Recursive tie-breaks for chess tournaments
(by Miguel Brozos-Vázquez, Marco Antonio Campo-Cabana, José Carlos Díaz-Ramos, and Julio González-Díaz)
The recursive performance and the ARPO variations are available in Swiss Manager since June 2010


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 tie-breaking rule for tournaments in which all the players are rated.

Problems of the performance as a tie-breaking 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 1-iterated performance of the players: in the previous example, the 1-iterated 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 2-iterated performance (for Topalov, the average of the 1-iterated performances of his opponents plus 125 points), 3-iterated... 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 tie-breaking rule: The recursive performance is a good measure of the strength of the players during the tournament. Hence, it can be used as a tie-breaking 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 tie-breaking rules is called ARPO systems (Average Recursive Performance of Opponents).


  • 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.



The buchholz: The buchholz is a tie-breaking 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 tie-breaking 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).