You are now a university professor, please describe the Elo ranking in detail, express it with a formula
2. Points calculation formula
Ra: Player A’s current Rating
Rb: Player B’s current Rating
Sa: actual win/loss value, win=1, tie=0.5, loss=0
Ea: expected win/loss value of player A, Ea=1 /(1+10^[(Rb-Ra)/400])
Eb: The expected outcome of player B, Eb=1/(1+10^[(Ra-Rb)/400])
because the E value is also Estimated, then Ea+ Eb=1
The Elo ranking system is a method for calculating the relative skill levels of players , typically used in competitive games and sports. Dr. Arpad Elo originally designed this system to evaluate chess players.
The main idea of the Elo system is: if A beats B in a match, then the system increases A's rank and B's rank decreases. However, the magnitude of the change depends on their expected winning percentage . If A is an advanced player and beats junior player B , then A's rank will increase slightly and B's rank will decrease slightly. However, if B beats A, then A's ranking will drop significantly and B's ranking will rise significantly.
The basic formula of the Elo ranking system is as follows:
1. Calculate the expected score for each player:
Expected win rate: Ea = 1 / (1 + 10 ^ ((Rb - Ra) / 400)) 
, similarly, Eb = 1 / (1 + 10 ^ ((Ra - Rb) / 400))
Here, Ra and Rb represent the current rankings of A and B, respectively. Ea and Eb are the expected odds of winning A and B.
2. Ranking update:
New 
Ra = old Ra + K * (Sa - Ea), new Rb = old Rb + K * (Sb - Eb)
Here, K is the K coefficient (generally 10 to 60, depending on the specific situation ), Sa and Sb are the results of the game (if A wins, Sa=1, Sb=0; if B wins, Sa=0, Sb =1; if there is a tie, Sa=Sb=0.5).
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where R'_A and R'_B are the updated Elo ratings, S_A and S_B are the actual scores (1 for a win, 0.5 for a draw, 0 for a loss), and K is the coefficient affecting the magnitude of the rating change.
Selection of K factor
The choice of K-factor depends on a number of factors, including the player's experience level and the importance of the game. Choosing an appropriate K value can help the Elo scoring system better reflect the actual strength of the player.
Higher K values (e.g. 40) may make a player's rating more volatile , more suitable for beginning or intermediate players , so that their rating can reflect their skill improvement more quickly . Whereas a lower K value (such as 10 or 20) may result in a more stable rating and is more suitable for advanced players.
In some cases, the K value can be adjusted based on the player's current rating or the type of match . For example, with chess, a junior player might have a K of 40, an intermediate player might have a K of 20, and a top player might have a K of 10.
Remember, all of these are only approximate ways of trying to quantify player strength. In reality, any game has a certain degree of randomness, so any scoring system cannot completely and accurately reflect the player's strength . The Elo rating system is just an attempt to assign a reasonable rating to each player using historical match data . At the same time, since the outcome of each game affects the player's Elo score, this system also brings an additional strategic layer to the game . Players need to make a trade-off between maintaining or improving their Elo score and trying new strategies . "
The above is the basic principle and formula of the Elo ranking system.
3. Example description
If the current rating of player A is 1500, and the rating of player B is 1600,
the estimated win-loss value of player A is Ea = 1/(1+10^[(1600-1500)/400]) ≈ 0.36 and the
estimated value of player B is The win-loss value Eb = 1-Ea = 1-0.36 = 0.64
Assuming that player A wins, the actual win-loss value is Sa = 1.
The final score of player A is R'a = 1500 + 32*(1-0.36) = 1500+20.5 = 1520. Player A wins by 20 points and player B loses by 20 points.
Assuming that player B wins, the actual win-loss value is Sa = 1. The
final score of team B is R'b = 1600 + 32*(1-0.64) = 1600 + 11.52 = 1612. Player B wins by 12 points and player A loses by 12 points.
PS: The ^ symbol represents a power operation.
4. The purpose of the ladder system
1. To protect novices from being abused by experienced players, so that there are no novices in the master game.
2. Create a competitive and fair game match to maximize the player's game fun.
3. No need to wait too long to find an opponent to enter the game.
V. Specific examples
11 Example of the ladder of the battle platform.
The following formula is a good illustration of the calculation method of ladder points, and the calculation of Si is based on the ELO algorithm. Although the 11 ladder still has bugs, it has to be admitted that it is indeed a great design.
What is the principle of dividing by 400 in the formula
The division by 400 portion of the formula used by Elo's ranking system is intended to map the base-10 logarithmic scale to the appropriate game or competition scale . For chess, Arpad Elo chose 400 because he found it to be an appropriate value that would allow a player ranked 400 higher to beat a player ranked 400 lower with a probability of about 0.75.
The specific reasons for choosing 400 include: at that time, the ranking of chess players was around 2000, and he hoped to have an appropriate range so that the ranking gap between masters and novices would not be too large or too small . And the number 400 can meet his needs.
However, in other applications this value may vary. For example, an online game might choose a larger or smaller value to accommodate a wider player base and faster game speeds.