StarCraft II: Ladder math

Solution 1:

It looks like an ordinary statistical calculation. The numerator with $\Pi_{g=1}^G$ is a likelihood or probability density, presumably of some outcomes for games $1$ to $G$. The denominator with $\int \Pi \dots$ is the integral of the numerator over all outcomes; it is a normalization constant to ensure the total probability of all results is $1$. Everything in the formulas is a calculation of (z-scores in) independent normal distributions, so they have a fairly simple probability model for how a player's ranking parameters drive the game outcomes.

The goal of the calculation might be to calculate a player's set of ranking parameters $y$ (a vector of numbers measuring strength, speed, skill, wins, or whatever interpretation the quantities have for the game) that maximize the conditional probability $P(g_j | y)$ of having observed the game outcomes $g_i$ for $i = 1$ to $G$. In other words, Maximum Likelihood Estimation of a player's parameters from game data. I can't read everything in the formula -- can you post a larger magnification? -- but the $\theta_{1,g} - \theta_{2,g}$ look like a measure of how one side of the game performed compared relative to the other, such as a difference in number of points, or a measure of how the sides were expected to perform relative to each other, given their ratings. Alternatively, $P(g_j | y)$ could be a Bayesian "posterior" distribution on $y$ in light of the game outcomes, so that the formula is a rule for updating the rankings given some game results. Here $\Phi(\theta_0 + \gamma_i + \psi_{i,0})$ can be understood as implying an initial rating, where the distribution of skills in the player population is assumed to be normal.

One can also infer from the formula that either they are doing the wrong calculation (after $G$ new games), or the big formula is actually a summary of what has happened after $G$ separate re-estimation steps, one after each game (so that in a single step there is no product involved, and the formula would involve only the ratings just before the game, and the game outcome). The probabilies they are computing for the $G$ games are of the form "what is the chance the player had performance at least $x$ in game 1, and at least $y$ in game 2, $z$ or better in game 3, ...". This is not the correct way to assess the probability that the whole set of $G$ game results is, collectively, above a certain level of performance. But if the parameters are re-estimated after every game, and the older game outcomes forgotten, then "chances of a victory at least $X$ big" is the only thing you can do, so this would account for the shape of their formula.

Now, as the old joke goes: "and by the way, what's StarCraft?".