Rating, Ranking: Massey's regression

Class 2

January 29th, 2020

DSTA honors Kobe Bryant (1978-2020)

Introduction

Definition

A measure of value of the subject, as objective and replicable as possible.

E.g., temperature.

Abilities are normally

latent
hard to measure
time-dependent
place-dependent

yet, abilities are also

hard to trascend (revert-to-mean effect)
relatively easy to perceive and project

Example: Football

-Hard to guess the single score $\Longrightarrow$ entertainment value

-easy for experts to guess the long-term effect (cue RTTM) $\Longrightarrow$ different levels of enjoyment

Low scoring creates randomness

Formalisation

1-dimensional ranking

$P:$ players, $|P|=n$

$T:$ time instants

$r: P \times T \rightarrow \mathbb{R}$

For a given rating func. $r$ :

ρ : P \times T \to [0.. | n |]

$\rho: P \times T \rightarrow [0..|n|]$

ρ (p) = k \leftrightarrow | {p_{j} : r (p_{j}) < r (p_{i})} | = k

$\rho(p) = k \leftrightarrow |\{p_j: r(p_j) < r(p_i)\}| = k$

δ (p_{i}, p_{j}, t) = | r (p_{i}, t) - r (p_{j}, t) |

$\delta(p_i, p_j, t) = |r(p_i,t)-r(p_j,t)|$

$\delta$ captures both similarity and distance

multi-dimensional ranking

Multi-dim. rating:

r_{m u l t i} : P \times T \to R^{d}

$r_{multi}: P \times T \rightarrow \mathbb{R}^d$

Often:

r_{m u l t i} (p_{i}, t) : f (r_{1} (p_{i}, t), \dots r_{d} (p_{i}, t))

$r_{multi}(p_i,t) : f(r_1(p_i,t), \dots r_d(p_i,t))$

Pareto dominance:

$p_i$ dominates $p_j$ (at time t) if on every dimension $x$

r_{x} (p_{i}, t) \geq r_{x} (p_{j}, t)

$r_x(p_i,t) \ge r_x(p_j,t)$

Rating in games

Ratings in games

score-based games are better-suited to create ratings
yet effect of time and hardness of the proposed test match could be hard to assess.

Should games keep user ratings?

Yes:

-feeling of improvement

-a gauge for new features

-leads to rankings:

–better matchmaking $\Longrightarrow$ entertainment value

–fraud/anomaly detection?

No: game prowness as social ranking?

«Le spectacle n’est pas un ensemble d’images, mais un rapport social entre des personnes, médiatisé par des images»

Guy Debord, La Société du spectacle (1967), Thèse 4.

-a reflection of US culture?

a turn-off for people who don’t feel competive?
turns-off casual users?

Sport ranking/estimation

Domain

n teams play each other in a tournament
final scores are recorded, e.g., PSG-Ajax: 2-0.
predict the score for a match in the future.

-focus on predicting the score difference (eg, 2-0=2)

[In]credible Assumptions

to each team a latent variable ``strenght’’ is assigned

numerical ratings determine a ranking among teams

and a prediction $Pr[a\rightarrow b] = \frac{\rho(a)}{\rho(a)+\rho(b)}$

strenght/rating is immutable during the tournament

strenght/rating imbalance determines score difference:

r_{i} - r_{j} = s_{i} - s_{j}

$r_i - r_j = s_i - s_j$

Let $\vec{p}$ be the all-season points difference vector ( $\sum p_i=0$ )

X_{m \times n} \cdot r_{n \times 1} = p_{m \times 1}

$X_{m\times n}\cdot r_{n\times 1}=p_{m\times 1}$

[\begin{matrix} 0 & 0 & + 1 & 0 & - 1 & 0 \\ 0 & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ \\ 0 & - 1 & 0 & + 1 & 0 & 0 \end{matrix}]

$\begin{equation*} \begin{bmatrix} 0 & 0 & +1 & 0 & -1 & 0 \\ & 0 & \ddots & \ddots & & \\ & \ddots & \ddots & \ddots & \ddots & \\ & \ddots & \ddots & \ddots & \ddots & \\ & & \ddots & \ddots & \ddots & \\ 0 & -1 & 0 & +1 & 0 & 0 \end{bmatrix} \end{equation*}$

$X_{m\times n}$ with $m>>n$ is overconstrained:

Massey’s ratings

Data preparation

M = X^{T} X

$M\ =\ X^TX$

$M_{n\times n}$ is squared, semidefinite and positive

However, the rows sum to 0 and cols. are not independent

0/ $\infty$ solutions ensue…

[\begin{matrix} n - 1 & 0 & - x & 0 & - y & 0 \\ n - 1 & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ \\ 0 & - z & 0 & - w & 0 & n - 1 \end{matrix}]

$\begin{equation*} \begin{bmatrix} n-1 & 0 & -x & 0 & -y & 0 \\ & n-1 & \ddots & \ddots & & \\ & \ddots & \ddots & \ddots & \ddots & \\ & \ddots & \ddots & \ddots & \ddots & \\ & & \ddots & \ddots & \ddots & \\ 0 & -z & 0 & -w & 0 & n-1 \end{bmatrix} \end{equation*}$

$m_{i,i}=n-1$ is the numbers of games i played,

$m_{i,j}$ is the no. of matches bet. i and j

Massey…

drops the last row/match
replaces it with a row of 1s, and sets $p_n=0$

(all ratings, positive and negative, will sum to 0)

$\overline{M} = M$ minus the last row, plus a row of 1s

$\overline{p} = p$ minus the last scalar, plus a 0

now $\overline{M}$ is non-singular and invertible
solves

\bar{M} r = \bar{p}

$\overline{M} r = \overline{p}$

i.e.,

r = ({\bar{M}}^{T} \bar{M})^{- 1} {\bar{M}}^{T} \bar{p}

$r = (\overline{M}^T\overline{M})^{-1}\overline{M}^T\overline{p}$

Close enough?

r = ({\bar{M}}^{T} \bar{M})^{- 1} {\bar{M}}^{T} \bar{p}

$r = (\overline{M}^T\overline{M})^{-1}\overline{M}^T\overline{p}$

The MSE solution to Massey’s formula is regression

Conclusions

Points to focus on

rating and rating is the fun side of Data Science!

latent variables that represent non-measurable skills
those leave in a feature space possibly separated from the data space
yet they may get a numeric estimate, and inform our predictions
Massey regresses on the latent variables

Further readings

Rating, Ranking: Massey's regression Class 2 January 29th, 2020