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Another ingenious approach is taken by chesstempo.com, a chess training site. Just as in chess itself the ratings of players are determined by pairwise comparisons (games between players), they pair players up against problems. If they solve the problem, the rating of the problem goes down, the rating of the player goes up. Players are given problems close to their ratings, which keeps everyone happy. I believe they use Glicko to track uncertainty in the rating.

Chapter 22 of David Barber's "Bayesian Reasoning and Machine Learning" (he makes it available online) does a nice (perhaps brief) job of explaining the progression through the Rasch model, the Bradley-Terry-Luce model and Elo.

As an aside, the way they chesstempo generate the exercises is also cute. The tactical chess problems are positions taken from high level (human) games fed into a chess engine which identifies blunderous moves where there is a single distinctly best way to respond. The challenge is to find that best move. Because they are taken from real games, they have the appearance and feel of real positions, which is important; many people believe pattern recognition is an important part of chess mastery. Apparently they've built up nearly 40000 such tactical exercises.

How awful, I just saw [ja27 17 hours ago]. I've even managed to describe it in almost exactly the same way..

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