1. Introduction

There are a wide variety of tournament formats. Two of the most popular—Swiss and knockout—each have structural limitations that raise concerns about their fairness, equality of opportunity, and strategic manipulability. In Swiss chess tournaments, for example, color imbalance (i.e., unequal numbers of games played with White or Black) can significantly affect outcomes. This arises because, as we discuss in more detail in Section 3, playing with White gives an advantage to a player, which is similar to a team’s home advantage in soccer. In addition, the pairings in chess may give rise to the “Swiss Gambit,” wherein players benefit from an early loss by facing weaker opponents in later rounds. By comparison, knockout systems are unforgiving in that a single early loss can eliminate strong players, perhaps due to a minor injury or simply having a bad day.

To address these issues, we introduce a novel system of matching and scoring players in tournaments, called Multi-Tier Tournaments, illustrated for chess by the following rules:

1. Players are divided into skill-based tiers, based on their Elo ratings.
2. Starting with one or more mini-tournaments of the least skilled players (Tier 1), the winner or winners—after playing multiple opponents—move to the next-higher tier.
3. The winners progress from lower tiers to a final tier of the best-performing players from lower tiers, which also includes players with the highest Elo ratings.
4. Performance in each tier is given by a player’s Tournament Score (TS), which depends on his/her wins, losses, and draws (not on his/her Elo rating).

Elo [1] ratings are a predictor of a match between two players, based on the difference in the players’ ratings. Thus, when a low-rated player defeats a high-rated player, the Elo score of the low-rated player significantly increases and the high-rated player’s score significantly decreases. On the other hand, the more similar the players’ ratings are, the less the players’ ratings are affected by the outcome of their match. In this manner, Elo ratings are self-correcting, providing a measure of the strength of players. For more details, see https://en.wikipedia.org/wiki/Elo_rating_system. Like Elo, the tournament score (TS) we propose, according to rule 4, depends on a player’s wins, draws, and losses, but only for players in each mini-tournament (more on this later).

Whereas a player’s Elo rating determines in which mini-tournament he/she starts play, TS and its associated tie-breaking rules determine whether a player moves up to higher tiers and, in the final mini-tournament, wins the tournament. This combination of players’ past Elo ratings and current TS’s provides a fair and accurate measure of the player’s standing among the players in the tournament.

We apply a variation of Multi-Tier Tournaments to the top 20 active chess players in the world (as of February 2024). Using a dataset of 1209 head-to-head games that we collected, we illustrate the viability of giving lower-rated players the opportunity to progress and challenge higher-rated players. We also discuss the application of Multi-Tier Tournaments to baseball, soccer, and other sports that emphasize physical rather than mental skills.

The International Chess Federation [2, Chapter C.04.1] prescribes several rules for organizing tournaments, which are distinctive in two major respects: (i) they have a fixed number of rounds, which is significantly less than the number of participants; (ii) they match players against opponents with similar scores after each round. Although the complete list of rules is extensive, the following are especially relevant to our paper:

1. Two players do not play against each other more than once.
2. The difference between the number of Black and the number of White games played by each player is not greater than two.
3. No player plays the same color three times in a row.

Rule (1) promotes diversity and reduces the potential bias arising from repeated matchups of the same players. Rule (2) promotes “color fairness,” which may be undermined by the first-mover advantage of White in chess if one player plays more games with this color. (To mitigate this bias, Brams and Ismail (2021) proposed a “catch-up” rule to put White and Black more on a par.) However, rule (2) allows a player to have one extra White game. Our analysis in Section 3 indicates a significant bias favoring the players who had an extra White game. For example, 66% of the top 10 finishers in major Swiss tournaments over the past decade had an extra White game. Rule (3) addresses “psychological fairness,” because consecutively playing with the same color may confer a psychological advantage (when playing with White) or disadvantage (when playing with Black).

As we will show, our approach not only satisfies rules (2) and (3), and other standards of fairness, but it also better balances the distribution of colors than the currently used Swiss system. However, because it allows players to move into higher and higher levels of competition, it is possible for two players who meet early in a tournament to be later opponents. We do not think this is a serious violation of fair play, because knockout tournaments that satisfy (1)—by eliminating competitors after one loss—are not necessarily fair to players in which a player’s lapse in one game can be more than made up for by wins in the other games that he/she plays. We believe it is multiple games, at each tier in a tournament, that should be the principal determinant of who stays in and who is eliminated.

What is distinctive about our approach is its simplicity. It does not depend on indicators of performance other than wins, losses, and draws in multiple contests; where draws are not allowed or possible, this indicator can be dropped. (We propose, however, rules for breaking ties when players have identical TS’s. These are specific to chess and will need to be revised in applying Multi-Tier Tournaments to other games and sports.) This makes it applicable to very different games and sports and, thereby, a general way of ranking competitors and choosing winners in tournaments.

2. Selecting and scoring players in mini-tournaments and determining winners

(i) Selection algorithm

Before players can be matched and scored in a tournament, they must be selected for a mini-tournament. Players in the first mini-tournament of Multi-Tier Tournaments (Tier 1) are defined as the subset of players who have the lowest Elo ratings. For example, if there are 20 players in a tournament, they might be the 8 players with the lowest ratings. Each of these players plays against the 7 others in Tier 1 and receives a Tier Score (TS), based on his/her wins, losses, and draws (more on this later, in which we relax the assumption that every player plays against every other player in a mini-tournament).

We first introduce the notation and give a more formal statement of our proposal. A Multi-Tier Tournament is given by the tuple , where represents the set of players with , and denotes the initial ordered Elo ratings, with being the lowest rating and the highest (ties broken randomly). (Players who do not have an Elo rating are assigned an initial rating, which is usually 1500.) The tournament is organized into tiers, with tier 1 as the lowest tier and tier as the highest. Each tier, except the final one, consists of a subset of players , where gives the number of players in each tier . The number of rounds in each tier is given by , and represents the number of players who progress to the next tier. Naturally, for every , we assume .

In every tier except the final one, the top players at the end of rounds advance to the next-higher tier (if there is one). Players are allocated to tiers as follows:

1. i. Players are initially ranked based on their Elo ratings given in the sequence
2. ii. Tier 1 consists of the lowest-rated players with Elo indices . The top players in this tier advance to tier 2.
3. iii. Tier 2 initially consists of players, whose ranks are given by the indices:

The top players advance to tier 3.

1. iv. For higher tiers except the final tier: Tier initially has players with Elo indices:

The top players advance to tier .

1. v. Final tier : The tournament winner is the player with the highest Tournament Score in this tier.

Example 1.

To illustrate how players move up from an 8-player mini-tournament in a 20-player tournament, assume that the 2 players with the highest TS’s in Tier 1 move up to Tier 2, joining the 6 players with the next-highest Elo scores. (Like Tier 1, Tier 2 would now have 8 players.) At the conclusion of the Tier 2 mini-tournament, the 2 players with the highest TS’s in Tier 2 would move up to Tier 3, joining the 6 players with the highest Elo scores, so now Tier 3 also has 8 players. The Tier 3 players then play a third and final mini-tournament, the winner of which becomes the overall winner of the tournament.

Altogether, 8 + 6 + 6 = 20 different players participate in up to three rounds of play. In Tier 1, each of the 8 players plays against the 7 other players, and the 2 players with the highest TS’s move up to Tier 2. Next, each of the 8 Tier 2 players plays against the 7 other players in Tier 2, and the 2 players with the highest TS’s move up to Tier 3. These 8 players play a final mini-tournament, and the player with the highest TS wins the tournament.

Players from Tier 1 who move up to Tier 2 play a total of 7 + 7 = 14 games. If they succeed in Tier 2 and move up to Tier 3, they play a total of 7 + 7 + 7 = 21 games. But if a player is not one of the top two in either Tier 1 or Tier 2, he/she is knocked out of the final competition.

The 6 players with the highest Elo scores, who start in Tier 3, play only 7 games, whether they win or lose in this mini-tournament. Only the single player with the highest TS in Tier 3 wins the tournament. Thus, a player starting out with a low or even moderate Elo rating, and therefore from a lower tier, means that he/she must play, and perform better, in more games than players who start out in the highest tier.

(ii) Scoring algorithm and tie-breaking

In the mini-tournaments of each tier, each player i receives a Tier Score (TS_iJ) against all his/her opponents in subset of his/her tier (as we will illustrate shortly, this may not be all members of his/her tier), according to the following formula:

TS_iJ varies from −1 (for a player who loses all his/her games) to +1 (for a player who wins all his/her games). Note that TS_iJ is based only on i’s wins, losses, and draws within a given tier. A player’s TS does not carry over from one tier to another.

TS_iJ differs from Elo ratings in not being a dynamically changing measure of player i’s strength, which may go up, down, or stay the same after each of a player’s matches. Instead, TS_iJ is a summary measure of i’s strength against his/her opponents in each of the mini-tournaments in which he/she plays (subscript J in TS_iJ is the subset of these players). For example, a player could win a mini-tournament and move up to the next-higher tier, but in the latter mini-tournament he/she might not do so well and so be eliminated from the tournament.

Notice that the denominator of TS_iJ includes draws as well as wins and losses; the greater the number of drawn games, the less the absolute value of TS_iJ. If the numerator is positive, this lowers the value of TS_iJ (more drawn games make TS_iJ less positive). If the numerator is negative, this raises the value of TS_iJ (more drawn games make TS_iJ less negative). Thus, more drawn games hurt a winning player but help a losing player, as one would expect. If the numerator is 0, the number of drawn games is irrelevant.

Put another way, winning a game increases TS_iJ more than drawing a game, and drawing a game increases TS more than losing a game. We think this is what a fair-scoring algorithm should do.

But what if the number of players in a mini-tournament is too great for each player to play a game against every other player in his/her starting tier? This will be the case when the number of players in a mini-tournament is greater than about 10, in which case we assume that players play against only a proper subset of opponents. (One could add more tiers to solve this problem, but this may make the tournament unduly lengthy if each tier takes a few days to complete. Thus, if there are 10 players in a mini-tournament, and each player plays against 3 opponents a day, the mini-tournament will take three days to complete against the 9 opponents of each player.) Before addressing the question of choosing such a subset, we suggest four rules to break ties when two or more players have the same TS_iJ’s at the end of a mini-tournament.

When two players, A and B, have identical TS_iJ’s after play concludes in any tier, we propose the following rules to break ties:

1. (i) If A and B play each other in a mini-tournament and A defeats B, A is ranked above B. If they draw or do not play a game against each other, go to step (ii).
2. (ii) If A wins more games than B, A is ranked above B. (Note: A may also lose more games than B, but if B has more draws, it is possible that A and B may have the same TS_iJ.)
3. (iii) If A and B win the same number of games but A takes an average of fewer moves to defeat his/her opponents, A is ranked above B.
4. (iv) If (i), (ii), or (iii), in that order, fail to break a tie in TS_iJ’s between A and B—but their tie must be broken to determine which players go onto the next-higher tier or which player wins the tournament—use a random device to break the tie (e.g., by tossing a coin).

We think (i), (ii), and (iii) will break almost all ties between two players in a mini-tournament. (Tie-breaking rule (iii) is perhaps least transparent. But we think defeating an opponent more quickly is consistent with being a stronger player.) These tie-breaking rules, except for (iv), are consistent with the better performance of A or B. They can be extended to break ties among more than two players, and thereby to determine, if necessary, which player or players remain in the competition, move to a higher tier, or win the final mini-tournament and, therefore, the tournament.

3. Practical concerns in chess tournaments

4. Application of multi-tier tournaments to top 20 chess players

In Table 1, we list the top 20 active chess players, based on their Elo ratings as of February 2024. As in Example 1, we divide players into three tiers: Initially, Tier 1 comprises the 8 players at ranks 1–8; Tier 2 comprises the 6 players at ranks 9–14; Tier 3 comprises the 6 players at ranks 15–20. Later, the two highest scorers in Tiers 1 and 2 move up, respectively, to Tiers 2 and 3, who then each have 8 members who compete against each other.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Elo ratings of the top 20 players.

https://doi.org/10.1371/journal.pone.0328826.t001

Like Example 1, there is no need to construct fair subsets of players, because each of the 8 players in each tier can play against every one of the 7 other players in its tier to determine which 2 players move up to a higher tier. But unlike Example 1, the number of games played by each member of a tier against others in it is not 1, as in Multi-Tier Tournaments, but may vary radically, from 0 (there are seven pairs of players who play no games against each other) to 71, as in the case of the Carlsen (#1) – Aronian (#19) pair (Carlsen won 12, lost 8, and drew 51 games against Aronian).

It was not appropriate to calculate, for each tier, TS_iJ of player i against all other players J in his/her tier, because this measure is too much influenced by the outlier pairs in a tier, especially those pairs which happen to play many games against each other (perhaps because they are geographically proximate or are invited to, and choose to participate in, many tournaments). To remedy the problem of combining the wins, draws, and losses of such pairs with those pairs playing far fewer games, we normalize TS.

For every two players, A and B, in a tier, we calculate A’s Normalized Tier Score (NTS) in his/her tier as follows:

with if A and B did not play any games. We illustrate the aggregation problem if we had used the formula for TS in section 2, TS_iJ (where subscript J indicates all opponents of i). Assume A plays a large number of games against B but many fewer games against the other 6 players, C, D, E, F, G, and H. In that case, the AB pair would have had an unduly large effect when we sum all the wins, draws and losses of A’s opponents, including B, using TS_iJ.

But if we first normalize values of TS between −1 and +1 for each of A’s opponents, as does, their summation does not give unduly large weight to the AB pair. Then, taking the average of NTSs, we obtain a summary measure of how, hypothetically, A would do against all his/her opponents in his/her tier, but now based on varying numbers of games A has played against his/her opponents over 20 years (see next paragraph)—instead of just one game in a single tournament that never occurred.

We used head-to-head contest data from 1209 games—excluding so-called rapid and blitz games that require somewhat different skills—that were played between May 2004 and February 2024 that is available from the following website that archives historical chess results (https://www.chessgames.com). Each matchup is based on an average of 14.4 games, with Tier 1 players averaging 6.6 games against each other, Tier 2 players averaging 10.1 games against each other, and Tier 3 players averaging 26.4 games against each other. Higher-ranked players tend to be older and are invited to more tournaments. (These are the average figures for each tier before winners from the lower tiers are added to the higher tiers.)

As Table 2 shows, MVL and Aronian have the greatest TS’s in both Tier 1 and 2. Hence, they advance to Tier 3. Consistent with Carlsen’s dominance in chess for over ten years, he is the clear winner in this Multi-Tier Tournament, with an average TS of 0.22 in Tier 3. Nakamura and Caruana share the second and the third places, respectively. Notably, despite starting the competition in the lowest tier, MVL and Aronian perform better than three Tier 3 players.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. A three-tiered tournament with top 20 players grouped into three tiers of 8, 6, and 6 players, based on their Elo ratings. The asterisk indicates the overall winner^a.

https://doi.org/10.1371/journal.pone.0328826.t002

5. Other approaches to the design of fair tournaments and limitations

FIDE has approved several versions of Swiss tournaments, including the Dutch, Burstein, Lim, and Dubov systems. These systems introduce changes to either the first-round pairing or later-round pairings under the Swiss system. For example, the Dubov system uses the average rating of the opponents to create pairings within the same score groups. However, this system still relies on the current score to match players and, unlike our system, it does not necessarily improve the overall average of opponents’ ratings or other desired aspects of matches (for a detailed overview, see Held [5]). The most commonly used version is, in fact, the Dutch system, which matches players based on Elo ratings for the first round and then uses the players’ updated scores for following rounds. For instance, in a six-player tournament with players ranked from 1 to 6, with 1 being the highest-rated, the first-round pairing would be 1–4, 2–5, and 3–6 according to the Dutch system.

While sharing similarities with league systems often used in sports like soccer, Multi-Tier Tournaments differ in several key aspects. First, unlike leagues, which run simultaneously over extended periods, Multi-Tier Tournaments progress sequentially within a single time frame so that the players in the lowest tier have an opportunity to win the entire event. Second, leagues use a single or double round-robin format, whereas Multi-Tier Tournaments, similar to Swiss tournaments, are mainly designed for tournaments with a large number of players and a small number of rounds.

Recently, UEFA updated the format for the Champions League. Similar to both Swiss and Multi-Tier Tournaments, teams play fewer games than in a round-robin, and like Multi-Tier Tournaments, the format incorporates a degree of fairness by having teams play half their matches at home and half away. However, there are also key differences between this format and Multi-Tier Tournaments. Multi-Tier Tournaments use cardinal Elo ratings for fairer pairing allocation, whereas UEFA uses a complex set of criteria, including random allocation and ordinal rankings, to determine pairings. In addition, Multi-Tier Tournaments do not include a knockout component, whereas the UEFA Champions League does.

Various metrics to evaluate tournament success have been explored in the literature. Scarf et al. [6] analyzed UEFA Champions League formats via simulations and found the round-robin to be the fairest in aligning pre- and post-tournament rankings. Similarly, Appleton [7] and Sziklai et al. [8] conducted simulations to assess different tournament formats, focusing on the win probability of the best teams and players. Key contributions in this field also include Glenn [9] and Searls [10], who both focus on the probability of the best players and teams winning in different tournament designs. As for Swiss tournaments, Olafsson [11] explored algorithmic properties of different tournament designs, whereas Fuhrlich et al. [12] and Sauer et al. [13] proposed a Swiss-based pairing algorithm that produced improvements in properties like color fairness over FIDE’s Swiss systems. For a review of the tournament design literature, see the recent survey by Devriesere et al. [14].

Tournament ranking systems have been theoretically explored by several researchers. Rubinstein [15] axiomatized the standard points-based system, Csato [16] studied Swiss system chess team tournaments, and Arlegi and Dimitrov [17] examined fairness in knockout tournaments.

Tiebreaking in tournaments typically follow two approaches. One involves a final contest to resolve ties, as seen in tennis tiebreakers, soccer penalty shootouts, and some American football overtime games. These methods, and their fairness, have been the subject of several studies. Che and Hendershott [18], Brams and Sanderson [19], and Granot and Gerchak [20] focus on bidding to determine which team picks a favored option. Additionally, studies, for example, by Apesteguia and Palacios-Huerta [21], Brams and Ismail [22], Brams et al. [23], Cohen-Zada et al. [24], Anbarci et al. [25], and Lambers and Spieksma [26] focus on determining a fairer ordering of play in tiebreak contests. (For further empirical analysis of the first-mover advantage in penalty shootouts, see Kocher et al., [27], Arrondel et al., [28], Rudi et al. [29], and Kassis et al., [30]). The second approach analyses different rules for tiebreaking in Swiss chess tournaments [31] and the FIFA World Cup group stage [32]. For an overview of tiebreaking methods, see [33; Chapter 1.3].

The concept of strategyproofness, or incentive compatibility, in tournament rules has also been a subject of interest. Selected contributions in this area include works by Pauly [34], Kendall and Lenten [35], Brams et al. [23], Dagaev and Sonin [36], Csato [33,37], Anbarci et al. [25], and Guyon [38].

Finally, the literature on the maximum number of consecutive games that can be played as White or Black in tournaments dates back to De Werra [39], who studied round-robin soccer tournaments. In this context, playing at home is similar to playing as White, and playing away is similar to playing as Black. For a review of the relevant results, see Goossens and Spieksma [40].

6. Conclusions

In contrast to Swiss tournaments, Multi-Tier Tournaments prioritize determining the pairings as fairly as possible prior to the start of games in a tier. In doing so, Multi-Tier Tournaments minimize, and eliminate whenever possible, biases related to the number of games played as White and Black (color fairness) and the number of consecutive games with each color (psychological fairness).

We showed how Multi-Tier Tournaments group players into tiers, and even within tiers, fairly. They also ensure that lower-rated players are afforded the opportunity to advance through the tiers, based on their performance. At the same time, they give a break to higher-rated players by allowing them to advance by playing fewer games.

Compared to knockout tournaments, where a single loss results in elimination, Multi-Tier Tournaments allow for a player’s loss in one game to be countered by his/her winning in subsequent games. Although our application focused on chess, Multi-Tier Tournaments are applicable to other games and sports where Elo or other scoring or rating methods are used, including tennis, soccer, and the big three of American sports—baseball, basketball, and football.

In fact, one may consider a season of play in a sport like baseball a tournament of sorts, wherein teams play about the same number of games against teams in their league over the course of a season. Their records in these leagues determine which teams go into the playoffs, which might be considered mini-tournaments themselves, requiring winning, for example, 2 out of 3, 3 out of 5, or 4 out of 7 games.

In most sports leagues, the number of games that teams play over a regular season does not depend on how they are rated (it is different leagues that reflect different levels of skill). Only in the playoffs are the numbers narrowed down, as in Multi-Tier Tournaments, when the best-performing teams or players ascend to higher tiers as poorer teams are eliminated. Multi-Tier Tournaments show how this competition can be structured in a fair and systematic way.

Supporting information

Acknowledgments

We are grateful to the referees for several helpful comments.