In my previous post I ran a simulation of 1000 tournaments with 45 games of King of the Hills with repeats among 50 of the world’s top 100 Scrabble players, to see how many repeats potentially may happen at worst with that format. Andrew Fisher asked about the details on individual players and how those change based on their expected wins. The plot below charts the maximum repeats for each individual player, against their expected wins.

My initial suspicion was that players with highest and lowest win expectations are at the most risk of repeats. But it seems from the chart above that the players on the extremities of expectations are not the victims of multiple repeats. The graph itself doesn’t illustrate much else with its odd whale-like shape. One can discern that the incidence of highest repeats happen both on the left and right side of the chart, and there is a gap separating the left and right albeit asymmetrically. But other than that, it seems we can’t really infer from this. So I explored another angle: does a player’s placing in the tournament, another proxy for player strength, determine the maximum number of repeats?

I used median of the rank, rather than the standing at the end of the tournament, to reflect better overall strength throughout the tournament. As expected, the extremes – top and bottom 2 – bear the brunt and are quite likely to have repeats of 15 times or more. Player in the range between top 10 and bottom 10 are very unlikely to experience repeats more than 7 times. From the chart, 4 repeats (the string of bigger dark red circles) seem to be what one can expect to experience. It’s interesting to note that although very rare, there are instances of players playing a same opponent only twice.

This chart however is a hindsight – based on the outcome of the tournament. Can one predict the number of repeats based on pre-tournament strength? I plotted the highest numbers of repeats against seeding below to explore.

The chart above is more shapeless than the rank-based chart, without the distinct U shape. It seems majority of the time everyone regardless of seeds can expect to play at least one opponent **four times**, with slightly lower chance but still pretty common to play a 4- and 5-peat, and even a repeat up to 10 times seem almost equally likely for everyone. In short: initial seed doesn’t really predict much the number of repeats you’ll have.

Apart from repeats, some players are interested to have a variety of opponents – hence the dislike of repeats for some. I repeated the explorations above to see if either median of rank or seeding has much to do with the number of opponents one will play, as shown in the charts below.

As expected, the results are somewhat inverses of the charts on repeats. After all, more repeats should mean fewer opportunities of playing other opponents. In the top chart, most players between top 10 and bottom 10 are likely to play **26** different opponents, while those in top and bottom 5 can expect fewer than 20 different opponents. So if you want variety, aim to perform in the middle.

When compared to initial seeding, again there is not much that can be discerned, majority are expected to play about 26 opponents whatever the seeding is. However, an anomaly stands out: the top seed disproportionately plays at most 14 different opponents most of the time, and never more than 29 different opponents. This begs further investigation, so I plotted the performance of every seed to see number of times they play different number of opponents. The result stunned me.

While most players regardless of seed seems to follow the same pattern, the top seed is clearly different. Which is when I realised: the top seed in my simulation was always the world number one Nigel Richards. If anyone has anything to complain about facing homogeneous opponents in 45 pure KOTH with repeats, it’s Nigel. Which brings me back to the first chart.

The anomalous gap in between the right and left part of the chart? That was because Nigel almost single-handedly forms a separate chunk to the right, with win expectations way ahead of everyone else. In other words, **Nigel breaks the statistical visual analysis**. He’s so far ahead of everyone else that one has to be careful when generalizing inferrences from a sample data that includes him.

I will be running a separate simulation for a hypothetical second division with 100 players drawn from players ranked 101 to 300 in the world ranking. With no large rating gaps there, I expect that simulation will show a more uniform chart than the above. Nonetheless, I think the conclusion will be the same: that initial seeding / expected wins do not have that big an impact on number of repeats or distinct opponents if there are no big rating gaps. What would be of interest is whether the higher number of players would reduce the repeats.

To end, here’s a few more fun imaginary tidbits from the virtual tournaments in the simulation:

- Nigel won 60.1% of the 1000 tournaments. This may sound a bit lower than what some of us would predict (and which may be due to my imperfect modelling of the spread in the simulation), but it is still amazing that one person can single handedly win more tournaments than the rest of the 99 next best players combined.
- Apart from Nigel only 42 other people managed to win a tourney, 14 of which were one-time winners. (
*Note that the percentages above may differ for most players on the same win since not everyone play the same number of tourneys. Only Nigel was in all 1000 of the simulation, while the others were randomly slotted in, resulting in most people having around 500 tourney participation.*) - The lowest rated person to win any of the tournament was 71st-ranked Stewart Holden, with 1964 rating. It’s exciting to know that outsiders do stand a chance even in the face of such advantage by Nigel. Even if Stewart managed to do it just once in 493 attempts 🙂
- Nigel’s lowest final position was 28th – one of only 23 times that he was not in the top 10.
- Nigel had one tournament where he was number 1 in 44 out of the 45 rounds. One other player managed that incredible feat: Paul Gallen, whom after beating Nigel in the first round, reached top position after winning his second round and stayed atop till the end.
- The biggest winning margin was 12.5 games, when Nigel had a 39.5-5.5 +3069 record, ahead of Pakorn Nemitrmansuk at 27-18 +1586

Your graphical displays are wonderful, and your comments and explorations are astute.

When you do 1000 sims, you use a different set of 50 players each time. This smooths out anomalies due to composition of the field, which may have advantages.

However I believe it is more revealing to use the same set for the 1000 sims. This should have the Nigel cloud nicely rearranged as a vertical line, when the horizontal axis is prior seeding, and may give sharper images for the other displays.

Thanks Barry. You are right that using the same set of players will reveal more on specific effect of the pairings.

You are right also that the different set I used can smooth out anomalies, which was why I chose to use different sets. This exercise started off as simulation of Causeway 2016 premier division, and the set was selected more for that purpose than general investigation.

I may compare both fixed and random sets in my next simulation for the 100 players in division 2.

Great analysis, Ricky, but I just noticed one problem when I was building on the outputs for my model – always having Nigel in the sample inflates Nigel’s winning percentage. According to my model (which sadly doesn’t allow for ties yet) & simulations, if Nigel gets drawn with the same probability as other players, he wins 31.7% of the tournaments, which is, correctly, close to half of the 60.1% you predict. Because Nigel isn’t in every sample, everyone from 1st to 30th ranking won at least once in my simulation:

Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Wins 317 84 79 82 65 56 37 27 25 25 17 20 13 20 16 11 14 16 13 1 5 3 4 1 6 2 3 1 3 1

[Tidbits: 2/3 of the players ranked 31-60 won at least one iteration. The “outside chance” person who won in my simulation is the 89th-ranked Vannitha! 64th ranked Jesse Matthews also made the list. As I was using the outdated data available on Aardvark, I didn’t make the list 🙁 ]

So the conclusion should be that according to his rating, Nigel should win 60.1% of the tournaments _he plays in_ involving a random draw of 50 players in the top 100, KOTH-no-repeats. That’s still a great achievement, but of course making the assumption that the ratings reflect accurate win percentages.

While the 31.7% estimate is less meaningful, you can look at two players’ wins to roughly compare how much more likely a player is to win the tournament versus another (e.g., if each player is sampled around the same number of times, Nigel is about 4 times as likely to win compared to PG, CBB or Komol, and more than a 10-to-1 favourite compared to Beevers, May or Panupol). To get a more accurate ratio, more simulations need to be run (I’m trying for 5000 now). But of course, to be the most accurate, we should use a fixed sample as Barry suggests.

Meanwhile, time to work myself into the top 30… 🙂

As you said, Nigel won 60.1% of the tournaments he played in. Similarly other players % show how many % of tournaments they played in they won e.g. Adam’s 8.2% is of the tournaments he played, otherwise it should show 4.2%. I have a note to that in the second bullet, I guess it’s not clear enough.

Using 31.7% will then require me to update everyone’s chart also (Adam 4.2%, Paul Gallen 4.0% etc) but it’s meaningless unless I ensure everyone play the same number of random tournaments, which is complex.

My aim was not to simulate who is more likely to win in a fixed pairing. My guess is for that it would approximate closely based on expected wins. Number of repeats / max repeats was what I could not figure out a formula for.

Due to it’s randomness, I’m sure even lower-ranked player than Vannitha can win it (Vannitha had a median rank of 6 in one of my tourneys, so she’s up there) given more samples.

Jesse won too in my sim, so I guess that makes him a real contender 😉 he should stop commentating and start playing!

Ah, I saw the note but missed the detail. But then the probability for other players then becomes their probability of winning, conditional on them playing and Nigel playing! Which, of course, doesn’t allow for meaningful comparison.

Ugh, started a new thread by accident. I meant comparison v.s. Nigel – comparison with non-Nigel still OK