TITLE: Why does the Elo rating system work?
QUESTION [22 upvotes]: The Elo rating system is used to rank players in games such as chess. I can find plenty of explanations online of how to compute someone's Elo rating, how to actually crunch the numbers in practice, but I can't find a single clear conceptual explanation of what the rating is supposed to mean and why.
The only information I can find is that apparently the Elo rating of two players allows you to calculate the odds that one player will win against the other. But every page I've been able to find that talks about this just drops the formula for how to calculate these odds on you and says "there you go, that gives the probability of winning", without explaining why. Wikipedia mentions something about the assumption that "chess performance is normally distributed", but doesn't go any further.
What is the underlying probabilistic model for two-player games that the Elo system is based on? What are its basic assumptions, and what is the proof, from those assumptions, that the Elo system does indeed allow you to calculate win probabilities?

REPLY [2 votes]: Not enough reputation to comment, and 3 years late to the discussion, but ... I give an answer to this question in this post.  
Yes, the win probabilities come from a strange $400\log_{10}$ log-odds, and you could further argue that the ratings themselves are the weights of this logistic regression, which we are doing streaming/online updates on using game outcomes (like a stochastic gradient descent step).
You could instead interpret Elo as an AR(1) autoregressive model (like this talk (opens PDF)), which would probably help explain 538's "autocorrelation" term, really to maintain stationarity.