probability of accuracy or inaccuracy in initial ranks is itself
a setting in the convergence heuristic for a maximum likelihood.

In retrospect, all of the players who had scored a 5-1 over six
rounds should be treated identically when arriving at round 7.
If rank mismatching in round 4, does not correspond to mismatching
in round 3, or in round 5, then all players were not treated equally.
The pairing algorithm cannot award preferance to the significance of
certain early rounds over later rounds. Each round in a tournament
must contribute an equal amount of decision-making factors to the
final results of that tournament. Equal rights for players also means
that effect of each round is identical to effect of any other round.
In practice, later rounds in a tournament tend to more closely match
up the players than the early rounds. A tournament is regarded as
more successful if the early rounds are more closely matched as well.
The disequal treatment among players is sometimes not discovered
until later in the tournament, as Accelerat also converges upon reality.
Good pairing algorithms will attempt to correct past disequal treatment,
as Barry Phease has mentioned. Once getting his MMS grouping and
number of times players have been paired up or down, Barry then pairs
down those who have the -highest- SOS, other things being equal.

> The same can be observed in a KO where the first round's pairing is
> decided randomly.

We reject random pairing out of hand, as competitively uninformative.
Pairing is a strategy for most rapid convergence upon maximum likelihood.
Random methods could just as well diverge as converge upon that ML.

>> Later this inequality is manifested when
>> conducting tiebreaker decisions based upon the flawed logic of SOS.

> Within the tournament, SOS depends on MMS, but not on ranks.

We have not yet heard the definition for MMS.

>> What you say is now considered acceptable will, upon much closer
>> examination, be in the future considered unacceptable even in Europe.

More like a prognostication, or even a prophecy. Want to bet ?

>> One (unstated) set of tournament axioms should instead include:
>>
>> (1) - Define "rank" as initial rank and adjusted by an Accelerat
>> (2) - Define "anomalies" as pairings among rank differences
>> (3) - Pairings for each round should minimize anomalies
>> (4) - Pairings for the entire tournament should minimize the
>> sum (or sum of squares) of anomalies over all rounds
>> (5) - Playoffs should occur among players of nearby ranks

> This seems to be possible as a (part of a) set of axioms. How
> different sets of axioms compare to each other and in which context of
> more fundamental set of axioms has not been researched so far.

>> (6) - Tiebreaker subpoints are accorded based upon SODOS
>> and then SODOSODOS, and then SODOSODOSODOS ...
>> which is a recursive entry for each successive round.

> What do you like about SODOS and its refinements?!

We were measuring "defeat chains" to resolve round robin ties.

>> The spectre of anomalies in the pairing scheme is far worse
>> than the problem of pairing twice during a (large) tournament.

> Opinions differ on this.

Correct opinions will not differ significantly.

>> Good tournament software first finds
>> examples of (3), then projects likely outcomes of hypothetical
>> round results to examine possible occurances for all successive
>> rounds. By preliminary searching of the tournament space an
>> estimated measure (4), for rounds N+1, N+2, N+3 ..., may be
>> obtained prior to actual publishing of the pairings for Round N.
>> Viable alternatives for Round N pairings may thereby be examined
>> in order to achieve both goals (3) and (4). This converges upon
>> the axiomatic ideal which is the maximum likelihood estimate for the
>> most probable tournament winner, followed by a maximum likelihood
>> estimate for next most probable winner (second place), and so forth.

> This is a possible amendment of axioms for a tournament. How it
> compares to other axioms needs to be researched. Similar axioms are
> possible if maximum likelihood is not applied to ratings but solely to
> MMS within the tournament.
>
> In some tournaments with few rounds and SOS used as the first
> tiebreaker for the final result list, such an approach has been used
> by a few tournament organizers in the last rounds to allow all still
> top MMS players equal principle chances to win the tournament. More
> research is required to compare such to tournaments where the final
> results are not distinguished by tiebreakers.

SOS is a dubious tiebreaker. In some circumstances SOS can
be entirely inverted to indicate the opposite of what was assumed.

>> ... the "window catchment" of this "MacMahon" was set too wide.
>> [ ... ]
>> at the end of the tournament, there were no playoffs between
>> players #3-#4, #5-#6, #7-#8.

> How to evaluate that and compare it to other axiomatic contexts?
> This is a serious drawback of MacMahon tournaments with too few
> rounds indeed. A, say, 10 rounds tournament does much better in
> approaching a round-robin for the final few top players.

It appears that TD software for the Toyota-Pandanet Brno 2005
was expecting a ten round tournament but called to a halt at six.

>> Once it was clearly understood that maximum likelihood
>> was the original goal,

> Was not approaching round-robin for the final top few players an
> important goal?

Even in a round-robin tournament there can be round-robin ties.
The tiebreaker criteria needs to inform us of maximum likelihood. An
SOS, or SOSOS, measure is not as closely correlated to the maximum
likelihood as the "sum of defeat chains" (up to N-1 deep, where N is
the number of rounds in the tournament), using SODOS^(N-1) ...

- regards
- jb

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