The Problem with Tiebreakers

As an Arbiter and TD, I have committed a lot of time to different tiebreakers, since I wanted to find the fairest tiebreaker. However, I found out that there are flaws in almost every system. In this list I am writing about these problems. Some problems are dependent on pairing systems, and some problems are repetitive. I also believe that a person, that is lower rated in the beginning should not get a (dis-)advantage to a higher rated player, so the argument “But it’s more impressive for the lower rated player to get the same result as the higher rated player so the lower rated one should be on top” doesn’t count. Also note, that most tiebreakers are discussed in a Swiss-system-competiotion. (I won't discuss the following tiebreakers: Fore Bucholz, Tiebreakers specific to Team events and Koya-System)

ARO (Average Rating of Opponents)
Explanation: This one is self-explanatory. You take the rating of the player’s opponents and get the average.

The problem with this one is that a player cannot control it. This tiebreaker is completely based on the pairings and a number that represents the opponent’s strength. However, the rating can be misleading. Imagene you have two players: Player A Rating: 1600 and Player B Rating: 1800 now they play against players C & D  (Rating for those two is irrelevant)

Player A - Player C; Result: 0-1

Player B - Player D; Result: 0-1

This of course is a very simple representation of the situation.

Now you could say: "Ok Player D has played the stronger opponent so his win is worth more" However what I didn't say is, that Player A is 10, just started playing and their rating is 6 months old and, in this time, they improved to a level of 2000. Player B meanwhile is 80 and had a decline in rating and now plays at a level of 1700.

This means that Player C had a stronger opponent, but the ARO does not account for that. Another issue, at least at the lower levels, is players without a rating. Those players get a pseudo-rating. This rating once again is not representative of the actual strength of the player.

The other thing is that players cannot influence this tiebreaker. Sure they can win and may get stronger opponents so the tiebreaker is higher but even there is the chance of  getting a lower rated opponent than someone else for example. 

  For this "simulation" we will look at player A & B. I always let the stronger player win, except in the game Player A - Player B Result: 0.5-0.5. So, both players played the best possible tournament while still tying. However, Player B still won on tiebreaker, just because he had stronger opponents, which Player A could not influence.

Buchholz (cut-1, cut-2, etc.) (BH)/Sum of Buchholz of Opponents(SBO)

Explanation: The sum of the scores of each of the opponents of a player. (cut-1, cuts out the opponent with the lowest score and cut-2 cuts out the higest and lowest or the two lowest scores)

Ah yes, as a German (Idk if other countries use it as much as us) I have quiet the experience with this one and even though I won more prizes due to Buchholz than I lost, I hate this one, since I think it is no good system to break ties. The main Issue I have with this one is, the rule for players that drop out. Since it would be unfair to give their opponents 0 BH for the following rounds every one of their opponents gets 0.5 points/round that the player misses. Sometimes this is just unrealistic. Picture the following scenario: Player A drops out after 5/9 Rounds because he has just 1 point. In game 5 he played against Player B (also 1 Point) but not against Player C (also 1 Point) who played against Player D (also 1 Point). Now player B gets 1+4*0.5=3 Points form Player A. Player D keeps on playing and scores a total of 2.5 Points. So, Player C gets 2.5 Points. In the unlikely scenario that they have otherwise the same BH-score this seems unfair to Player C since he did not get those save .5 points/round. (same with games where one went unpaired: you get your own score at the time +0.5 points per following round). Another thing is again due to the pairings. Picture this scenario: 

Last round of a 5 Round Tournament:

Pre Round Tabel

Opponents Player A: Player C (1P), Player D (2P), Player E (3P), Player F (3P)

Opponents Player B: Player G (1P), Player H (2P), Player I (3P), Player J (3P)

relevant pairings for Round 5:

Player A 0.5-0.5 Player B (Both get 4.5 Buchholz from each other)

Player C-H play other opponents. 

Player I - J 

From these pairings arises a problem since the opponents of player A don't play each other he can get 4 BH from this round. But since Player I and Player J Play each other, Player B can get 3 BH at most). So he has a disadvantage. 

The SBO tries to fix these issues by taaking the added Buchholz of the opponents, but it just depens the issue becaus all the others remain but the chance of two opponents of a players opponents playing eachother, thus lowering the possible achivable SBO-points, is higher.

Direct Encounter (DE)

Explanation: If all the tied players have met each other, the sum of points from these encounters is used to produce separate standings. The player with the highest score is ranked first among the
tied players, and the others follow according to the separatestandings.

If the tied players have not played all the games against each other, but one of them is bound to be at the top of the separate standings whatever the outcome of the missing games, that player is ranked first among the tied players - the same applies to the second rank when the first is assigned this way; and so on.

Here it is a bit more difficult to find criticism. I personally think this is one of the best to use. But there are still problematic things like if there are too many players to declare a "winner". Also, the black, white debate. If two players are tied and player A (w) beat player B (b), B may argue, with my full support, that he had a disadvantage due to playing black in that game. Also, if they drew their game the tiebreaker is useless (Unless you use the "Norway chess" Scoring system).

Games one elected to play (GE)

Explanation: The number of rounds diminished by the number of half-point-byes, zero-point-byes or forfeit loss-es that a player had in the tournament. 

This one does make sense. However, it is a restrictive one and I discourage you from using it in Swiss Tournaments. People who have a good reason to take a bye for a round have a massive disadvantage, however if you want to use it, at max use it as 3rd or 4th tiebreaker. However, one change I would like to see is a version where you count number of half-point-byes, zero-point-byes but not forfeit losses. This way players who forfeit get punished. 

Number of games with/won with Black (BPG/BWG)

Explanation: Each game played over the board with the Black pieces counts
one (unplayed games do not count)./Each win achieved over the board with the Black pieces counts
one (unplayed games do not count)

Now these are just the worst ones in Swiss Tournaments and Round-Robin tournaments with 1+2n rotations. Because there is just no way for players to change which color they have more often so they either have just a clear advantage or they have more opportunities to score. And even if they have as many black games as white games it don't think it is a good tiebreaker since it just counts the performance with one color so 4 wins with black are worth more than 2 wins with white and black each.

Number of wins (WIN)

Explanation: Each win, including unplayed wins, counts one.

Here is something to discuss what is worth more: one win or two draws 

this i think is the most important thing with this Tiebreaker. There is not much else I can say. 

Tournament Performance Rating (TPR) & Perfect Tournament Performance (PTP)

Explanations:

• TPR: Computed adding to ARO a number (called rating difference - it may be negative) resulting from the conversion of the achieved fractional score (the number of points divided by the number of opponents - excluding any results from unplayed games) as described in Article 8.1 of the FIDE Rating
  Regulations
• PTP: This is the lowest rating that a player should have in order to receive a rating variation of zero after meeting all the opponents faced during the tournament. The full rating scale is used in this computation (i.e. no ±400 cut)

I decided to put them together because they face very similar issues. Of them is that as in ARO the rating of opponents sets a limit to how high your tiebreaker-score can go so in the direct encounter the lower rated opponent has a disadvantage. Also, the inclusion of pseudo-ratings. One thing I would add is, that even with a 0.1 difference the one with the higher score is put on top, so there should be a range added where players still count as tied (Same thing with the Average Performance Rating of Opponents).

Sonneborn-Berger

Explanation: The sum of the scores of the opponents a player has defeated (including by forfeit in round-robin tour-naments) and half the scores of the opponents with whom he has drawn. When a
player meets the same opponent more than once (e.g. in a double round-robin or in a double Swiss), the games are counted together.

This one faces similar problems as the Bucholz so read that and transfer critic over here. However, I must say that it makes a better job at being fair, due to the combination with the players own record so he can influence the tiebreaker.

(Sum of) Progressive scores (PS)

Explanation: After each round a player has a certain tournament score. This tie-break is computed adding the scores of the player at the end of each round.

Example of 7 Round Tournamet

This is one of my favorites. However, it is still flawed. Since in Swiss system tournament the first half and the second half play against each other players in the second half appear to have a more difficult opponent. But the reason I still like this one is that it shows who played higher. For example: Before the last round Player A has 5 points and Player B has 4 points. Therefore, player A normally gets an opponent that also has 5 points and player B gets one that has 4 points. Player A loses, player B wins. I would argue that player A should be ranked higher, because he has played an opponent that had a higher score and should therefore be seen as stronger independent of the rating of any player.