TL;DR

Iâ€™ve become curious about how to organize tournaments (in the sense of allocating games in it) when games have more than two players.

In these days of Coronavirus, thereâ€™s been a surge in online playing, and board games make no exception. In particular, BoardGameArena is a very nice place to play online, and they saw about a 6x increase in their traffic (which gave them a few issuesâ€¦).

One thing that always tickled me is that there is a tournament system. Alas, as it is today their system only caters for two-players games, i.e. even when games would allow for additional players, instances in tournaments only allow two. This is sub-optimal in a lot of games that I like (e.g. Tokaido) and that are better played with three or more players.

# Two-players tournaments are easyâ€¦

â€¦ and there are a lot of ways to set them up. One of the easiest ways is to get a number of players that is a power of 2, then half them at each round with direct eliminations.

One consequence of direct-elimination matches is that half of the people will play only a single game in the tournament. Which might be good for the competition, less for people who wants to play đź¤¨

In the two-players space the answer to this issue is round robin tournaments, in which each participant plays against each other. This is the same as, for example, sport leagues (although often in this case they play two matches against each other participant).

# Direct elimination for more players?

One easy extension is to organize $N$-players games in direct-elimination matches, assuming that there are $N^p$ players. This means having $p$ rounds. Alas, this has the same drawback of the direct elimination tournaments for two-players games, with the additional negative aspects that:

• at each round, more people end playing at the same time ($\frac{2}{3}$ for three-players games, $\frac{3}{4}$ for four-players games, and so onâ€¦)
• more players are needed for the same number of rounds (e.g. a three-rounds tournament for two-players games would require $2^3=8$ participants, for three-players games would require $3^3=27$ participants, i.e. more than three times).

One possibility is to let more players through each round. For example, letting two players pass to the next round in four-player games basically means keeping the same structure as in two-player direct-elimination tournaments:

 1-1 1-2 1-3 1-4   1-5 1-6 1-7 1-8

2-1 2-2           2-3 2-4

winner


At any rate, this does not solve the need for people to play more!

# k-players leagues?

Another solution would be to find a way to extend the â€śleagueâ€ť approach to $k$-players games (out of a total population of $n$ players participating in the tournament).

One straightforward way to do this is to form all $n \choose k$ sub-sets of $k$ players out of all $n$ participants, where each of them will play in ${n-1} \choose {k-1}$ matches. This might mean a few too many matches though: a tournament with 8 players overall and 4-player matches would mean 70 matches overall, where each player competes in 35 of them. Ooops.

One observation is that many of those matches areâ€¦ redundant. From the example with 4-players matches out of 8 total participants, we have the following matches (among others):

1 2 3 4
1 2 3 5
1 2 3 6
1 2 3 7
1 2 3 8


They are somehowâ€¦ pretty similar, in that players 1, 2, and 3 are playing five matches with each other inside.

Hence, while itâ€™s interesting that each participant plays against every other one at some time, we can probably remove a lot of the redundant games and enjoy the tournament. In general, we should aim for an arrangement where we have a limit on how many times the same people play at the same table.

# Block Designs

One answer to the challenge in the previous section is to leverage Block Designs. This is the definition for t-designs (slightly adapted):

Given a finite set $X$ (of $v$ elements called points) and integers $t$, $k$, $r$, $\lambda \geq 1$, we define a $t$-design $B$ to be a family of $k$-element subsets of $X$, called blocks, such that any $x$ in $X$ is contained in $r$ blocks, and any subset of $t$ of distinct points is contained in $\lambda$ blocks. The number of elements in family $B$ is $b$.

Uh? Translated in tournamentese:

• we have a set $X$ of $v$ participants to the tournament;
• we want to organize matches with $k$ players inside;
• each player competes in $r$ matches;
• we want that $t$ players compete in exactly $\lambda$ matches in which all are present at the same time.

In case we want to limit the number of times pairs of players compete at the same table, we set $t = 2$, deal with 2-designs, and call them BIBD (Balanced Incomplete Block Designs). Actually, block design usually refers to 2-designs, and we will stick with them in the following.

Easy right? Next problem pleaaaaase!

Well, not so fast.

# Block Designs for arranging matches?

There are several different ways to create BIBDs, not all totally amenable for tabletop games. For example, consider the BIBD induced by the Fano plane (numbers are player identifiers, from 0 up to 6):

(1, 3, 5)
(0, 3, 4)
(2, 3, 6)
(0, 1, 2)
(1, 4, 6)
(0, 5, 6)
(2, 4, 5)


It might be applicable to a tournament of 7 players for games that accept 3 players at a time. Everyone plays against anybody else, but only once. Yay!

There is a big defect though. As any other block design based on finite projective planes, it has the characteristic that any two blocks always share exactly one point. Which means: if you want to limit the number of games that are played at the same time, e.g. so that each player is only on a single table at the same time (like a real-time table would more or less require), then you can only play a very limited number of games at the same time. In particular, if you limit to one game per player at most, at any time only a single game can run, where three participants play and four wait.

# Finite affine planes to the rescue

On the other hand, BIBDs induced by finite affine plaines do not have this limitation. Itâ€™s easy to get one from a finite projective plane: just get rid of a block and all items in it, and what youâ€™re left with is a finite affine plane. In the Fano plane case, letâ€™s get rid of the group (0, 5, 6), as well as players 0, 5, and 6:

(1, 3, 5) -> (1, 3)
(0, 3, 4) -> (3, 4)
(2, 3, 6) -> (2, 3)
(0, 1, 2) -> (1, 2)
(1, 4, 6) -> (1, 4)
(0, 5, 6) ->
(2, 4, 5) -> (2, 4)


This should look familiar: itâ€™s basically a round-robin arrangement for four players, which allows two real-time games to go on at the same time, for a total of 3 rounds:

1st round: (1, 2) (3, 4)
2nd round: (1, 3) (2, 4)
3rd round: (1, 4) (2, 3)


This is quite amenable now:

• everybody plays against each other, but no more than once
• everybody plays at each round
• everybody plays a reasonable amount of games

# Alternative paths of investigation

Another promising path for investigating is to explore the so-called Social Golfer Problem (see also here), which is formulated as follows:

The task is to schedule $g \cdot p$ golfers in $g$ groups of $p$ players for $w$ weeks such that no two golfers play in the same group more than once.

The goal is to find the minimum number of weeks $w$ where this can happen, and Iâ€™m also not sure at the moment if this also requires that every golfer still plays with every other oneâ€¦ but it seems promising to find out something to code in the future, and also to overcome some rigidity in the schemas that we will investigate in the short time.

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