ELO-Rating

It is an excellent method for calculating relative skill levels of players represented as numeric values.

Understanding ELO

• Consider a PvP match b/w two players, player A and player B.
• Player A has ELO rating $R_a$ and player B has ELO rating $R_b$.
• ELO ratings of individual players reflect their relative skill levels based on performance in past matches.
• Now, a player defeating a much weaker player is not the same as defeating a much more stronger player. ELO rating takes this into account and is reflected in updated ratings of both players.
• After each match, ELO rating of both players is updated based on a formula.
• Originally, it only considers the outcome of a match.

Formula to calculate New Rating of a Player

$$ R^{'}_a = R_a + K \times (S_a - E_a) $$

where,
$R^{'}_a: \text{New Rating of player A}$
$R_a: \text{Original Rating of player A}$
$K\ : \text{Scaling factor}$
$S_a: \text{Actual score or outcome of the match}$
$E_a: \text{Expected score or outcome of the match}$

Scaling Factor: $K$

• It determines how much influence each match can have on the ELO ratings of the players.
• It can be varied according to matches, leagues, competitions, player rating levels, etc.
• Generally, can be set to $32$.

Actual score: $S_a$

• Represents whether a player won or lost or drawn the match.
• Generally, it takes one of the three values:
  • $1$ for a win
  • $0$ for a loss
  • $0.5$ for a draw
  • i.e. $S_{a} \in [0, 1]$

Expected score: $E_a$

• This is where magic of ELO ratings happen.
• It represents the probability that the player will win a match.
• It is calculated based on ELO ratings of two players.
• Formula:

$$ E_a = \frac{Q_a}{Q_a + Q_b} \text{, where} \begin{cases} Q_a = 10 ^ {R_a / c} \\ Q_b = 10 ^ {R_b / c} \\ c = 400 \text{ (generally)} \end{cases} $$

On solving by substituting values of $Q_a$ and $Q_b$,

$$ E_a = \frac{1}{1 + 10 ^ {\frac{R_b - R_a}{c}}} \text{, where } c = 400 $$

$\therefore E_a \in [0, 1]$

• As $0 \leq S_a \leq 1$ and $0 \leq E_a \leq 1$, maximum change in a player's rating can be $K$.

Initial ELO Rating

• It is the rating that a new player has when he starts playing the game.
• It must be significantly larger than $K$ factor, to allow a player to play, and lose enough matches before eventually winning matches, so that his ELO rating would finally start rising up.
• Generally, can be set to $1000$.

Considering Points scored by players

But, why?

• Consider a game, where the only thing which matters is not just the outcome of game, i.e. win or loss.
• Let's say:
  • In 1st game, player A won over player B by 5-1 score.
  • In second game player C won over player B by 2-1 score.
• In this scenario, player A had a much more dominant win than player C.
• Thus, player A much have a better rating inrease than player C.

Method 0

• In this, we are considering the effect of points scored by individual players on the change in rating.
• We only consider the fact whether the player has won or lost or drawn the match.
• This is same as original ELO Rating.

Method 1

• We replace $S_a$ with the fraction of points scored by player A divided by total score.
• i.e.,

$$ S_a = \frac{P_a}{P_a + P_b} $$

where,
$P_a: \text{Points scored by player A}$
$P_b: \text{Points scored by player B}$

• This is more intuitive and we don't need to update the main rating change formula.
• But, it gives less control and predictability on the rating change.

Method 2

• In this, we keep $S_a$ same as outcome of match, i.e., $S_a \in$ { $0, 0.5, 1$ }.
• And we extend rating update formula with another scaling factor to which acts as a bonus for the amount of scored points.
• New Formula:

$$ R^{'}_a = R_a + K \times (S_a - E_a) + p \times L \times \left( \frac{P_a}{P_a + P_b} \right) $$

where,
$L: \text{Second scaling factor}$

$$ p = \begin{cases} 1 & \quad \text{if } S_a - E_a \gt 0 \\ -1 & \quad \text{if } S_a - E_a \lt 0 \\ 0 & \quad \text{otherwise} \end{cases}$$

$$p = \frac{S_a - E_a}{|S_a - E_a|}$$

• $L$ can be set to 16, and can be varied as well.
• This method is less intuitive, but gives much more control and predictability of the ratings.
• Using this, maximal change that a rating can have is $K + L$.

Implementation

• I have implemented all of the above formulas in ELO class in Python and Java.

Class ELO

• It has a constructor and two public methods.
  • constructor(): To create an instance of class ELO
  • elo(): It can be called after a match with ratings of both players and match result. It returns the updated ratings of the players.
  • elo_with_points(): It is similar to elo(), but this is with consideration of points scored by the two players.

Constructor

• It takes 3 arguments:
  1. k_factor: Value of $K$ factor. (int, default $32$)
  2. c_value: Value of $c$ number. (int, default $400$)
  3. l_factor: Value of $L$ factor. (int, default $16$)
• Constructor instantiates an object of ELO class with given parameters.
• In practice, it can be called multiple times for different leagues, arenas, matches, championships, competitions, etc.

elo() method

• This is the rating update function which only considers the outcome of a match.
• It takes 4 arguments:
  1. rating_a: Current rating of player A. (float)
  2. rating_b: Current rating of player B. (float)
  3. outcome: Outcome of a match. (int)
• outcome must $\in$ {0, 1, 2} where,
  • outcome = 0 $\implies$ Match is drawn
  • outcome = 1 $\implies$ Player A won the match
  • outcome = 2 $\implies$ Player B won the match
• It returns the new ratings of the players in the form of tuple/array of float point numbers. e.g. (1400.0, 1550.0)
• It must be called after a match with the required parameters to get the new ratings of the players.

elo_with_points() method

• This is the rating update formula which takes into consideration the individual points scored by the players.
• It takes 5 arguments:
  1. rating_a: Current rating of player A. (float)
  2. rating_b: Current rating of player B. (float)
  3. points_a: Points scored by player A. (float)
  4. points_b: Points scored by player B. (float)
  5. method: Method to use for consideration of points scored. (int, default 2)
• method must $\in$ {0, 1, 2} where,
  • method = 0 $\implies$ Just consider the outcome of a match.
  • method = 1 $\implies$ Consider fraction of points at the place of $S_a$.
  • method = 2 $\implies$ Uses the L factor for consideration of points.
• It returns the new ratings of the players in the form of tuple/array of float point numbers. e.g. (1400.0, 1550.0)
• It must be called after a match with the required parameters to get the new ratings of the players.

Dependencies

• Python: No extra libraries used.
• Java: No extra libraries used.

Files

1. ELO.py: Python implementation of ELO class.
2. ELO.java: Java implementation of ELO class.