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Dr Ian Plummer

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Ties in Swiss Tournaments

Kevin Carter writes

It is usual when running a Swiss event to aim for n+2 rounds, where 2n is the next highest power of two above the number of participants. This is said to optimise the probability of a single winner.

23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256

To test this I have written a computer simulation of a Swiss tournament for up to 32 players.

It shows, for instance, that in a 20-player competition run over 7 rounds (n+2 where 2n=32), there is a 62% chance of an outright top and a 26% chance of a 'resolved tie' - that is two or more players on the same number of wins but one has beaten the other(s). So, this gives an 88% probability of a single winner. The other 12% are 'unresolved ties' - typically three players equal on wins and each having beaten one of the other two.

In this example n+2 rounds are superior to n+1, at only 60%, and n+3, at 68%. However, nine rounds (n+4) proves to be slightly better for a 20-player event, at an 89% probability of a single winner.

Running the simulation over several different player numbers and rounds, we find that up to 24 players there is generally a double peak optimum, at about 90%, for n+2 and n+4. Above 24 players interestingly the double peak moves to n+3 and n+5. For instance, for 32 players 10 rounds are optimum, at 92%, just ahead of 8 rounds at 88%, with 7 rounds (n+2) down at only 69% and 9 rounds (n+4) at 74%.

The above results all assume each player in every game has a 50% chance of winning. In reality this will not be so. In a handicap tournament we would generally expect a shallow normal distribution of abilities and in a level event a skewed distribution.

The simulation was re-run for both of these distributions. Surprisingly the results did not vary a great deal - an improvement of 2-4% being typical.

So, what should be done about the one in ten of all Swiss events that do not yield an outright winner? I have seen counting net points and shooting at the peg, but both are terrible solutions. Mathematically the fairest would seem to be trying to ascertain which of the tied players has faced the strongest opposition, for instance by totalling the number of wins by the opponents of each.

Regards, Kevin Carter; 27.5.98

Author: Kevin Carter
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Updated 28.i.16
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