RECURSIVE PERFORMANCE
The performance:
Recall, by means of an example,
the functioning of the performance. Suppose
we have a tournament in which Topalov scores 6
points in 9 rounds. According to the FIDE tables,
this means that Topalov has played at a level 125 points above his opponents. Hence, his performance
will be the average elo of his opponents plus 125 points. According to FIDE, the performance is a good
measure of the strength of the players in the
tournament. Thus, proposing it as a tiebreaking
rule for tournaments in which all the players are
rated.
Problems of the performance as a tiebreaking rule:

It depends too much
on the initial ratings of the players. This is a
problem, since the strength exhibited by a
player in a tournament might be very different
from his elo.

All the players of the tournament must be rated.
Recursive performance: The recursive
performance follows the same idea as the performance,
but does not have any of the aforementioned problems.
In general, the performance of a player is a better
indicator of his strength during the tournament than
his own elo. This suggests calculating the 1iterated
performance of the players: in the previous
example, the 1iterated performance of Topalov would
be calculated as the average of the performances of
his opponents plus 125 points. This will be a better
measure of Topalov's real performance in the
tournament than the standard performance. With this
idea in mind we can define the 2iterated
performance (for Topalov, the average of the 1iterated
performances of his opponents plus 125 points), 3iterated...
The recursive performance is just the limit of this
process; that is, the infinitely iterated
performance.
Advantages of the recursive performance:

It does not depend
on the initial elos.

If there are unrated
players, we can asigns each of them an
arbitrarily chosen rating. This choice does not
affect the final ranking proposed by the
recursive performance.
The recursive performance as a tiebreaking rule:
The recursive performance is a good measure of
the strength of the players during the tournament.
Hence, it can be used as a tiebreaking rule in the
same way that the buchholz uses the scores. For each
player we can calculate the average of the recursive
performances of his opponents, or the average
excluding the worst, or the best and the worst...
This family of tiebreaking rules is called ARPO
systems (Average Recursive
Performance of Opponents).
Comments:

The unplayed games are not taken into account to
calculate the recursive performance.

When using systems like ARPO without the worst,
the two worst... each unplayed game counts as
one "worst rival".
¿How to verify that the recursive performance
is correctly computed? Though the calculations
are difficult, verification is easy. It suffices to
calculate the performance of the players but using
the recursive performances instead of the elos. By
doing this, the difference between the recursive
performance and the new performance will be the same
for all the players. If this happens, the recursive
performance is correctly computed.

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RECURSIVE BUCHHOLZ
The
buchholz:
The buchholz is a tiebreaking rule that
consists of calculating, for each players, the sum
of the points of his opponents and then order the
players in accordance with these sums.
Problems of the buchholz
as a tiebreaking rule:

It might be the case
that in a tournament two players score 6 points;
one of them after having been leading the
tournament and the other one after winning
several rounds in a row at the end of the
championship. In this case, it seems clear that
the 6 points of the first player should be worth
more than the ones of the latter. Yet, buchholz
system cannot distinguish between these two
players.

The buchholz system
is very sensitive to withdrawals, byes,... being
this specially important if we take into account
that that the latter are becoming very popular
in international opens.
A first improvement: In order to solve the
second of the above problems, we might just work
with the average of the points of the opponents
instead of using their sum. By doing this, there is
no need to worry about the corrections for byes,
withdrawals,... as it happens with the recursive
performance, unplayed games are not taken into
account.
Recursive buchholz: The idea of the recursive
buchholz is, essentially, to iterate the buchholz to overcome the other problem
mentioned above (remarkably, this is done in a similar way as with the recursive
performance: at each iteration not only the points of the opponents of each
player are taken into account, but also his own points). 
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