Friday, 4 January 2013

Predicting Countdown's 30th Anniversary Tournament (Part 1)

To celebrate its 30th anniversary Countdown is hosting a very special tournament. 41 past champions have been invited back for a battle royale of letters and numbers. With such a high quality field, picking a likely winner is no easy task, and so I thought I'd take a statistical approach to predicting who might come out on top. If you just want to see the results, I've put them in a separate post, but if you're interested in some of the stats behind them read on...

Before I could start I needed some measure of how good each player is. Fortunately, a lot of them play a very similar game at, which calculates a rating based on their online play (you may be familiar with similar systems in, for example, the chess world). From here we get ratings for 28 of the 41 players, which is good, but still leaves us with some work to do.

One of the competitors - series 65 champion Graeme Cole - has produced some statistics for all those taking part. (Along with a handy diagram of the tournament structure.) One of these stats will serve as a proxy measure of ability: 'max percentage'. This is the proportion of the total possible points available players actually scored during their televised appearances. Obviously there are lots of reasons why these data aren't ideal, but they're the best we've got, and for those players who do play online they correlate pretty well with their ratings. Fitting a simple linear regression model of ranking against max percentage (which worked surprisingly well) we can then estimate ratings for those players who don't play online. The figure (click for a bigger version) summarizes the relationship: it's fairly noisy but good enough for a bit of fun.

Now that we have an estimate of every player's skill, the next challenge is estimating the probability of, say, a player with a rating of 1600 beating a  player with a rating of 1500. Once again, apterous data lend a hand, with tens of thousands of recorded games giving an insight into just this sort of question. This time, a logistic regression model does the trick, allowing us to estimate the relationship between the difference in ratings between two players and the probability that each player will win. For example, a player who is rated 100 points higher than their opponent would be expected to win about 63% of the time, shooting up to 93% for a 500 point advantage.

From here, it's relatively straightforward to calculate the probability that each player will be crowned champion. To find out what they are just go to part 2.

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