Chapter 4 — Guide Every FIDE tiebreak system explained with step-by-step examples: Buchholz, Sonneborn-Berger, ARO, APRO and 24 more. Free interactive tiebreak calculator included.
Two players finish a tournament on the same number of points. Who wins? This is where tiebreak systems come in. ChessPairings.org supports 28 different tiebreak systems — the most complete implementation available in free tournament software. This chapter explains each major system with real numbers so you can choose the right tiebreaks for your event and understand exactly how they work.
Tiebreaks are a necessary compromise. In a perfect world, every tie would be resolved by additional games — but that's rarely practical in a multi-round Swiss event. So we use mathematical criteria based on the games already played to separate players with equal scores.
The controversy? Every tiebreak system has weaknesses. Buchholz rewards players who faced strong opponents, but you can't choose who you play. Sonneborn-Berger can wildly favor players who were lucky in whom they beat. Direct encounter is the fairest criterion in theory — but it only works when the tied players actually faced each other.
The key insight: no tiebreak system is perfectly fair. The goal is to choose systems that are transparent, announced in advance, and appropriate for your tournament type.
Tiebreak criteria must be announced before round 1 begins. Changing tiebreaks mid-tournament is not allowed under FIDE regulations. If you forget to announce them, the Chief Arbiter must apply the FIDE default order for that tournament type.
The August 2024 revision of C.07 introduced some important changes worth knowing:
Unplayed games now score ½ point for tiebreak purposes (not the actual result). If a player receives a bye or a forfeit win, their opponents treat that game as if the player scored ½ when calculating Buchholz and similar systems. This reduces the distortion caused by byes and forfeits.
ARO and APRO (Average Rating of Opponents, and its performance-based variant) were formally codified as recognized FIDE tiebreak systems in the 2024 revision.
The default recommended order for Swiss tournaments (when not specified in regulations) is now:
| # | Tiebreak system | Notes |
|---|---|---|
| 1 | Buchholz Cut-1 | Removes the weakest opponent score |
| 2 | Buchholz | Sum of all opponent scores |
| 3 | Number of wins (incl. forfeits) | Rewards decisive play |
| 4 | Direct Encounter | Head-to-head result if they played |
| 5 | Rating | Last resort: higher rated player places first |
Let's use our 8-player, 5-round Alekhin Memorial tournament to demonstrate each tiebreak. After 5 rounds, the final scores are:
| Player | Rtg | R1 | R2 | R3 | R4 | R5 | Score |
|---|---|---|---|---|---|---|---|
| Fischer | 2200 | 1 vs Spassky | ½ vs Tal | ½ vs Kasparov | 1 vs Petrosian | 1 vs Karpov | 4.0 |
| Kasparov | 2180 | 1 vs Karpov | 1 vs Petrosian | ½ vs Fischer | ½ vs Tal | ½ vs Spassky | 3.5 |
| Tal | 2150 | 1 vs Botvinnik | ½ vs Fischer | 1 vs Karpov | ½ vs Kasparov | ½ vs Lasker | 3.5 |
| Petrosian | 2120 | 1 vs Lasker | 0 vs Kasparov | 1 vs Spassky | 0 vs Fischer | 1 vs Botvinnik | 3.0 |
| Spassky | 2080 | 0 vs Fischer | ½ vs Botvinnik | 0 vs Petrosian | 1 vs Lasker | ½ vs Kasparov | 2.0 |
| Karpov | 2050 | 0 vs Kasparov | 1 vs Lasker | 0 vs Tal | 1 vs Botvinnik | 0 vs Fischer | 2.0 |
| Botvinnik | 1990 | 0 vs Tal | ½ vs Spassky | 1 vs Lasker | 0 vs Karpov | 0 vs Petrosian | 1.5 |
| Lasker | 1960 | 0 vs Petrosian | 0 vs Karpov | 0 vs Botvinnik | 0 vs Spassky | ½ vs Tal | 0.5 |
We have a tie between Kasparov and Tal at 3.5 points. We need tiebreaks to determine 2nd and 3rd place. Let's calculate each one.
Sum of the final scores of all opponents you faced. The idea: if you beat strong players (who went on to score well), your win was worth more than beating someone who lost all their other games.
Why it works: it rewards being paired against difficult opponents. It's the most widely used tiebreak in Swiss chess worldwide.
Kasparov's opponents: Karpov (2.0) + Petrosian (3.0) + Fischer (4.0) + Tal (3.5) + Spassky (2.0) = 14.5
Tal's opponents: Botvinnik (1.5) + Fischer (4.0) + Karpov (2.0) + Kasparov (3.5) + Lasker (0.5) = 11.5
→ Kasparov wins the tiebreak with 14.5 vs 11.5. Kasparov is ranked 2nd, Tal 3rd.
Notice what happened: Kasparov's opponents had a much higher combined score than Tal's, largely because Kasparov faced Fischer, Tal, and Petrosian — three of the top four finishers — while Tal played against Botvinnik and Lasker, who finished near the bottom.
Same as Buchholz, but the lowest opponent score is removed before summing. Buchholz Cut-2 removes the two lowest scores.
Why it's FIDE's top recommendation: it reduces the "unlucky bye" problem. If you were given a bye in round 1 (which counts as facing a player with 0 score), that would unfairly crush your Buchholz. Cutting the minimum score softens this distortion.
Kasparov: opponents scored 2.0, 3.0, 4.0, 3.5, 2.0 → remove lowest (2.0) → 3.0 + 4.0 + 3.5 + 2.0 = 12.5
Tal: opponents scored 1.5, 4.0, 2.0, 3.5, 0.5 → remove lowest (0.5) → 1.5 + 4.0 + 2.0 + 3.5 = 11.0
→ Kasparov still wins: 12.5 vs 11.0. Same result in this case, but the margin is more stable.
ChessPairings.org computes all 28 tiebreak systems automatically after every round.
For each game you won: add your opponent's final score in full. For each game you drew: add half your opponent's final score. For each game you lost: add nothing.
Why it's used in Round Robin: it rewards not just winning, but winning against players who performed well. A draw against the tournament leader is worth more than a win against the last-place finisher.
Kasparov's results:
Won vs Karpov (2.0) → +2.0 | Won vs Petrosian (3.0) → +3.0 | Drew vs Fischer (4.0) → +2.0 | Drew vs Tal (3.5) → +1.75 | Drew vs Spassky (2.0) → +1.0
Kasparov SB = 2.0 + 3.0 + 2.0 + 1.75 + 1.0 = 9.75
Tal's results:
Won vs Botvinnik (1.5) → +1.5 | Drew vs Fischer (4.0) → +2.0 | Won vs Karpov (2.0) → +2.0 | Drew vs Kasparov (3.5) → +1.75 | Drew vs Lasker (0.5) → +0.25
Tal SB = 1.5 + 2.0 + 2.0 + 1.75 + 0.25 = 7.50
→ Kasparov wins again: 9.75 vs 7.50.
Sonneborn-Berger is the standard tiebreak for Round Robin tournaments (like the candidates cycle, or any closed invitational event). It's less suited for Swiss because in a Swiss, not everyone plays everyone — so the variance is too high.
The average FIDE rating of all opponents you faced. Simple and intuitive: if two players tied, the one who played against higher-rated opposition ranks higher.
Why it's useful: it doesn't depend on opponents' results during the tournament — only their rating. This makes it stable and resistant to "luck" in who won what later in the event. Downside: it doesn't reflect whether opponents performed well or poorly.
Like ARO, but uses each opponent's performance rating in this tournament rather than their pre-tournament FIDE rating. An opponent who outperformed their rating counts for more; one who underperformed counts for less.
Why it's more sophisticated: it captures the actual strength opponents demonstrated, not just their historical rating. It's computationally heavier and only meaningful when enough games have been played.
Kasparov's opponents: Karpov (2050) + Petrosian (2120) + Fischer (2200) + Tal (2150) + Spassky (2080) = 10,600 ÷ 5 = ARO 2120
Tal's opponents: Botvinnik (1990) + Fischer (2200) + Karpov (2050) + Kasparov (2180) + Lasker (1960) = 10,380 ÷ 5 = ARO 2076
→ Kasparov wins: ARO 2120 vs 2076.
If the tied players played each other during the tournament, use their mutual result. The player who won their head-to-head game ranks higher. If they drew, or didn't play each other, this criterion is skipped.
Conceptually the fairest criterion — it directly answers "who beat whom?" But in Swiss, tied players don't always face each other, so it's often inapplicable. It's best used as a secondary criterion after Buchholz.
In our Alekhin Memorial, Kasparov and Tal played each other once (Round 4).
Round 4: Kasparov (White) drew Tal → Kasparov scores ½, Tal scores ½
Total DE: Kasparov 0.5 — Tal 0.5 → Tie! This criterion is skipped.
We proceed to the next tiebreak in the order.
Count the number of games won. Among players with the same total score, the one who won more games (with fewer draws) ranks higher. Forfeits won are included in the 2024 FIDE regulations.
Why it's popular: it rewards decisive play over drawing. A player with 4 wins and 0 draws (4.0 points) ranks above a player with 2 wins and 4 draws (4.0 points). Some players argue this unfairly penalizes solid drawing players.
Kasparov: Won vs Karpov (R1), Won vs Petrosian (R2) = 2 wins
Tal: Won vs Botvinnik (R1), Won vs Karpov (R3) = 2 wins
→ Still tied at 2 wins each! Move to the next criterion: Rating.
Kasparov (2180) vs Tal (2150) — Kasparov ranks 2nd by rating as last resort.
ChessPairings.org supports all 28 FIDE-recognized tiebreak systems. Here's a reference overview of the most important ones beyond what we've covered in detail:
| System | Best for | How it works |
|---|---|---|
| Median Buchholz (MB) | Swiss | Removes both highest and lowest opponent score before summing. More stable than plain BH. |
| Recursive Buchholz | Swiss | Uses Buchholz of each opponent's Buchholz (second-order). Very robust but complex. |
| Progressive Score | Swiss, Rapid | Sum of cumulative scores after each round. Rewards fast starters. |
| Koya System | Round Robin | Score in games against players who scored ≥ 50%. Rewards performance against the stronger half. |
| Number of Blacks | Swiss | Player who had more Black games ranks higher. Rarely decisive. |
| Cumulative Opponent Score | Swiss | Running total of opponent scores round by round, cumulated. Similar motivation to Progressive. |
| Performance Rating | Both | Estimates the rating a player performed at during the event. Not always calculable precisely. |
| Rating (pre-event) | Both | Higher rated player ranks first. Last resort only. Never use as primary tiebreak. |
| Lot (draw) | Both | Random drawing. Absolute last resort. Must be announced in advance. |
| Tournament type | #1 | #2 | #3 | #4 |
|---|---|---|---|---|
| Swiss (standard) | Buchholz Cut-1 | Buchholz | Wins | Direct Encounter |
| Swiss (rapid/blitz) | Buchholz Cut-1 | Buchholz | ARO | Wins |
| Round Robin | Direct Encounter | Sonneborn-Berger | Koya | Wins |
| Team Swiss | Match Points | Board Points | Buchholz (match) | Direct Encounter |
| School / beginner | Wins | Direct Encounter | Rating | Lot |
For most club and school events, a simple order of Buchholz Cut-1 → Buchholz → Wins → Direct Encounter covers 99% of situations. Keep it simple — the more exotic the tiebreaks, the harder they are to explain to players and parents.
Always print the tiebreak order on the tournament regulations sheet and post it on the notice board before round 1.
ChessPairings.org includes an interactive tiebreak calculator at chesspairings.org/en/tiebreaks. It supports all 28 tiebreak systems and allows you to:
Enter any tournament's results manually or import from a TRF file, then instantly see the rankings under any combination of tiebreak systems. You can compare how rankings would change between systems — useful for choosing your tiebreak order before the tournament.
If you're running a live event in ChessPairings.org, tiebreaks are recalculated automatically after every round. No manual calculation needed.
Enter results and compare all 28 systems instantly. Free, no account required.
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