I've been watching a lot chess events recently and find the myriad of tournament formats quite interesting, with each system having its own pros and cons. I also play a lot of squash and contract bridge, and have always been fascinated by the dynamics of tournaments like squash ladders and Swiss teams events in bridge. I've had a go at devising some formats of my own and thought it would be nice to document these here. Most of these are quite natural ideas, so I wouldn't be at all surprised if they've already been invented (although I haven't come across them if they have). The main idea is what I've called a *Fibonacci Tournament*, in which competitors jostle to reach the apex of a pyramid and hold this position against attacks from their rivals.

As an introduction, here's a list of some common formats you might encounter in various sports and games:

**Knockout:**competitors are paired off (either randomly or according to some seeding), the losers being eliminated and the winners going through to the next round, until only one competitor remains. This format works best if the number of competitors is an integer power of 2 (e.g. 16, 32, etc.). Examples are extremely common and include the*FA Cup*in football, or the*Wimbledon*tennis tournament.**Pros:**simple; better competitors tend to meet in the later stages.**Cons:**very harsh – one bad performance can end your tournament.**Round-robin:**this is in the form of a league where each competitor plays every other competitor. Points are awarded for each win (or draw) and the overall winner is the competitor with the most points at the end of the competition. Examples include the*Premier League*in English football, and most other team sports.**Pros:**very fair.**Cons:**may require lots of matches; lots of one-sided matches in a mixed-ability field.**Double-elimination:**this format is common in American sports and is similar to the knockout format, except that each competitor gets two 'lives'. Competitors all start the competition in the*winners' bracket*, are knocked down to the*losers' bracket*the first time they lose, and are knocked out completely the second time they lose. The winner is the last competitor remaining in the competition.**Pros:**not as harsh as a straight knockout.**Cons:**not all competitors play the same number of games; the final may or may not have to be repeated, depending on how the competitors arrived there.**Swiss-system:**this format is suitable for a large field of competitors (too large for a round-robin) where a predetermined number of games will be played. Points are awarded for each win (or draw), just like in a round-robin. In each round of the competition, matches are determined such that each competitor faces an opponent with a similar overall score, so that the leading competitors tend to face off against one another, making for higher quality matches. Competitors are not usually permitted to face the same opponent twice. The winner is the competitor with the highest score after a set number of rounds. Examples include the*FIDE Grand Swiss*in chess and various bridge events.**Pros:**works well when lots of competitors.**Cons:**some competitors can have easier matches than others to gain the same number of points; no rematches; it is common for competitors to finish the tournament level on points, so some criteria for breaking ties is required.**Ladder:**competitors are ordered, with each competitor being assigned to a rung on a ladder. Competitors can move up the ladder by challenging a higher-ranked competitor. In the simplest version, competitors simply swap places if the lower-ranked competitor should beat the higher-ranked one. The aim is to get as high up the ladder as possible. More refined rules are also possible (with multiple competitors per rung) and there may be a limit to how many rungs competitors can climb at a time. This sort of competition is common in squash and tennis circles.**Pros:**informal.**Cons:**the challenge system means that some competitors may play more than others; the competitor at the top of the ladder at the end of the competition may not be the overall best competitor.**Hybrid:**these events are a mix of different formats. A very common example would be a round-robin competition from which the highest ranked competitors qualify for a subsequent knockout tournament. An example of this latter format is the*UEFA Champions League*in football.**Pros:**flexible.**Cons:**feels arbitrary; early good results may become irrelevant in the later stages of the competition.

In my opinion, the best thing about a knockout tournament is that the best competitors tend to play each other in the final stages; the worst thing about it is that one poor performance can end a competitor's chances. A round-robin is much fairer, since form will tend to even out, but the final stages of the competition can be anticlimatic, and mismatches will be common. Here are some ideas to get the best of both worlds.

## Nested-Robin

This is an extended version of a standard round-robin but with the best competitors playing each other more often towards the end of the competition, making for a more exciting climax to the tournament.

This tournament requires the number of competitors to be an integer power of 2 (e.g. 8, 16, etc.). A round-robin is played between the competitors, with point awarded for wins and draws in a standard way (e.g. 2 points for a win, 1 for a draw). At the end of the initial round-robin, the league is split into two halves: the best performing competitors go into the upper half, the worst performing into the lower. Competitors in the lower half are either eliminated completely, or can continue in their own sub-competition (to order the lower half) that has no bearing on the overall tournament winner. The competitors in the upper half then begin a second round-robin competition between themselves, carrying forward the points already accumulated. At the end of this second round-robin, each league is again partitioned into two, and this process repeats until a further split leaves just one competitor – the winner – remaining. The key idea is that the points from the previous round-robin carry forward to the next, meaning that every game counts, but that the best competitors end up playing each other multiple times toward the end of the competition, making for a more exciting finish.

For instance, if there were 16 competitors, then each would play an initial 15 games in the first round-robin, 7 in the second, 3 in the third and 1 in the last, making 26 games in total. If there are $2^n$ competitors, then $2^{n+1} - n - 2$ games will be required to determine a winner. This is the main disadvantage of this format: the number of games required is even more than for a standard round-robin! However, it works well for smaller fields.

## Rolling Knockout

This is like a standard knockout competition except that competitors simply go back to the 'first round' when they lose a match, rather than being eliminated entirely. The idea is to reach the 'final' and stay there for as long as possible, since the points for a win increase dramatically as competitors progress through the rounds. It's called a *Rolling Knockout* because the 'final' is not actually the final match in the tournament; in fact, the competition can roll on indefinitely with a final taking place in every round. The winner of the tournament is the competitor with the most points after a predetermined number of rounds.

Again, the number of competitors must be an integer power of 2 (i.e. $2^n$, where $n$ is an integer). Each competitor belongs to one of $n$ tiers, $0, \ldots, n-1$, and all competitors start off in tier 0. In each round, each competitor is randomly drawn against another competitor in the same tier, $t$. If they win the game, then they earn $2^t$ points and are promoted to the next tier up (subject to the maximum tier being $n-1$); if they lose the game, they do not accumulate any points and go back to tier 0. Draws are not permitted in this format, and should be resolved by some form of tiebreak (e.g. a penalty shootout).

For example, if there were 8 competitors, then there would initially be 4 tier 0 matches (quarter-finals). The 4 winners would each gain 1 point and advance to tier 1 (semi-finals); the 4 losers would stay on tier 0. In the second round, there will be two quarter-finals and two semi-finals. The 2 winners of the quarter-finals (tier 0) would each gain 1 point and advance to the semi-finals (tier 1) in the next round; the 2 winners of the semi-finals (tier 1) would each gain 2 points and advance to the final (tier 2); the 4 losers would go back to the quarter-finals. In the third round (and all subsequent rounds), there will be two quarter-finals (tier 0), one semi-final (tier 1) and one final (tier 2). The 2 quarter-final winners will again earn 1 point each and advance to the semi-final; the 1 semi-final winner will earn 2 points and advance to the next final; the winner of the final will earn 4 points and also qualify for the next final (i.e. remain in tier 2); the 4 losers will once more go back to the quarter-finals.

The advantage of this format is that it has the purity of a knockout, with the best competitors rising to the apex of the pyramid in logarithmic time (for a competition with $N=2^n$ competitors, it takes $n-1$ wins to reach the final). In addition, the best competitors will repeatedly meet at the higher tiers, leading to high-quality games and giving ample opportunity for rematches and grudge matches. However, since a competitor who loses is not immediately eliminated, it removes a lot of the randomness of a standard knockout.

Disadvantages of this format include the necessity to for the number of competitors to be an integer power of 2 and the need for each game to end in a decisive result (no draws). For a larger field, it could also be argued that the sending losers all the way back to tier 0 is overly harsh. These defects are addressed by the *Fibonacci Tournament* (see below) which is a refinement of the *Rolling Knockout*.

## Fibonacci Tournament

This is a refinement of the *Rolling Knockout*. Once again, the objective is to reach the apex of competition and hold this position against all comers. The main difference with the *Rolling Knockout* is that losing competitors go back two rounds, rather than all the way back to the 'first round', which makes mistakes less costly. It is also more flexible because it can handle draws and works with any number of competitors.

Like in the *Rolling Knockout* format, competitors are partitioned into tiers, with all competitors starting the competition in tier 0. This time there are $T$ tiers in total, where the value of $T$ depends of the number of competitors (see below). At the start of each round, all the competitors are ranked in a ladder such that no competitior of a lower tier appears above a competitor of a higher tier. Within each tier, the order in the ladder is randomised. Games are now assigned as follows: the first competitor (at the top of the ladder) plays the second; the third plays the fourth; the fifth plays the sixth; etc. If there are an odd number of competitors, then the competitor unlucky enough to be at the bottom of the ladder skips that round. After each game, the following adjustment is made before the next round:

- Winners (in tier $t$) gain $2^{t+1}$ points and move up 1 tier (up to a maximum of $T\!-\!1$).
- Losers move down 2 tiers (down to a minimum of 0).
- Drawing competitors (in tier $t$) gain $2^t$ points and move down 1 tier (down to a minimum of 0).

The draw for the next round then takes place and the competition continues in this way for a set number of rounds, at which point the winner is the competitor that has accumulated the most points.

The number of tiers, $T$, depends on the number of competitors, $N$. A general formula for $T$ will be given below. For now, here is a table with the first few values:

Number of competitors | Number of tiers |
---|---|

2 | 1 |

3-5 | 2 |

6-10 | 3 |

11-18 | 4 |

19-31 | 5 |

32-51 | 6 |

52-84 | 7 |

85-137 | 8 |

138-224 | 9 |

225-364 | 10 |

365-590 | 11 |

591-957 | 12 |

### Advantages

- Works for any number of competitors.
- All competitors play the same number of matches.
- The tournament quickly stratifies into a pyramid structure, so that the best competitors rise to the top. This ensures consistently competitive matches at all levels.
- The idea of striving to reach and occupy the apex of the pyramid is inherently exciting.
- The top of the ladder can be reached in logarithmic time ($T$ steps), so it is well-suited to competitions with a large number of competitors.
- The best competitor will win in the longterm: the penalty for having a single poor performance is to drop down two tiers, which can be regained with two successive wins (rather than going all the way back to tier 0, as in the
*Rolling Knockout*format). - It can handle draws in a sensible way (see below).
- The tournament can be tailored to any number of matches, depending on the requirements of the organisers.
- The tournament can be split into smaller sub-tournaments, with perhaps a prize for each, and then an overall grand prize for the competitor with the most cumulative points at the end of the tournament season. Competitors' tiers could be optionally carried over between tournaments and competitors could leave or join the event (with new competitors starting out on tier 0).
- The scale by which points are awarded means that ties between competitors at the end of the tournament will be less common compared with the
*Swiss-system*. - The random element of the draw means that competitors may play each other repeatedly. This will be especially common in the higher echelons of the pyramid, where the best competitors can expect to meet each other multiple times during the tournament.

### A note on draws

An alternative to the outlined scheme would be for competitors to remain at the same tier following a draw. However, I believe it is better to discourage draws wherever possible, treating them as 'less than half a win', in some sense (such as in football, where 3 points are awarded for a win and just 1 for a draw). Since competitors gain 1 tier for a win, and drop 2 for a loss, dropping 1 tier for a draw seems like the correct approach to encourage attacking play (since a draw is then not a 'good' outcome for either competitor). An exception could be made if there is some bias which makes it more likely for one of the two competitors to win. For instance, in chess there is a distinct advantage to the player with the white pieces. In this case, I would suggest that a player drawing with the white pieces drops by 1 tier, whereas a player drawing with the black pieces remains at the same tier for the next round. Alternatively, matches in a chess tournament could be best of two games, one with each colour.

### Further notes:

- A
*Fibonacci Tournament*with exactly 8 competitors (and no draws permitted) is exactly the same as a*Rolling Knockout*with the same number of competitors. - For games like chess, where one player takes the white pieces and the other the black ones, I would suggest the following scheme to determine which colour pieces each player should have: if they played with different colours in the previous round, then they swap to the opposite colour in this round; if they played with the same colour in the previous game, then who gets which colour is determined randomly. This scheme means that, on average, the competitors play with the same colour in two successive games 25% of the time.

### Why Fibonacci?

Why is this called a *Fibonacci Tournament*? It's because for a competition with $N$ competitors and $T$ tiers (and no draws allowed), the mean number of competitors, $\bar{N}_t$, in each tier obeys:

$F_m$ is the $m$-th Fibonacci number, where $F_0 = 0$, $F_1 = 1$ and $F_{m+2} = F_{m+1} + F_m$. In other words, the Fibonacci numbers form a sequence $0, 1, 1, 2, 3, 5, 8, 13 \ldots$, where each number in the sequence is the sum of the preceding two. For instance, in a competition with 7 tiers and 66 competitors, we expect there to be 26 competitors in tier 0, 16 in tier 1, 10 in tier 2, 6 in tier 3, 4 in tier 4, 2 in tier 5 and 2 in tier 6. This is only 'on average': sometimes there will be a more than two competititors in the top tier; sometimes there will be fewer. Fibonacci numbers grow exponentially according to the following approximate formula:

$$ F_m \approx \frac{\varphi^m}{\sqrt{5}} $$which becomes increasingly more accurate as $m$ increases. $\varphi$ is the *Golden Ratio*, given by:

which means that, for large $m$, each tier will have (on average) about 62% more competitors in it than the one above.

The format can work perfectly when there are no draws and the number of competitors in each tier, $t$, is twice the Fibonacci number, $F_{T - t}$. Consider a tournament with 5 tiers and 24 competitors, such that 10 are in tier 0, 6 in tier 1, 4 in tier 2, 2 in tier 3 and 2 in tier 4. This partition of the competitors is stable and remains the same in all subsequent rounds: competitors can move between tiers but the number of competitors in each tier is fixed. As it happens, the requirement for all competitors to start the competition on an equal footing, in tier 0, means that this precise configuration can never be reached exactly, at least in this particular example. However, the average (mean) occupancy of each tier will tend towards this distribution, given enough time. The reason for this is that the Fibonacci numbers exhibit the following relationship:

$$ 2 F_m = F_{m+1} + F_{m-2} $$which ensures that this is a stationary distribution of competitors (derivation left to the interested reader).

### Choosing $T$

For a competition with $N$ competitiors, $T$ is chosen in such a way that the mean occupancy of the highest tier is approximately 2 (these two competitors can be thought of as being in the 'final' for that round of the competition):

$$ T = \textrm{round} \left[ \frac{\log{(N+2)} + \log{\sqrt{5}} - \log{2}}{\log{\varphi}} \right] $$In a competition that runs over a long period of time (perhaps indefinitely), competitors may join (starting at tier 0) or leave as the competition progresses. In this case, $N$ effectively varies from round to round and the maximum attainable tier, $T\!-\!1$, varies accordingly. Depending on the requirements of the competition, competitors may be permitted to skip rounds. In this case, they are not included in the random draw for the round in question and do not contribute to $N$. However, they maintain their existing tier should they return to the competition in a later round. Competitors who enter a round at a tier exceeding the current maximum tier (because the number of active competitors has reduced) maintain this tier (i.e. it is not reduced to the current maximum) but cannot increase it any further following a win.

## Fibonacci League

This competition has a same structure to a Fibonacci Tournament. In each round, competitors are divided into leagues (of given size, $M$) instead of pairs. Each league is resolved via a round-robin. Upon the conclusion of each league, some competitors will be promoted (to a league one tier higher), some relegated (to a league two tiers lower) and some either remain in the same tier or drop down by one tier. An advantage of this system is that fixtures are known a few games in advance, making planning easier if travel is involved. The value of $T$ should be chosen to ensure that the expected number of competitors in the top tier is approximately $M$.

## Fibonacci Ladder

This competition is a bit like a ladder tournament, except with points accumulated for each match played. All entrants start at tier 0 and can challenge any other entrant. The rules are very similar to a Fibonacci Tournament. A major difference is that there is no maximum tier. Instead, the following rules apply to the outcome of a match between two competitors, where competitor #1 is in tier $t_1$ and competitor #2 is in tier $t_2$:

- The winner gains $2^{\min(t_1, t_2)+1}$ points. If the winner's opponent is from the same tier or a higher tier, then the winner moves up 1 tier; if the winner's opponent is from a lower tier, then the winner remains in their current tier.
- The loser moves down 2 tiers (down to a minimum of 0).
- Following a draw, both competitors gain $2^{\min(t_1, t_2)}$ points and move down 1 tier (down to a minimum of 0).

This format is similar in nature to a squash ladder. It suffers from the same disadvantage that some competitors may challenge more than others, so not all competitors play the same number of games. Also, it would be possible to 'game the system' to some extent, if enough competitors were to collude. An advantage that the Fibonacci ladder has over a standard squash ladder is that points are accumulated over time, meaning that a competitor's overall performance can be measured in some way. In this regard, the system has some similarity with the Masterpoints system used in bridge, which rewards frequency of play as well as quality.

## Squash Tournament

Here's a fun idea for a squash tournament, following the Fibonacci framework, in which matches are played at regular intervals (e.g. monthly). Each match is first to 25 points, scored with English scoring (in which only the server can score a point), and is won by the first player to reach at least 25 points and be at least 2 points clear of their opponent. The matches in each round are drawn in the same way as for a Fibonacci tournament. Points are awarded as follows:

- A losing player is awarded a point for each point they scored in the match, up to a maximum of 25. The awarded points are then multiplied by 2 for each tier above 0 the player has attained. They then drop down by two tiers (down to a minimum of 0).
- A winning player is awarded 50 points, minus the the number of points their opponent scored in the match, down to a minimum of 25. The awarded points are then multiplied by 2 for each tier above 0 the player has attained. They then gain one tier (up to a maximum tier of $T\!-\!1$).

For example, Alice (tier 2) beats Bob (tier 3) by a score of 25-13. Alice gains $2^2 \times (50-13) = 148$ points and moves up to tier 3; Bob gains $2^3 \times 13 = 104$ points and moves down to tier 1. In the same round, Charlie (tier 1) beats Daisy (tier 1) by a score of 29-27. In this case, both players score $2 \times 25 = 50$ points (since the match went into overtime), but Charlie moves up to tier 2 and Daisy moves down to tier 0.

The winner of the tournament is the player who has accumulated the most points at the end of a certain number of matches.