






Recursive tiebreaks for chess tournaments 
(by Miguel BrozosVázquez,
Marco Antonio CampoCabana, José Carlos DíazRamos, and Julio
GonzálezDíaz) 


Good properties of
the recursive methods
 The recursive methods use
all the information of the tournament to break the ties.
That is, they do not only use the information concerning the
opponents of the tied players (as performance and buchholz
do), but also the information about the opponents of the
opponents, about the opponents of the opponents of the
opponents...
 Because of the previous
point it is difficult that to players remain tied after
applying a recursive tiebreak. The latter is very unlikely
to happen unless the tied players have exactly the same
opponents (and in such a case it seems reasonable that our
tiebreaks do not break the tie).
 The recursive methods, as
far as we conceive them, do not take into account unplayed
games. Hence, there is no need to take arbitrary decisions
concerning byes, withdrawals,...

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Shortcomings of the
recursive methods
 Lot of people criticizes the
fact that a computer is needed in order to calculate the tiebreak
in the tournament.
 In the same lines, it is
also criticized that it will be difficult for the players to
verify (and understand) the tiebreaks at the end of the
tournament.
 Up to 4 or 5 rounds might be
needed for the methods to be convergent. Hence, intermediate
standings prior to that round cannot incorporate the tiebreak.

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Specific good properties of the recursive performance
 It does not depend on
the initial elos of the players.
 It is not needed that all
the participants in the tournament are rated. If there is an
unrated player we can assign him an arbitrarily chosen
rating. The final ranking proposed by the recursive
performance does not depend on this choice.
 It inherits the good
properties of the formula to calculate the elo.

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Specific shortcomings of the recursive performance
 It inherits the bad
properties of the formula to calculate the elo.
 As a particular case of the
previous point, the recursive performance is very sensitive
with respect to extreme results. That is, if there are
players which end up the tournament with 0% or 100% or the
points, then this players might influence too much the tiebreaks
of the rest of the players of the tournament (unfavoring or
favoring decisively the players that have played against
them). Also the buchholz system and the performance exhibit
this problem, but the recursive performance is even more
sensitive to it.
 Another possible
inconvenient is that, since the performance is very
sensitive to the presence of players with 0% or 100% of the
points, as far as there are many players like these in the
tournament, the recursive performance varies a lot from
round to round. That is, the recursive performance might
exhibit a lot of variability as far the number of rounds is
small with respect to the total number of players.

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Common problems of
most tiebreaking rules.
Nonideal tournaments: byes, forfeits, withdrawals,...
 As we have
already said in the previous point. The tiebreaking
rules that are based in measures of the strength
exhibited by the opponents of each players share
the problem of being sensitive to extreme
results of some players.
 On the other
hand, most tournaments are nonideal; there are
byes, withdrawals,... that can affect the
behavior of most tiebreaking rules. One must be
specially careful in
tournaments that grant byes of 0.5 points because the
recipients of such byes might be favored by the tiebreak.
Suppose that we have a tournament with 9 rounds in which a
player receives 2 byes in the first two rounds and then
scores 6 points in the remaining 7 games. Suppose as well
that this player ends up tied for the first place. When
calculating the tiebreak, this player will have 6 points
out of 7 whereas the remaining tied players would probably have 6.5
out of 9; the former being favored when applying the elo
formula to calculate the performances. Indeed, as a extreme
situation, even a player that looses the first
game of the tournament without showing up can
mend his initial mistake, finish tied for the
first place and win in the tiebreak because of
his higher percentage of points in the played
games.
 One extreme case
even more worrying is the following. Consider a
tournament in which players A and B reach the
9th and last round of the tournament with 8 and
7 points respectively. No other player has a
chance of winning the tournament. In this case,
player A can almost ensure for himself the
victory by not showing up in the last round. By
doing so, the recursive performance of A would
be computed as 8 points out of 8 games (instead
of 8 out of 9 if he had played the last game and
lost it). Hence, he might enormously benefit by
not playing the last game. One way to avoid
these speculations (which, to a lesser extent,
can also appear with other tiebreaking rules) is
to take all the last round games as played even
if there are players that do not show up.

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Our
recommendation:
ARPO system (average recursive performance of opponents)
without the best and the worst.
 When using the average of
the recursive performance of the opponents instead of the
recursive performance itself, the undesirable behavior with
respect to the players that have received byes or that do
not show up in a given round disappears.
 Besides, when removing the
worst and the best opponents from the average, we also
ensure that the "extreme players" do not influence
decisively the final standings.

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