Chess Pairings: A Review of Monrad, Swiss, and Berger

Chess pairings typically do not employ randomization. The exception is the first round in Berger and Monrad, where the players are drawn randomly into an ordered list that then determines who faces whom in the opening round. From that point onward, the process is entirely deterministic. In Swiss pairings, chance plays no part at all, since the starting order is seeded, usually by rating. Thus, the algorithm contains no built-in element of randomness. It is the players themselves who introduce variation through their more or less unpredictable results.

The Monrad system is named after the Danish chess organizer Poul Monrad, who during the mid-20th century developed and popularized this simplified variant of Swiss pairings. It became widespread in Denmark, Norway, and Sweden, particularly in school competitions, weekend tournaments, and larger amateur events, because it is simpler than the complete Swiss system: players are paired strictly according to their current score, without the more complicated FIDE restrictions.

In Monrad (simplified Swiss), a certain unfairness can initially arise, as one top player may face a strong opponent while another receives a significantly weaker one. For this reason, elite tournaments prefer some form of Swiss pairing. Another drawback is that a player may be assigned the same colour for three consecutive rounds, and in rare cases even more. In a seven-round tournament, roughly 34% of the players are likely to experience this at least once, and in a nine-round tournament the proportion rises to about 45%. This is a significant weakness of the Monrad system if it is used in elite tournaments. In amateur tournaments, however, it may not matter as much, since wins and losses there depend less on which colour one gets and more on who makes the first mistake.

The above applies to Danish-Swedish Monrad. According to FIDE’s official rules, a player may not have the same colour three times in a row, nor may they have a colour balance that deviates by more than ±2. (Dutch Swiss follows this rule, except in rare cases.) The Norwegian Monrad system is less strict: a player may never have more than three consecutive games with the same colour, and if possible no player should have more than 50% + 1 games with the same colour (see here). These rules are not absolute and may, in exceptional cases, be overridden.

I have created a program in JavaScript for Danish-Swedish Monrad here: Monrad Pairing System. The program introduces a new tie-break method as an alternative preset: for each game, take the opponent’s final score as a decimal between 0 and 1 (for example, 0.8 for 80% of the maximum score). Square it to amplify the differentiation. Then multiply by the result against that specific opponent, yielding full credit for a win and half credit for a draw. The second term provides partial compensation for a loss — especially against a successful player. Finally, multiply by one hundred to avoid decimals. I provisionally call it WNT, a name derived from the letters of my surname. Until now, all tie-break systems (Buchholz, Sonneborn-Berger, etc.) have been designed around the principle of being easy to calculate by hand. The result is a coarse tie-breaking mechanism. There is no reason to preserve that legacy today.

WNT Score = 100 × ∑_i [ (P_i / N ) × (r_i + P_i / (2N ) ) ]

Alternative form:


∑_i = the sum over all opponents you have faced
r_i = your result against opponent i  (1, ½, 0)
P_i = opponent i ’s final score in the tournament
N = total number of rounds in the tournament
P_i / N = the opponent’s score share (score ratio expressed as a decimal between 0 and 1)

This method provides a fairer reflection of player performance than traditional tie-break systems. When players finish tied on points, they are separated based on the quality of the opposition they have faced. WNT consistently rewards those who actually play on the top boards. Those who face and score points against the tournament’s best players are rewarded quadratically and disproportionately higher than those who gather their points on the lower boards, rendering the tournament strategy known as the “Swiss gambit” virtually useless in practice. In traditional Buchholz, a player who suffers an early loss and picks up points on lower boards against weaker opposition can often still obtain a surprisingly high Buchholz score, as opponents’ points are summed linearly. In WNT, thanks to the squaring, this gap is widened substantially. At the same time, facing a strong opponent yields a return even in defeat.

Unlike truncated systems (such as Buchholz Cut), there are no artificial thresholds. Every half-point an opponent fights for increases a player’s WNT, but it is encounters at the top boards that ultimately make the difference. Sonneborn-Berger was originally developed for round-robin tournaments (all-play-all) and is less suited for Swiss-system and Monrad tournaments. Although WNT is designed specifically for the latter, in round-robin tournaments the algorithm reduces mathematically to the same relative ranking produced by the classic Sonneborn-Berger system, making WNT applicable across all tournament formats.

WNT integrates the advantages of opponent strength or schedule difficulty (Buchholz) and individual performance or win quality (Sonneborn-Berger) into a single, unified metric, rather than treating them as separate, sequential filters where the latter rarely comes into play — such as in this cumbersome tie-break hierarchy: direct encounter → Buchholz Cut 1 → Buchholz (full) → Sonneborn-Berger → most wins / most games with black. It allows the tournament organizer to display a live WNT score throughout the tournament, a meaningful measure that can only increase. Thanks to WNT’s high mathematical resolution, the likelihood of two players finishing with exactly the same decimal score is practically negligible. (Consider the following two PDF files of a simulated Monrad tournament, in which the final standings are determined by WNT and Buchholz, respectively: file 1, file 2.)

Example: a five-round tournament

• A win against a strong player (4/5) yields the maximum return: 112 points. (Result bonus: 100 × 1 × 0.8 = 80. Opposition bonus: 50 × 0.64 = 32. Total: 112.)

• A draw against the same player yields 40 + 32 = 72 points.

• A loss yields 0 + 32 = 32 points. (In other words, you still receive 32 points simply for having faced top-level opposition.)

• A win against an average player (2.5/5) yields 62.5 points.

• A win against a player near the bottom of the standings (1/5) yields 22 points.

Thus, the system places a substantial premium on both facing and scoring against strong opposition. Wins against top performers are rewarded disproportionately more than wins against weaker opponents, while even losses to elite players retain significant value because opponent strength remains part of the calculation. This strongly incentivizes competitive performance on the top boards rather than point accumulation against the lower half of the field.

Tie-break systems reflect an important principle: the final score alone does not tell the whole story. Two players may finish a tournament on the same score, even though one has fought through considerably stronger opposition than the other. To ignore this difference is to treat two fundamentally different achievements as equivalent. Tournament points alone are therefore too crude a measure of success.

The score shows how many points you earned. The tie-break shows how difficult they were to earn.
 

My online version of the Norwegian system is available here: Monrad Tournament Manager. The final standings are ranked according to Buchholz (Cut 1 or 2). It can be used online, but the page itself can also be downloaded and run locally. The Danish-Swedish version uses either WNT, cumulative (progressive) score, or Buchholz (the opponents’ total score) to differentiate players with the same final result. Cumulative score is defined so that early wins yield a higher tiebreak, which favours an aggressive playing style. (Starting with a win and then a loss gives 2 cumulative points after two rounds, whereas two draws give 1.5. In a sense, this can be considered fair.)

Berger means that every player meets every other player, either once or twice. It was formalised by Johann Berger in the late 19th century. It is the fairest of all pairing systems, which makes it ideal for Candidates tournaments and elite events. Players alternate colours automatically. I have created a Berger system for 3-16 players according to FIDE specifications. It can be used directly in the browser, but can also be downloaded and run locally: Berger Tournament System.

Scheveningen is a team-play format in which two teams of equal size meet in a full cross-pairing: every player in Team A faces every player in Team B over n rounds: Scheveningen Chess Manager. It takes its name from a tournament in Scheveningen, the Netherlands, in 1923, where the format was used in a team match between two groups of grandmasters. The system is fair and deterministic (no arbitrary pairing is required) and it therefore spread quickly. This format can also be used within a club by dividing the players into two evenly matched teams. It then becomes both an individual tournament and a team match, which can be quite stimulating.

As a tie-break method, the program uses Koya as the primary criterion and Sonneborn-Berger as the secondary criterion. The Koya score is the sum of the points a player has scored against opponents who finished with at least 50% of the maximum possible score. Koya thus measures how well a player has performed against the strongest opposition in the tournament, whereas Sonneborn-Berger provides a more fine-grained and comprehensive measure based on the combined results of a player’s opponents.

The Swiss system originated in Switzerland in the late 1800s, primarily to handle large chess tournaments where a round-robin format was impractical. Early Swiss events were informal and used simple rules: players with the same score were paired together, much like the modern Monrad system. Swiss systems are today constructed in various ways. These include double-accelerated Swiss, Haldane acceleration, and FIDE acceleration (now rare). Certain variants, however, perform very poorly, among them Nordic Swiss (see here). A typical problem for intermediate players is that they oscillate between overly strong and overly weak opposition: in one round they face a significantly stronger player, in the next a clearly weaker one, and then once again an overly strong opponent. This oscillation diminishes the enjoyment of the tournament and produces an uneven tournament experience. In my accelerated Swiss system, this does not occur: Swiss Tournament System. The program includes a safeguard that makes it highly unlikely for a player to receive the same colour three times in a row.

Accelerated Swiss Pairing (Winther acceleration):

Round 1: Players are divided into four groups (quadrants). The highest-ranked player in Q1 plays White against the highest-ranked player in Q2, and correspondingly for Q3 and Q4. (For uneven groups, sizes are adjusted automatically, and with an odd number of players, the lowest-ranked gets a bye).

Round 2: Highest-seeded winner plays Black against the highest-seeded winner in the other half. Second highest-seeded plays Black against the next in the other half, and so on.

Round 3+: Standard Swiss pairings begin. Players are sorted by score, WNT, and initial seeding. Pairings are generated top-down.

Colour allocation: The player with fewer White games receives White. If equal, the lower-seeded player receives White. Absolute colour balance is guaranteed (colour difference can never reach 3). If colour balance conflicts with the 2-in-a-row rule, up to three games in a row with the same colour is permitted.

This system is well suited for open tournaments. Elite players have little reason to object, as they face lower-ranked players in the first two rounds. At the same time, it avoids the oscillation from which intermediate players often suffer in other Swiss pairings. The program uses WNT as a secondary sorting criterion after score when pairing players. This supports the objective of pairing players who have faced opposition of comparable strength. A favourable WNT places a player higher within the score group, potentially leading to stronger future opponents.

Having the winners in the second half receive White against the winners in the first half in round two is, in my view, a good idea — all players will be pleased. From round three onward, players with similar ratings meet each other more consistently. Generally, top players prefer Black against weaker opposition and White against stronger. The program accommodates this by granting White to the lower-ranked player when the number of White games played is tied. The program minimizes the number of players who receive Black in the first two rounds. Since the higher-seeded player is normally assigned White in the first round, this situation cannot occur for players in the first and third quarters of the initial ranking list. An option exists to randomize players within each quadrant, ensuring greater variety in first-round pairings for the same group of club players.

The fundamental problem of Swiss pairing is that it attempts to achieve a fair tournament within a limited number of rounds, without being able to employ the ideal round-robin format. Many Swiss variants fall short in this respect, especially when the intermediate bracket oscillates between overly strong and overly weak opposition, or when the colour allocation becomes unreasonable. The proposed variant is an attempt to combine simplicity with stability: the first two rounds provide a controlled acceleration where elite players meet lower-ranked opponents. From round three onward, players with similar ratings meet each other with increasing frequency, lending the tournament a natural progression.

In my Club Swiss System, players are divided into three rating groups, each randomized independently. This combines the randomness of the Monrad system with the sporting equity of Swiss pairings, creating a format well suited to chess clubs. For the first two rounds, players from the strongest group are paired against those from the middle group, while the weakest group competes internally. From the third round onward, pairings are made across the entire field.

My Berger-Swiss Tournament System represents an attempt to combine the predictability of round-robin groups with the flexibility of Swiss pairing. Players are initially grouped by rating, with exchanges between adjacent Berger groups to broaden opposition. Depending on the number of players, a residual field may be needed for those who cannot be accommodated in complete round-robin groups; this field uses Swiss pairing from the start. After five or seven rounds, all active players join a common Swiss field, retaining their scores and histories. Fixed early schedules simplify planning and allow postponed games. The later Swiss phase brings players with similar scores together across the original groups. Ties are resolved using WNT, which combines opponents’ scores with individual game results.

Overview of Pairing Systems

FIDE Swiss (“Dutch”) is the international gold standard for chess tournaments and the official system prescribed by FIDE for most open Elo events as of 1 February 2026. Its purpose is to achieve the fairest pairing possible. The algorithm, which is highly complex (see here), was first published in 2023-2024 by FIDE’s Technical Commission and was established as the official system in 2026, effective from 1 February 2026. It replaces the earlier Dutch (2009) and Dutch (2017), which were used in some tournaments but never became a global standard. It is a modern algorithm developed by FIDE to reduce colour imbalance, reduce float chaos, increase determinism and reproducibility, and avoid predictability in the top standings. FIDE’s requirement that the system be deterministic and reproducible means that the same input yields the same pairing, no “human judgement” may occur, and no subjective choices are allowed. In practice, it is impossible to pair Dutch (2026) by hand, as the procedure is extremely cumbersome and computationally intensive.

The reason the Dutch system has become the norm is how it balances sporting fairness, colour allocation, and a clear differentiation of the participants. The system has the strictest colour rules of all Swiss methods. A player may never have the same colour three rounds in a row (with exceptions only in extreme cases in the very last round). The algorithm also strives to ensure that no player ever has more than one additional white or black game compared to their opponent. Many other Swiss systems allow greater imbalances, which often leads to players being disadvantaged with several black games in decisive stages. The Dutch system is held to deliver the highest standard of sporting integrity: it maximally protects colour allocation and ensures that players with the best results meet each other towards the end of the tournament.

I have created an experimental version here: Dutch Pairing System. By all indications, it appears to be working correctly. A tournament with 32 players over nine rounds was simulated. GPT-Sol-6.1 and Claude-Sonnet-5.5 analyzed the tournament files to verify that the pairings in each round complied with FIDE specifications. Both models concluded that the pairings were correct in all nine rounds. A PDF containing their analyses is available here.

In the Dutch system, having two consecutive black games is normal. Three black games in a row is generally forbidden, with rare exceptions in the last round. Four black games in a row is never possible. A player may in some cases finish the tournament with two more black games than white. A colour difference of -3 or more is something the system actively tries to avoid for top scorers and their opponents. Players are ranked by score and then by TPN (Tournament Pairing Number). This follows the same principle as classical Monrad: (1) seeding, (2) strict pairing by order, (3) no acceleration rules.

From the player’s perspective, the system is significantly more unpredictable than classical Monrad. The final pairing is often impossible to foresee. The algorithm attempts to optimise across 16 different quality criteria. Players with the same score do not necessarily meet each other, and one may face someone with a completely different TPN than expected. Colour allocation is complex and not intuitive.

The Dubov Swiss system is an original method developed by the Russian mathematician Eduard Dubov. The system, which is approved by FIDE, actively strives to ensure that all players finishing on the same score have faced opponents with as similar an average rating as possible. The aim is to reduce the risk that a player reaches a prize-winning position thanks to luck in the pairings and thus unusually weak opposition compared with competitors on the same score.

However, the method is less intuitive for players and spectators. In a conventional Swiss system, the players ranked first and second will naturally meet as the tournament approaches its end. In the Dubov Swiss, two players in clear shared lead may very well not meet, simply because the system judges that one of them needs to face a higher-rated opponent in order to balance his ARO value (Average Rating of Opponents). FIDE Dutch, by contrast, is designed to produce a clear winner through top-group elimination, whereas the Dubov system is designed as a statistical measurement instrument: its goal is to give the entire field as fair and comparable an assessment of playing strength as possible. The Dubov system is mathematically elegant but has remained a niche option for practical and psychological reasons. It is, however, ideal for tournaments aimed at further qualification, such as the Candidates Tournament. I have created a pairing program that follows FIDE’s specifications: Dubov Swiss Pairing System.

The Burstein Swiss System was developed in 1993 by Almog Burstein (born 1950), an Israeli international arbiter (IA), chess organizer, and long-time FIDE official. The core philosophy is to equalize the strength of opposition among players (or teams) in the same score group. It was famously adopted by FIDE for team competitions and served as the pairing system used in Chess Olympiads from Moscow 1994 through Dresden 2008. It remains one of the officially recognized pairing systems in the FIDE Handbook.

The Burstein system uses a hybrid method. It begins with Dutch seeding: the first half of the scheduled rounds, rounded down and capped at four, are paired using the FIDE Dutch system. In subsequent rounds, players are divided into scoregroups and ranked within each bracket by Buchholz, Sonneborn-Berger, and then ascending TPN. Pairings generally match the highest-ranked available player with the lowest-ranked compatible player. Floaters (players paired against an opponent from a different score group) and any pairing-allocated bye (unpaired round) are selected according to prescribed priorities, while avoiding repeat opponents, respecting absolute colour restrictions, and minimizing unmet colour preferences. My implementation is here: Burstein Pairing System.

The Lim Swiss System is named after its developer, Professor Lim Kok Ann of Singapore (1920-2003), a former FIDE Secretary-General and long-time chess organizer. Approved by the FIDE General Assembly in 1987, it is one of FIDE’s officially recognized Swiss pairing systems.

Unlike the standard Dutch system, which pairs scoregroups sequentially from the top downward, the Lim system employs a two-sided, outside-in approach: pairing begins with the highest scoregroups working downwards towards the median, then alternates to the lowest scoregroups working upwards, pairing the median scoregroup last. The algorithm strives to pair players with equal or similar scores while strictly regulating colour balance and floater classification. FIDE’s Olympiad team pairing system incorporates similar convergence principles adapted for teams. My implementation is here: Lim Pairing System.

© Mats Winther, 2026.

References

‘Buchholz system’. Wikipedia.
https://en.wikipedia.org/wiki/Buchholz_system

‘General Regulations for Competitions. Annex 1: Details of Berger Table’. FIDE Handbook.
https://handbook.fide.com/chapter/C05Annex1

‘General Rules and Technical Recommendations for Tournaments. 04. FIDE Swiss Rules. C.04.3 FIDE (Dutch) System’. FIDE Handbook.
https://handbook.fide.com/chapter/C0403202602

‘General Rules and Technical Recommendations for Tournaments. 04. FIDE Swiss Rules. C.04.4 Other FIDE-approved Pairing Systems. C.04.4.2 Burstein System’. FIDE Handbook.
https://handbook.fide.com/chapter/C040402202602

‘General Rules and Technical Recommendations for Tournaments. 04. FIDE Swiss Rules. C.04.4 Other FIDE-approved Pairing Systems. C.04.4.1 Dubov System’. FIDE Handbook.
https://handbook.fide.com/chapter/C040401202602

‘General Rules and Technical Recommendations for Tournaments. 04. FIDE Swiss Rules / C.04.4 Other FIDE-approved Pairing Systems / C.04.4.3 Lim System’. FIDE Handbook.
https://handbook.fide.com/chapter/C0403202602

‘General Rules and Technical Recommendations for Tournaments. 07. Play-Off and Tie-Break Regulations 07. Tie-Break Regulations’. FIDE Handbook.
https://handbook.fide.com/chapter/TieBreakRegulations032026

‘Norges Sjakkforbunds Monradsystem’. Sjakktuelt.
https://2000.sjakk.no/kunnskap/systemer/norges-sjakkforbunds-monradsystem/

‘Scheveningen system’. Wikipedia.
https://en.wikipedia.org/wiki/Scheveningen_system

‘Sonneborn-Berger score’. Wikipedia.
https://en.wikipedia.org/wiki/Sonneborn%E2%80%93Berger_score

‘Swiss-system tournament’. Wikipedia.
https://en.wikipedia.org/wiki/Swiss-system_tournament

Online pairing programs:

Berger Tournament System (Round-Robin).
https://mats-winther.github.io/berger-eng.htm

Berger-Swiss
https://mats-winther.github.io/berger-swiss.htm

Burstein Pairing System
https://mats-winther.github.io/fide-burstein-eng.htm

Club Swiss System
https://mats-winther.github.io/club-swiss.htm

Dubov Swiss Pairing System
https://mats-winther.github.io/fide-dubov-eng.htm

Dutch Pairing System
https://mats-winther.github.io/fide-dutch.htm

Lim Pairing System
https://mats-winther.github.io/fide-lim-eng.htm

Monrad Pairing System (Danish-Swedish model).
https://mats-winther.github.io/monrad-eng.htm

Monrad Tournament Manager (Norwegian model).
https://mats-winther.github.io/norwegian-monrad.htm

Scheveningen Chess Manager (two teams in cross-pairing).
https://mats-winther.github.io/scheveningen-eng.htm

Swiss Tournament System (Winther acceleration).
https://mats-winther.github.io/swiss-eng.htm