Recursive tie-breaks for chess tournaments
(by Miguel Brozos-Vázquez, Marco Antonio Campo-Cabana, José Carlos Díaz-Ramos, and Julio González-Díaz)
Good properties of the recursive methods Shortcomings of the recursive methods Specific good properties of the recursive performance Specific shortcomings of the recursive performance Common problems of most tie-breaking rules. Non-ideal tournaments: byes, forfeits, withdrawals,... Our recommendation: ARPO system (average recursive performance of opponents) without the best and the worst
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 tie-break. 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 tie-breaks 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,...
top
Shortcomings of the recursive methods Lot of people criticizes the fact that a computer is needed in order to calculate the tie-break in the tournament. In the same lines, it is also criticized that it will be difficult for the players to verify (and understand) the tie-breaks 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 tie-break.
top
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.
top
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 tie-breaks 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.
top
Common problems of most tie-breaking rules. Non-ideal tournaments: byes, forfeits, withdrawals,... As we have already said in the previous point. The tie-breaking 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 non-ideal; there are byes, withdrawals,... that can affect the behavior of most tie-breaking 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 tie-break. 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 tie-break, 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 tie-break 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.
top
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.
top