# StarCraft II: Ladder math

At the Blizzcon 2010, StarCraft II multiplayer panel, this stuff was supposed to explain the ladder matchmaking system. I look at this and go eh? what!?

Is any of this real? or are they just messing with me, because I can't make heads or tails of this humongous equation. Can anyone break it down for me? or at least explain the principle behind this, I don't get how these integrals, derivatives, Euler-functions and multiplication sums yield anything meaningful...

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Link? What is it supposed to be? As far as I can tell it's a bunch of normal distributions (e.g. treating a bunch of variables as if they were described by Bell curves) and you're trying to compute... something with them. –  Qiaochu Yuan Oct 27 '10 at 16:57
The nasty denominator is just there to normalise the top of the fraction, so to make it a density. It's hard to say more without knowing what all the variables are supposed to be. Also, what is $\Phi$? And how does this ladder mathmaking system work? –  Rasmus Oct 27 '10 at 17:32
@Rasmus: $\Phi$ is the integral of the standard (unit variance) normal distribution. –  T.. Oct 27 '10 at 17:34
Statistical equations have the nasty habit of introducing lots of symbols, numbers, and integrals. In the end though, it a pretty simple expression. Also, I dont think the phi's are meant to represent totient functions. –  crasic Oct 28 '10 at 6:28
It's a very elaborate way of saying "look, this stuff is complicated and patented, don't even try understanding it okay?" –  badp Dec 7 '10 at 20:32

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?".

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+1 for the joke –  Djaian Oct 27 '10 at 18:08
+1 because I couldn't understand a thing you said. Especially the joke. –  muntoo Oct 27 '10 at 23:11
The joke involves a man consulting a sequence of priests or rabbis from different sects about whether something is permitted (e.g., "is it OK with God if I get a Ferrari for Christmas"). The permissiveness of the sect is inversely related to the authority of the answer. The strictest one gives a doctrinally definitive answer whose conclusion is "absolutely not! And by the way, what's a Ferrari?". –  T.. Oct 28 '10 at 0:56
Thanks for a very insightful answer. I'll get back to you with some additional information from the panel. They go in depth about how they work with win percentages. Not sure how much you know about StarCraft, but each game is played as either one of three races or random. I believe the goal is to maintain a decent matchmaking depending on regional race win/loss ratio of differently skilled players. –  John Leidegren Oct 28 '10 at 6:40
I have no knowledge about Starcraft except what you provide, so the more the better. However, I think the formula may have left out some details, such as whether they are integrating over one triple of (theta, gamma, psi) or one per game. –  T.. Oct 28 '10 at 15:46