Pure KOTH with 50 vs 100 players

After my simulation of 1000 tourneys of 45-rounds pure King-of-the-Hill match between 50 of the world’s top 100 Scrabble players, Adam Kretschmer asked if having 100 players in a pure KotH tournament will reduce the likelihood of excessive repeats between two players.

To that end I ran another simulation for 45-rounds featuring 100 players as a comparison. The initial exercise was modeled after a hypothetical top division in the revived Grand Causeway Challenge tournament. For this simulation, I modeled it after a hypothetical second division, with 100 participants randomly pulled from among players ranked 101st to 300rd in the world based on WESPA ratings as of 18 January 2015, i.e. Odette Rio to Martin de Mello. This incidentally worked out nicely in terms of rating gap within a division, as the difference between player #101 to #300 (231 points) is comparable with that between player #2 to #100( 269 points). (Note: I compare the rating gap excluding Player #1, who was discovered to be a whale’s tail in the earlier simulation. The data below however still uses the full simulation which included his results.)

The hunch is that more players in the field will reduce the possible maximum repeats between two players. The simulation seems to be bearing this out somewhat.

50v100_Max repetitions (all tourney)

The most frequently occurring maximum repeats in the tournament in the second division shifted leftwards from that of the premier division, to 11 times rather than 13 times. One can expect in any tournament that there will be at least a pair of players who will get to play each other double-digit times. But overall, enlarging the field does reduce the chance of high-number of repeats.

However it is interesting to note that at the extreme top end, a worst-case of 26 repeats happened again. No tournaments in either divisions end up with fewer than maximum 7 repeats. So it seems the number of players would not affect the boundaries. My guess is these are more dependent on the number of rounds in the tournament (of course longer giving more chance of repeats), with the worst maximum repeats estimated at slightly below 60% of the number of rounds.

Now comparing the two divisions in terms of number of repeats for each player based on final result, the result seems to be strikingly similar.

50v100Max repeats vs median rankBoth left and right charts have the same extreme cases (26-game head-to-head for table-topper, 24-game duel for bottom-dweller), and below 10 repeats for the majority of the rest. There are two telling patterns to observe.

First, the bottom of the chart sinks slightly for Second Division. Meaning, while in Premier Division you’d be very unlikely to never play at least one other person 3 times, in Second Division it is much more frequent to see players who has maximum of one repeat (i.e. 2 games played) against a same opponent. In fact, three-peat is the most common experience for Second Division as opposed to 4-peat in Premier Division. So, good news for those who don’t like repeats.

Second, the chart in Second Division doesn’t scale up much from Premier Division, but rather it elongates. Rather than curving to maintain a U shape, its centre holds to form a flat shape, and its ends are practically identical to that of Premier Division. Meaning, you can view the field as comprising 3 groups: top 10, bottom 10, and everyone else. Changing the field size will not change the groups to top 15 or bottom 15; basically everyone outside top 10 and bottom 10 in a division can expect comparable number of repeats, while the special ones can expect more repeats.

In other words: enlarging the field does NOT reduce the number of people affected by excessive repeats in a division. What it does is reduce the percentage, and reduce the number of people with excessive repeats in a tourney. Instead of having top 10 and bottom 10 in two divisions of 50 playing more repeats, now you only have top 10 and bottom 10 of a combined division of 100 experiencing that.

While it seems things are more or less the same in terms of repeats, the picture is different in terms of number of different opponents played.

50v100Number of opponents vs median rank

Again, the Second Division’s chart is a horizontally-elongated version of the Premier Division’s. But it clearly shows the entire chart shifted higher by one band of 5: everyone – and I mean everyone, including top and bottom 10 – plays 5 more different opponents. Clearly a good news for those favouring variety.

To interpret this with relation to the earlier chart on maximum repeats: you will have a nemesis that you meet about as frequently regardless of the field size. But when you’re not meeting your nemesis, chances are your sideshow opponents will be more varied in a larger field than in a smaller one.

And same as before, initial seeding has no bearings on likely number of different opponents met.

50v100Number of opponents vs seedingRegardless of seeds, one can expect about 30-31 different opponents in a division of 100 with 45 rounds, as opposed to 25-26 for division of 50. And there is no clear outlier in Second Division, unlike the outlier due to the Nigel effect in the left chart. So it seems the Nigel effect is not bad enough to affect the rest of the field; it’s just him.

Some previous comments suggested that instead of randomly selecting players for every simulation, I should choose one set of 100 players and use them throughout the 1000 iterations. I plotted the same for these and the general finding, in terms of trend and expected number of repeats and different opponents, are the same. One difference is in the chart on the most number of repetitions in a division.

50v100_sameMax repetitions (all tourney)


Here, with the same set of 100 players throughout, somehow there is a spike where 13 repeat match-ups is the maximum in the most number of tournaments. Other trends seem to be the same. I have not yet found an explanation for this, and hence will leave it at that.

In conclusion: more players in the field doesn’t reduce the maximum number of repeats in a division, but does increase the chance of meeting more diverse opponents.

To close on the same note, here is the tile chart of the winners in the 1000 simulation where the set of players are not fixed. Feel free to cheer or jeer at players who don’t perform to their ratings expectations.

50v100Winners Treemap

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