With two-stage tournaments commonplace, one thing tournament directors have to live with – and most players dread – is the Play Off, and its sibling the Tie Break.
In Swiss opening stages, the ideal situation is that taking all the players with at least X wins gives a nice round number for the final knockout. This rarely happens, and even with more complex schemes that in theory guarantee it, one person not arriving on the day can break the pattern. In World Cup style groups, as well as leagues with promotion and relegation, players can end up tied with no obvious way to separate them, particularly in very small groups.
There are a number of ways to sort out such situations, which I’ll take a look at below. First, though, a quick observation; in face to face tournaments, there is almost always time pressure on the playoffs to produce a quick answer, as all other players have to wait and watch. In online tournaments and leagues, there’s more time to sort things out and we can sometimes adopt schemes that reduce the element of luck in the draw.
The easy cases!
Two players tied for one qualification place is easy – they just play a single match and the winner progresses (or avoids relegation).
Four players tied is also easy. With one player to progress from four, they play a simple knockout. With two to progress, they can be drawn together in pairs and the two winners qualify. With three to progress, you can resort to:
In this tie-break, your four competitors play a simple knockout, but the loser of each first-round game goes into the “final”, and the loser of that “final” is the one who fails to progress in the main tournament. Unlike the general method below, all four competitors have an equal draw.
The general method
With Swisses, you can end up with some really difficult numbers to sort out. Fortunately this method always works, although some lucky players will end up with twice the chance of others of getting through. That is usually necessary – I would hate to have to work an absolutely fair scheme and it might take a very long time to work out!
In the example above, I need three qualifiers from eight. So I start at the furthest right round in the drawsheet which can accommodate the qualifiers (the round of 4), and I mark the places they will end up – the three starred slots. For clarity, I’ve scrubbed out the section that doesn’t lead to these places.
Our truncated drawsheet now has rounds of 3, 6, 12, … and we move leftwards to the first round that has enough slots for all the players we’re tie-breaking. In this case the round of 12. We fill it up by drawing names at random and putting them first in the odd numbered slots, and then the even numbered ones – this automatically sorts out the correct number of byes (getting *that* wrong is a nightmare I don’t want to relive!). I’ve written them in in order A, B, C, … and we have a perfectly done draw.
As you can see, C, D, E and F only end up playing one game and A, B, G and H play two – someone had to be lucky, given the time constraint…
Three is the not-so-magic number
A number of times in the Online Championships and Charity Challenge, when using the world cup format, I’ve had to do a tie-break among three people. With plenty of time available, we can give all three a more equal chance in two ways.
The first is potentially long, but easy. Draw the three players in a random order; say, A, B, C. First A and B play a match. Suppose A wins; he then plays C. If A also beats C, then he’s clearly through. Otherwise C plays B, and we keep going with the winner of each match playing the person who’s sitting out next, until someone beats both their opponents in back to back games. The draw is not entirely even – A, B and C’s winning chances are in the ratio 5:5:4 – but it’s far more even than the 2:1:1 ratio of the general method.
This method can also tie-break two qualifiers from three, again by doing an anti-tournament – follow the same pattern but with the loser staying on after each match until someone has lost back to back matches, and they are the one eliminated.
The downside is that it may take a large number of matches to actually come to a conclusion. So to finish off the article, here’s a method that was shown to me by Rick Janowski, which achieves both goals – the draw introduces a minimal element of luck and you are guaranteed to only need two matches. Draw A, B and C again. A plays B in a match. Say A wins; A then plays C as if they were in a knockout where C had a bye, but A starts at 1-away 2-away Crawford. This gives C 32.2% chance to qualify, assuming equal opponents, and because A and B originally started with the same chance they therefore began at 33.9% – which is pretty close to ideal.
I hope I’ve provided some insight into the various forms playoffs can take – while there’s no format that works perfectly in all situations, there are a lot of options available to the TD to keep things moving and provide an enjoyable experience for all!