Chess ELO calculations
Based on the difference in ratings between the chess players and each of their opponents in our Project 1 tournament, calculate each player’s expected score (e.g., 4.3) and the difference from their actual score (e.g., 4.0). List the five players who most overperformed relative to their expected scores, and the five players that most underperformed relative to their expected score.
You’ll find some small differences in different implementations of ELO formulas. You may use any reasonably-sourced formula, but please cite your source. Due end of day Sunday 18-Feb-2024.
Image source (and potentially useful reference!): The Elo Rating System for Chess and Beyond. Feb 15, 2019. [video, 7m]
In this R Markdown, I will load the same files that have been created previously, two local files, and import the data as dataframes into RStudio.
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## parse_dateIn this section the three existing files that have been created in the project1 are loaded into RStudio. The same files will be loaded from the Data folder.
#Load three files
Game_DF <- read.csv("Data/Game_Data.csv")
Player_DF <- read.csv("Data/Player_data.csv")
Average_rating <- read.csv("Data/Average_rating_report.csv")
print("Three files are laoded")## [1] "Three files are laoded"In this section, the loaded data will be used to create the scores for individual players.
What is Elo, it is called after Arpad Elo a Hungarian-American physics professor. Elo is a rating score method for calculating the relative skill levels of players in specifically 2-player games like chess. Since is it for 2-payer, it also now used for other sports as well such as other 2-palyer games like board games and video games.
let’s start by understanding what Elo formula is, the following explains to how calculate the new score based on Elo rating after a game is given:
\[ \text{R}_{New} = \text{R}_{Old} + 32 \times (S - E) \]
where:
- \(\text{R}_{New}\) is the player’s new Elo rating,
- \(\text{R}_{Old}\) is the player’s old Elo rating,
-32 is the K-factor, which determines how much the rating changes after each game, K factor can change, but here I will be using 32 as suggested by the referred YouTube source.
- \(S\) is the actual score of the game (1 for a win (W), 0.5 for a draw (D), and 0 for a loss (L)),
and - \(E\), also known as \(P\), is the expected score of the game based on the player’s and opponent’s ratings.
The expected score \(P_\text{A}\) which predicts the outcome of the game of player A versus player B is calculated using the following formula:
\[ P_\text{A} = \frac{1}{1 + 10^{((\text{R}_\text{B} - \text{R}_\text{A})/400)}} \]
Now the probability of player A’s is calculated as below:
\[ P_\text{A} = { 10^{((\text{R}_\text{A} - \text{R}_\text{B})/400)}}\times P_\text{B} \]
where: - \(P_\text{A}\) is player A’s win probability, \(P_\text{B}\) is player B’s win probability, and \(R_\text{A}\) and \(R_\text{B}\) are player A and B ratings. The second formula is related to the first one by replacing \(P_\text{B}=1-P_\text{A}\).
So we start with the initial rating of the player as it is stored in Player_DF, and based on the result of each round we keep updating the score along the way toward the last round of 7. For each step also we calcualtes the probability of player winning rate, P.
We also define two simple function cal player_P for probability calculation and player_R for rating calculation.
player_P <- function(Player1_R, Player2_R) {
# This function calculate the propability of player 1 wining against player 2 based on the rate of player 1 and player 2.
# the equation is shown above is used
player_prob <- numeric(max(length(Player1_R),length(Player1_R))) #define the vector to the size of the maximum of length of the two input
#Define a logic vector to use for filtering the NA b=for both player1 and 2
NA_logic <- c(is.na(Player1_R) | is.na(Player2_R))
player_prob[!NA_logic] <- 1/(1+10^((Player2_R[!NA_logic]-Player1_R[!NA_logic])/400)) #calcualte the player_prob based on player1 and player2 _Rs
# Fill in NA values with the original Player1_R values
player_prob[NA_logic] <- NA
return(player_prob)
}
player_R <- function(Old_R, game_result , player_prob){
# This function calculate the new rating of player based their old rating, the acheived scope, and the probability of wining.
# the equation is shown above is used
#use switch to change the game staus to numer to calcualte the New_R
game_result_value <- sapply(unlist(game_result), function(x) {
switch(x,
"W" = 1.0,
"L" = 0.0,
"D" = 0.5,
"H" = 0.0,
"B" = 0.0,
"U" = 0.0,
0.0)
})
New_R <- numeric(length(Old_R)) #defien the vector
NA_logic <- is.na(player_prob) # check for player_ptob with NA which means no change to the current player or no information to updat the rating
New_R[!NA_logic] <- Old_R[!NA_logic]+32*(game_result_value[!NA_logic]-player_prob[!NA_logic]) # update rating for those with propability score
New_R[NA_logic] = Old_R[NA_logic] #thpse w/o score, copy the old one as new
return(New_R)
}
#now after the two functions are defined, we start with the first game and populating the player A vs player B rating for each game and correct the players rating based on the result of each game.
#Inizializing the DataFrame of the Win propability of each round, Win_P# and Rating for each round Rate_R#
dataframe_size <- nrow(Player_DF)
DF <- data.frame(
player_no = numeric(dataframe_size),
pre_rate = numeric(dataframe_size),
Win_P1 = numeric(dataframe_size),
Rate_R1 = numeric(dataframe_size),
Win_P2 = numeric(dataframe_size),
Rate_R2 = numeric(dataframe_size),
Win_P3 = numeric(dataframe_size),
Rate_R3 = numeric(dataframe_size),
Win_P4 = numeric(dataframe_size),
Rate_R4 = numeric(dataframe_size),
Win_P5 = numeric(dataframe_size),
Rate_R5 = numeric(dataframe_size),
Win_P6 = numeric(dataframe_size),
Rate_R6 = numeric(dataframe_size),
Win_P7 = numeric(dataframe_size),
Rate_R7 = numeric(dataframe_size)
)
#Assign the player_no
DF["player_no"] <- Player_DF["player_no"]
#Assign the pre rating
DF["pre_rate"] <- Player_DF["rate_pre"]
for (i in seq(ceiling(length(DF)/2)-1)){
Temp_1 <- Game_DF|> filter(round_no==i)|>
select(player1_no,player2_no,round_status) # find the palyer1 for the round
# Temp_2 <- Game_DF|> filter(round_no==i)|>
# select(player2_no)
# Temp_3 <- Game_DF|> filter(round_no==i)|>
# select(round_status)
#the Temp_1 and Temp_2 are the player number which is the row numer in DF, thus they are used to return the column values for even rows, i.e., row 2, whcoh is the previous rating for each player of the specific row number in Temp_1 or _2
DF[unlist(Temp_1["player1_no"]),i*2+1] <- player_P(DF[unlist(Temp_1["player1_no"]),i*2],DF[unlist(Temp_1["player1_no"]),i*2]) # calculate the wining peobability for player1_no
#the first player previous rating, with the status of the game and the previosuly calcualted win probability, is pass to funtion player_R to calcualte the new R
DF[unlist(Temp_1["player1_no"]),i*2+2] <- round(player_R(DF[unlist(Temp_1["player1_no"]),2*i],unlist(Temp_1["round_status"]),unlist( DF[unlist(Temp_1["player1_no"]),i*2+1])),0) # calculate the new ratign baased on the game status and round it
}
DF <- DF|>mutate(improvement = Rate_R7-pre_rate)
print("List of underperformed players ")## [1] "List of underperformed players "DF|>select(player_no,pre_rate,Rate_R7,improvement)|>arrange(desc(improvement)) |> head(10)## player_no pre_rate Rate_R7 improvement
## 1 1 1794 1874 80
## 2 2 1553 1633 80
## 3 3 1384 1464 80
## 4 4 1716 1780 64
## 5 5 1655 1719 64
## 6 6 1686 1734 48
## 7 7 1649 1697 48
## 8 8 1641 1689 48
## 9 9 1411 1459 48
## 10 10 1365 1413 48print("List of overperformed players")## [1] "List of overperformed players"DF|>select(player_no,pre_rate,Rate_R7,improvement)|>arrange(improvement) |> head(10)## player_no pre_rate Rate_R7 improvement
## 1 63 1175 1079 -96
## 2 53 1393 1313 -80
## 3 54 1270 1190 -80
## 4 55 1186 1106 -80
## 5 56 1153 1073 -80
## 6 57 1092 1012 -80
## 7 58 917 837 -80
## 8 59 853 773 -80
## 9 60 967 887 -80
## 10 62 1530 1450 -80