This assignment goal is to compare the chess player’s actual score with their expected score calculated using the ELO rating system’s probability formula. This analysis will then facilitate to classify players who most over performed and those who most under performed of course relative to their expected score.
To reach that goal:
1) Read the csv file that i obtained from project 1 as a Data frame in R Use the ELO rating system formula as followed: Expected_Score = 1/(1+10ˆ((Rb-Ra)/400)) where Rb represents the Average Opponents rating and Ra the player rating to calculate a “baseline” score for a 7-round chess tournament.
2) Use the mutate() function from Dplyr R-package to help us accomplish 2 tasks:
3) Finally, I will either use the slice_max() and slice_min() functions or Sort the column of the score difference by ascending and descending to rank the five most over performed and most under performed players.
library(dplyr)Warning: package 'dplyr' was built under R version 4.5.2
Attaching package: 'dplyr'The following objects are masked from 'package:stats':
filter, lagThe following objects are masked from 'package:base':
intersect, setdiff, setequal, unionHere i will load the final result data table that i obtained from the Chess tournament analysis in Project1 from GitHub repo . The dataset contains the Average player’s opponent rating and the Pre tournament player rating that will be useful to calculate each player’s expected score.
url<- "https://raw.githubusercontent.com/Pascaltafo2025/Week-5B--DATA-607/refs/heads/main/Final_tournament_PlayersInfo.csv"
Chess_Players_ratings <- read.csv(url)
head(Chess_Players_ratings,10) PlayerName State TotalPoints PreRating AvgOpponentRating
1 GARY HUA ON 6.0 1794 1605
2 DAKSHESH DARURI MI 6.0 1553 1469
3 ADITYA BAJAJ MI 6.0 1384 1564
4 PATRICK H SCHILLING MI 5.5 1716 1574
5 HANSHI ZUO MI 5.5 1655 1501
6 HANSEN SONG OH 5.0 1686 1519
7 GARY DEE SWATHELL MI 5.0 1649 1372
8 EZEKIEL HOUGHTON MI 5.0 1641 1468
9 STEFANO LEE ON 5.0 1411 1523
10 ANVIT RAO MI 5.0 1365 1554To perform those calculations, we will Use the ELO rating system formula as followed: Expected_Score = 1/(1+10ˆ((Rb-Ra)/400)) where Rb represents the Average Opponents rating(AvgOpponentRating) and Ra the pre-tournament player rating (PreRating). Once we find the expected score for each player, we will then compute difference from their actual score using a basic arithmetic calculation (Difference = actual score(Total Points) - Expected Score). NB: We should keep in mind that tournament was madeof 7 rounds.
# Expected score over 7 games and round the result to 2 decimal places
Total_rounds<- 7
Expected_Score_Total = Total_rounds * (1 / (1 + 10^((Chess_Players_ratings$AvgOpponentRating - Chess_Players_ratings$PreRating) / 400)))
Expected_Score <- round(Expected_Score_Total,2)
head(Expected_Score,10) [1] 5.24 4.33 1.83 4.86 4.96 5.06 5.82 5.11 2.41 1.76# Difference between player's actual score and expected score
Difference <- Chess_Players_ratings$TotalPoints - Expected_Score
Difference <- round(Difference,2) #Round to 2 decimal places.
head(Difference,10) [1] 0.76 1.67 4.17 0.64 0.54 -0.06 -0.82 -0.11 2.59 3.24# Add the expected score and difference columns to our initial players result dataframe.
Chess_Players_ratings_Update <- Chess_Players_ratings %>%
mutate(Expected_Score ,Difference
)
head(Chess_Players_ratings_Update,10) PlayerName State TotalPoints PreRating AvgOpponentRating
1 GARY HUA ON 6.0 1794 1605
2 DAKSHESH DARURI MI 6.0 1553 1469
3 ADITYA BAJAJ MI 6.0 1384 1564
4 PATRICK H SCHILLING MI 5.5 1716 1574
5 HANSHI ZUO MI 5.5 1655 1501
6 HANSEN SONG OH 5.0 1686 1519
7 GARY DEE SWATHELL MI 5.0 1649 1372
8 EZEKIEL HOUGHTON MI 5.0 1641 1468
9 STEFANO LEE ON 5.0 1411 1523
10 ANVIT RAO MI 5.0 1365 1554
Expected_Score Difference
1 5.24 0.76
2 4.33 1.67
3 1.83 4.17
4 4.86 0.64
5 4.96 0.54
6 5.06 -0.06
7 5.82 -0.82
8 5.11 -0.11
9 2.41 2.59
10 1.76 3.24First method
Here i will rank the players’ score difference in a descending order then extract the top 5 as my result.
# Sort players' Difference score in a descending order
Overperformers <- Chess_Players_ratings_Update [order(-Chess_Players_ratings_Update$Difference), ]
Top5_Overperformers <- head(Overperformers, 5)
# List of Final results
Top5_Overperformers[, -2] PlayerName TotalPoints PreRating AvgOpponentRating
3 ADITYA BAJAJ 6.0 1384 1564
10 ANVIT RAO 5.0 1365 1554
15 ZACHARY JAMES HOUGHTON 4.5 1220 1484
46 JACOB ALEXANDER LAVALLEY 3.0 377 1358
37 AMIYATOSH PWNANANDAM 3.5 980 1385
Expected_Score Difference
3 1.83 4.17
10 1.76 3.24
15 1.26 3.24
46 0.02 2.98
37 0.62 2.88Second method
We will be extracting the five players who most over performed using the “slice_max()” function.
# Extract Top five Overperformers
Top5_Overperformers <- Chess_Players_ratings_Update %>%
slice_max(Difference, n = 5)
Top5_Overperformers[, -2] PlayerName TotalPoints PreRating AvgOpponentRating
1 ADITYA BAJAJ 6.0 1384 1564
2 ANVIT RAO 5.0 1365 1554
3 ZACHARY JAMES HOUGHTON 4.5 1220 1484
4 JACOB ALEXANDER LAVALLEY 3.0 377 1358
5 AMIYATOSH PWNANANDAM 3.5 980 1385
Expected_Score Difference
1 1.83 4.17
2 1.76 3.24
3 1.26 3.24
4 0.02 2.98
5 0.62 2.88These players who most over performed significantly out punched their weight. For instance, ADITYA BAJAJ in particular Pre tournament rating was nearly 200 points below his average opponent rating. According to the logistic distribution implemented by Arpad Elo to make the ELO Rating system, if the difference in the ratings between two players is 200 points, then the expected score for the stronger player is 76%. However, ADITYA BAJAJ still scored 6/7, which borders on a huge statistical outlier.
First method
Here i will rank the players’ score difference in an ascending order then extract the top 5 as my result. Also, i can just use the descending order ranking and get the last five players using “tail”
# Extract top 5 of the players' score difference in an ascending order
underperformers <- Chess_Players_ratings_Update[order(Chess_Players_ratings_Update$Difference), ]
Top5_underperformers <- head(underperformers, 5)
Top5_underperformers[, -2] PlayerName TotalPoints PreRating AvgOpponentRating Expected_Score
62 ASHWIN BALAJI 1.0 1530 1186 6.15
25 LOREN SCHWIEBERT 3.5 1745 1363 6.30
30 GEORGE AVERY JONES 3.5 1522 1144 6.29
27 GAURAV GIDWANI 3.5 1552 1222 6.09
29 CHIEDOZIE OKORIE 3.5 1602 1314 5.88
Difference
62 -5.15
25 -2.80
30 -2.79
27 -2.59
29 -2.38# Extract the last 5 players' from the score difference in a descending order from the previous question
Top5_underperformers <- tail(Overperformers, 5)
Top5_underperformers[, -2] PlayerName TotalPoints PreRating AvgOpponentRating Expected_Score
35 JOSHUA DAVID LEE 3.5 1438 1150 5.88
27 GAURAV GIDWANI 3.5 1552 1222 6.09
30 GEORGE AVERY JONES 3.5 1522 1144 6.29
25 LOREN SCHWIEBERT 3.5 1745 1363 6.30
62 ASHWIN BALAJI 1.0 1530 1186 6.15
Difference
35 -2.38
27 -2.59
30 -2.79
25 -2.80
62 -5.15Second Method
We will be extracting the five players who most overperformed using the “slice_min()” function.
# Extract Top 5 Underperformers
Top5_underperformers <- Chess_Players_ratings_Update %>%
slice_min(Difference, n = 5)
Top5_underperformers[, -2] PlayerName TotalPoints PreRating AvgOpponentRating Expected_Score
1 ASHWIN BALAJI 1.0 1530 1186 6.15
2 LOREN SCHWIEBERT 3.5 1745 1363 6.30
3 GEORGE AVERY JONES 3.5 1522 1144 6.29
4 GAURAV GIDWANI 3.5 1552 1222 6.09
5 CHIEDOZIE OKORIE 3.5 1602 1314 5.88
6 JOSHUA DAVID LEE 3.5 1438 1150 5.88
Difference
1 -5.15
2 -2.80
3 -2.79
4 -2.59
5 -2.38
6 -2.38The result here present a dramatic outlier called Ashwin Balaji . Indeed, his Pre tournament rating was 1530 whereas the average opponent rating he faced was 1186 which rating difference was 344. According to the “algorithm 400” that gives the relationship between the rating difference Vs Probability of success, Ashwin Balaji had closed to 90% chance of success against his opponent yet scored just 1/7.
Source: https://en.wikipedia.org/wiki/Performance_rating_(chess)
These results showed that players coming in with modest Elo ratings consistently performed well above their weight racking up points against stronger opponents far more often than probability would predict. Meanwhile, many of the top-rated players who were expected to cruise through the lower-rated field failed unexpectedly, finishing with scores well below what their ratings suggested they were capable of.