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:

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:

  1. Winners (in tier $t$) gain $2^{t+1}$ points and move up 1 tier (up to a maximum of $T\!-\!1$).
  2. Losers move down 2 tiers (down to a minimum of 0).
  3. 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
21
3-52
6-103
11-184
19-315
32-516
52-847
85-1378
138-2249
225-36410
365-59011
591-95712

Advantages

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:

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:

$$ \frac{\bar{N}_t}{N} = \frac{F_{T-t}}{F_{T+2} - 1} $$

$F_m$ 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:

$$ \varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618 $$

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 $t_2$:

  1. 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.
  2. The loser moves down 2 tiers (down to a minimum of 0).
  3. 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:

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.