Introduction

This analysis builds on the chess tournament data from Project 1 to compare each player’s actual tournament score with the score that would be expected from the rating differences between that player and each opponent. The business question is: Which players performed most above or below what their pre-tournament ratings predicted? I use the standard Elo expected-score formula and compare expected points with actual points from the same played games. Official tournament totals are also reported, but unplayed/administrative rounds cannot be assigned an Elo expectation without an opponent.

For the expected-score calculation, I use the logistic Elo formula \(E_A = 1/(1 + 10^{(R_B-R_A)/400})\), where \(R_A\) is the player’s rating and \(R_B\) is the opponent’s rating. FIDE’s rating regulations similarly determine a scoring probability from the rating difference for each rated game: FIDE Rating Regulations. The logistic formula used here is also described in this Elo rating system reference.

Load Packages

I use tidyverse for importing, parsing, joining, reshaping, and summarizing the tournament data, and knitr for formatted tables.

library(tidyverse)
library(knitr)

options(dplyr.summarise.inform = FALSE)

Import the Tournament Data

To make the analysis reproducible, the tournament file is read directly from the public GitHub repository used for Project 1 rather than from a file stored on my computer.

data_url <- "https://raw.githubusercontent.com/chanicemcken/Data-607-Project-1/main/tournamentinfo.txt"

tournament_raw <- readLines(data_url, warn = FALSE)

head(tournament_raw, 10)

##  [1] "-----------------------------------------------------------------------------------------" 
##  [2] " Pair | Player Name                     |Total|Round|Round|Round|Round|Round|Round|Round| "
##  [3] " Num  | USCF ID / Rtg (Pre->Post)       | Pts |  1  |  2  |  3  |  4  |  5  |  6  |  7  | "
##  [4] "-----------------------------------------------------------------------------------------" 
##  [5] "    1 | GARY HUA                        |6.0  |W  39|W  21|W  18|W  14|W   7|D  12|D   4|" 
##  [6] "   ON | 15445895 / R: 1794   ->1817     |N:2  |W    |B    |W    |B    |W    |B    |W    |" 
##  [7] "-----------------------------------------------------------------------------------------" 
##  [8] "    2 | DAKSHESH DARURI                 |6.0  |W  63|W  58|L   4|W  17|W  16|W  20|W   7|" 
##  [9] "   MI | 14598900 / R: 1553   ->1663     |N:2  |B    |W    |B    |W    |B    |W    |B    |" 
## [10] "-----------------------------------------------------------------------------------------"

Parse the Player Records

The tournament file stores each player across two lines. The first line contains pair number, player name, total score, and seven round results. The second line contains state and rating information. I identify the 64 player rows, match each one to the following detail row, and extract the fields needed for the Elo analysis.

player_idx <- which(str_detect(
  tournament_raw,
  "^\\s*\\d+\\s*\\|"
))

stopifnot(length(player_idx) == 64)

player_lines <- tournament_raw[player_idx]
detail_lines <- tournament_raw[player_idx + 1]

player_fields <- str_split_fixed(player_lines, "\\|", 11)
detail_fields <- str_split_fixed(detail_lines, "\\|", 11)

players <- tibble(
  Pair_Number = as.integer(str_trim(player_fields[, 1])),
  Player_Name = str_squish(player_fields[, 2]),
  Actual_Score = as.numeric(str_trim(player_fields[, 3])),
  State = str_trim(detail_fields[, 1]),
  Pre_Rating = as.integer(
    str_extract(detail_fields[, 2], "(?<=R:)\\s*\\d+")
  ),
  Round_1 = str_trim(player_fields[, 4]),
  Round_2 = str_trim(player_fields[, 5]),
  Round_3 = str_trim(player_fields[, 6]),
  Round_4 = str_trim(player_fields[, 7]),
  Round_5 = str_trim(player_fields[, 8]),
  Round_6 = str_trim(player_fields[, 9]),
  Round_7 = str_trim(player_fields[, 10])
)

kable(
  head(players, 10),
  caption = "First 10 Parsed Player Records"
)

First 10 Parsed Player Records
Pair_Number	Player_Name	Actual_Score	State	Pre_Rating	Round_1	Round_2	Round_3	Round_4	Round_5	Round_6	Round_7
1	GARY HUA	6.0	ON	1794	W 39	W 21	W 18	W 14	W 7	D 12	D 4
2	DAKSHESH DARURI	6.0	MI	1553	W 63	W 58	L 4	W 17	W 16	W 20	W 7
3	ADITYA BAJAJ	6.0	MI	1384	L 8	W 61	W 25	W 21	W 11	W 13	W 12
4	PATRICK H SCHILLING	5.5	MI	1716	W 23	D 28	W 2	W 26	D 5	W 19	D 1
5	HANSHI ZUO	5.5	MI	1655	W 45	W 37	D 12	D 13	D 4	W 14	W 17
6	HANSEN SONG	5.0	OH	1686	W 34	D 29	L 11	W 35	D 10	W 27	W 21
7	GARY DEE SWATHELL	5.0	MI	1649	W 57	W 46	W 13	W 11	L 1	W 9	L 2
8	EZEKIEL HOUGHTON	5.0	MI	1641	W 3	W 32	L 14	L 9	W 47	W 28	W 19
9	STEFANO LEE	5.0	ON	1411	W 25	L 18	W 59	W 8	W 26	L 7	W 20
10	ANVIT RAO	5.0	MI	1365	D 16	L 19	W 55	W 31	D 6	W 25	W 18

The numeric extraction of Pre_Rating intentionally keeps the rating itself while dropping provisional-rating text that may follow it. This prevents values such as provisional ratings from being treated as nonnumeric.

Reshape the Round Results

The seven rounds are stored in separate columns, so I transform them into long format. Each row below represents one player-round observation. The result code (W, D, L, H, or U) is separated from the opponent pair number.

games <- players %>%
  pivot_longer(
    cols = starts_with("Round_"),
    names_to = "Round",
    values_to = "Round_Result"
  ) %>%
  mutate(
    Round = as.integer(str_remove(Round, "Round_")),
    Result = str_extract(Round_Result, "^[WDLHU]"),
    Opponent_Pair = as.integer(str_extract(Round_Result, "\\d+")),
    Game_Score = case_when(
      Result == "W" ~ 1,
      Result == "D" ~ 0.5,
      Result == "L" ~ 0,
      TRUE ~ NA_real_
    )
  )

kable(
  head(games, 14),
  caption = "Tournament Results in Long Format"
)

Tournament Results in Long Format
Pair_Number	Player_Name	Actual_Score	State	Pre_Rating	Round	Round_Result	Result	Opponent_Pair	Game_Score
1	GARY HUA	6	ON	1794	1	W 39	W	39	1.0
1	GARY HUA	6	ON	1794	2	W 21	W	21	1.0
1	GARY HUA	6	ON	1794	3	W 18	W	18	1.0
1	GARY HUA	6	ON	1794	4	W 14	W	14	1.0
1	GARY HUA	6	ON	1794	5	W 7	W	7	1.0
1	GARY HUA	6	ON	1794	6	D 12	D	12	0.5
1	GARY HUA	6	ON	1794	7	D 4	D	4	0.5
2	DAKSHESH DARURI	6	MI	1553	1	W 63	W	63	1.0
2	DAKSHESH DARURI	6	MI	1553	2	W 58	W	58	1.0
2	DAKSHESH DARURI	6	MI	1553	3	L 4	L	4	0.0
2	DAKSHESH DARURI	6	MI	1553	4	W 17	W	17	1.0
2	DAKSHESH DARURI	6	MI	1553	5	W 16	W	16	1.0
2	DAKSHESH DARURI	6	MI	1553	6	W 20	W	20	1.0
2	DAKSHESH DARURI	6	MI	1553	7	W 7	W	7	1.0

Special entries such as H and U do not contain a rated opponent pair number, so they are not treated as normal opponent matchups. I retain them in the data for validation, but only rounds with a valid opponent number are used to calculate Elo expected score.

Match Opponents to Their Pre-Ratings

The expected score for a game depends on both players’ ratings. I therefore create a lookup table using pair number and pre-rating, then join it back to each player’s opponent number.

opponent_lookup <- players %>%
  select(
    Opponent_Pair = Pair_Number,
    Opponent_Name = Player_Name,
    Opponent_Rating = Pre_Rating
  )

rated_games <- games %>%
  filter(Result %in% c("W", "D", "L"), !is.na(Opponent_Pair)) %>%
  left_join(opponent_lookup, by = "Opponent_Pair")

opponent_validation <- rated_games %>%
  summarise(
    Rated_Game_Records = n(),
    Missing_Opponent_Matches = sum(is.na(Opponent_Rating))
  )

kable(
  opponent_validation,
  caption = "Opponent Matching Validation"
)

Opponent Matching Validation
Rated_Game_Records	Missing_Opponent_Matches
408	0

stopifnot(opponent_validation$Missing_Opponent_Matches == 0)
stopifnot(!anyDuplicated(players$Pair_Number))
stopifnot(all(!is.na(players$Pre_Rating)))
stopifnot(all(!is.na(rated_games$Game_Score)))

Validate the Project 1 Opponent Matching

Before calculating expected scores, I validate the opponent matching using Gary Hua from Project 1. His listed opponents are pair numbers 39, 21, 18, 14, 7, 12, and 4. Their average pre-rating should be approximately 1605.

gary_check <- rated_games %>%
  filter(Player_Name == "GARY HUA") %>%
  select(
    Round,
    Player_Name,
    Pre_Rating,
    Opponent_Pair,
    Opponent_Name,
    Opponent_Rating
  )

kable(
  gary_check,
  caption = "Gary Hua Opponent Matching Check"
)

Gary Hua Opponent Matching Check
Round	Player_Name	Pre_Rating	Opponent_Pair	Opponent_Name	Opponent_Rating
1	GARY HUA	1794	39	JOEL R HENDON	1436
2	GARY HUA	1794	21	DINH DANG BUI	1563
3	GARY HUA	1794	18	DAVID SUNDEEN	1600
4	GARY HUA	1794	14	BRADLEY SHAW	1610
5	GARY HUA	1794	7	GARY DEE SWATHELL	1649
6	GARY HUA	1794	12	KENNETH J TACK	1663
7	GARY HUA	1794	4	PATRICK H SCHILLING	1716

stopifnot(nrow(gary_check) == 7)
stopifnot(identical(sort(gary_check$Opponent_Pair), sort(c(39L, 21L, 18L, 14L, 7L, 12L, 4L))))
gary_average <- mean(gary_check$Opponent_Rating)
stopifnot(abs(gary_average - 1605) < 1)

tibble(
  Validation = "Gary Hua average opponent pre-rating",
  Calculated_Value = round(gary_average, 2),
  Expected_Approximate_Value = 1605
) %>%
  kable(caption = "Project 1 Validation Check")

Project 1 Validation Check
Validation	Calculated_Value	Expected_Approximate_Value
Gary Hua average opponent pre-rating	1605.29	1605

This check is important because an incorrect opponent match would affect the expected probability for a game and therefore the player’s final expected tournament score.

Calculate Elo Expected Score for Each Game

For each rated matchup, I calculate the player’s expected score using:

\[ E_A = \frac{1}{1 + 10^{(R_B-R_A)/400}} \]

An expected value near 0.50 indicates an approximately even matchup. A value above 0.50 means the player was favored based on pre-tournament rating, while a value below 0.50 means the opponent was favored.

elo_expected <- function(player_rating, opponent_rating) {
  1 / (1 + 10 ^ ((opponent_rating - player_rating) / 400))
}

rated_games <- rated_games %>%
  mutate(
    Rating_Difference = Pre_Rating - Opponent_Rating,
    Expected_Score_Game = elo_expected(
      Pre_Rating,
      Opponent_Rating
    )
  )

kable(
  rated_games %>%
    select(
      Player_Name,
      Round,
      Pre_Rating,
      Opponent_Name,
      Opponent_Rating,
      Rating_Difference,
      Expected_Score_Game
    ) %>%
    head(12),
  digits = 3,
  caption = "Example Elo Expected-Score Calculations"
)

Example Elo Expected-Score Calculations
Player_Name	Round	Pre_Rating	Opponent_Name	Opponent_Rating	Rating_Difference	Expected_Score_Game
GARY HUA	1	1794	JOEL R HENDON	1436	358	0.887
GARY HUA	2	1794	DINH DANG BUI	1563	231	0.791
GARY HUA	3	1794	DAVID SUNDEEN	1600	194	0.753
GARY HUA	4	1794	BRADLEY SHAW	1610	184	0.743
GARY HUA	5	1794	GARY DEE SWATHELL	1649	145	0.697
GARY HUA	6	1794	KENNETH J TACK	1663	131	0.680
GARY HUA	7	1794	PATRICK H SCHILLING	1716	78	0.610
DAKSHESH DARURI	1	1553	THOMAS JOSEPH HOSMER	1175	378	0.898
DAKSHESH DARURI	2	1553	VIRAJ MOHILE	917	636	0.975
DAKSHESH DARURI	3	1553	PATRICK H SCHILLING	1716	-163	0.281
DAKSHESH DARURI	4	1553	RONALD GRZEGORCZYK	1629	-76	0.392
DAKSHESH DARURI	5	1553	MIKE NIKITIN	1604	-51	0.427

Calculate Each Player’s Expected and Actual Scores

The official tournament score may include half-point byes or other administrative results that do not have a rated opponent. Comparing that official total with an expectation computed only for played games would inflate or deflate the difference. For the primary ranking, both actual and expected points are therefore restricted to the same rated games. I retain the official score in a separate column for transparency.

expected_totals <- rated_games %>%
  group_by(Pair_Number) %>%
  summarise(
    Rated_Games = n(),
    Actual_Rated_Score = sum(Game_Score),
    Expected_Score = sum(Expected_Score_Game),
    .groups = "drop"
  )

player_performance <- players %>%
  select(Pair_Number, Player_Name, State, Pre_Rating,
         Official_Tournament_Score = Actual_Score) %>%
  left_join(expected_totals, by = "Pair_Number") %>%
  mutate(
    Rated_Games = replace_na(Rated_Games, 0L),
    Actual_Rated_Score = replace_na(Actual_Rated_Score, 0),
    Expected_Score = replace_na(Expected_Score, 0),
    Performance_Difference = Actual_Rated_Score - Expected_Score,
    Nonrated_Score = Official_Tournament_Score - Actual_Rated_Score
  ) %>%
  arrange(desc(Performance_Difference), Pair_Number)

kable(
  player_performance %>%
    mutate(across(c(Official_Tournament_Score, Actual_Rated_Score,
                    Expected_Score, Performance_Difference,
                    Nonrated_Score), ~round(.x, 2))),
  caption = "All Players: Actual vs. Expected Points in Rated Games"
)

All Players: Actual vs. Expected Points in Rated Games
Pair_Number	Player_Name	State	Pre_Rating	Official_Tournament_Score	Rated_Games	Actual_Rated_Score	Expected_Score	Performance_Difference	Nonrated_Score
3	ADITYA BAJAJ	MI	1384	6.0	7	6.0	1.95	4.05	0.0
15	ZACHARY JAMES HOUGHTON	MI	1220	4.5	7	4.5	1.37	3.13	0.0
10	ANVIT RAO	MI	1365	5.0	7	5.0	1.94	3.06	0.0
46	JACOB ALEXANDER LAVALLEY	MI	377	3.0	7	3.0	0.04	2.96	0.0
9	STEFANO LEE	ON	1411	5.0	7	5.0	2.29	2.71	0.0
2	DAKSHESH DARURI	MI	1553	6.0	7	6.0	3.78	2.22	0.0
52	ETHAN GUO	MI	935	2.5	7	2.5	0.30	2.20	0.0
51	TEJAS AYYAGARI	MI	1011	2.5	7	2.5	1.03	1.47	0.0
24	MICHAEL R ALDRICH	MI	1229	4.0	7	4.0	2.55	1.45	0.0
37	AMIYATOSH PWNANANDAM	MI	980	3.5	5	2.0	0.77	1.23	1.5
5	HANSHI ZUO	MI	1655	5.5	7	5.5	4.38	1.12	0.0
1	GARY HUA	ON	1794	6.0	7	6.0	5.16	0.84	0.0
4	PATRICK H SCHILLING	MI	1716	5.5	7	5.5	4.74	0.76	0.0
59	SEAN M MC CORMICK	MI	853	2.0	6	1.0	0.41	0.59	1.0
50	SHIVAM JHA	MI	1056	2.5	6	2.0	1.42	0.58	0.5
58	VIRAJ MOHILE	MI	917	2.0	6	1.0	0.43	0.57	1.0
61	JEZZEL FARKAS	ON	955	1.5	7	1.5	0.97	0.53	0.0
7	GARY DEE SWATHELL	MI	1649	5.0	7	5.0	4.58	0.42	0.0
60	JULIA SHEN	MI	967	1.5	5	1.0	0.60	0.40	0.5
38	BRIAN LIU	MI	1423	3.0	6	2.5	2.13	0.37	0.5
14	BRADLEY SHAW	MI	1610	4.5	7	4.5	4.18	0.32	0.0
36	SIDDHARTH JHA	MI	1355	3.5	6	3.0	2.70	0.30	0.5
28	SOFIA ADINA STANESCU-BELLU	MI	1507	3.5	7	3.5	3.31	0.19	0.0
62	ASHWIN BALAJI	MI	1530	1.0	1	1.0	0.88	0.12	0.0
34	MICHAEL JEFFERY THOMAS	MI	1399	3.5	7	3.5	3.44	0.06	0.0
40	FOREST ZHANG	MI	1348	3.0	7	3.0	2.94	0.06	0.0
23	ALAN BUI	ON	1363	4.0	7	4.0	3.94	0.06	0.0
6	HANSEN SONG	OH	1686	5.0	7	5.0	4.94	0.06	0.0
8	EZEKIEL HOUGHTON	MI	1641	5.0	7	5.0	5.03	-0.03	0.0
44	JUSTIN D SCHILLING	MI	1199	3.0	6	2.0	2.07	-0.07	1.0
56	MARISA RICCI	MI	1153	2.0	5	1.0	1.08	-0.08	1.0
12	KENNETH J TACK	MI	1663	4.5	6	4.0	4.11	-0.11	0.5
32	JOSHUA PHILIP MATHEWS	ON	1441	3.5	7	3.5	3.72	-0.22	0.0
16	MIKE NIKITIN	MI	1604	4.0	5	3.5	3.80	-0.30	0.5
57	MICHAEL LU	MI	1092	2.0	6	1.0	1.30	-0.30	1.0
21	DINH DANG BUI	ON	1563	4.0	7	4.0	4.32	-0.32	0.0
19	DIPANKAR ROY	MI	1564	4.0	7	4.0	4.33	-0.33	0.0
41	KYLE WILLIAM MURPHY	MI	1403	3.0	4	2.0	2.36	-0.36	1.0
55	ALEX KONG	MI	1186	2.0	6	1.0	1.44	-0.44	1.0
13	TORRANCE HENRY JR	MI	1666	4.5	7	4.5	4.95	-0.45	0.0
27	GAURAV GIDWANI	MI	1552	3.5	6	3.5	4.00	-0.50	0.0
49	MICHAEL J MARTIN	MI	1291	2.5	5	2.0	2.54	-0.54	0.5
18	DAVID SUNDEEN	MI	1600	4.0	7	4.0	4.59	-0.59	0.0
26	MAX ZHU	ON	1579	3.5	7	3.5	4.10	-0.60	0.0
39	JOEL R HENDON	MI	1436	3.0	7	3.0	3.62	-0.62	0.0
17	RONALD GRZEGORCZYK	MI	1629	4.0	7	4.0	4.66	-0.66	0.0
47	ERIC WRIGHT	MI	1362	2.5	7	2.5	3.19	-0.69	0.0
53	JOSE C YBARRA	MI	1393	2.0	3	1.0	1.72	-0.72	1.0
11	CAMERON WILLIAM MC LEMAN	MI	1712	4.5	7	4.5	5.34	-0.84	0.0
63	THOMAS JOSEPH HOSMER	MI	1175	1.0	5	0.5	1.43	-0.93	0.5
22	EUGENE L MCCLURE	MI	1555	4.0	6	3.5	4.48	-0.98	0.5
48	DANIEL KHAIN	MI	1382	2.5	5	1.5	2.53	-1.03	1.0
29	CHIEDOZIE OKORIE	MI	1602	3.5	6	3.5	4.60	-1.10	0.0
20	JASON ZHENG	MI	1595	4.0	7	4.0	5.13	-1.13	0.0
33	JADE GE	MI	1449	3.5	7	3.5	4.64	-1.14	0.0
64	BEN LI	MI	1163	1.0	7	1.0	2.27	-1.27	0.0
43	ROBERT GLEN VASEY	MI	1283	3.0	7	3.0	4.33	-1.33	0.0
45	DEREK YAN	MI	1242	3.0	7	3.0	4.37	-1.37	0.0
35	JOSHUA DAVID LEE	MI	1438	3.5	7	3.5	4.96	-1.46	0.0
31	RISHI SHETTY	MI	1494	3.5	7	3.5	5.09	-1.59	0.0
42	JARED GE	MI	1332	3.0	7	3.0	5.01	-2.01	0.0
54	LARRY HODGE	MI	1270	2.0	6	1.0	3.40	-2.40	1.0
30	GEORGE AVERY JONES	ON	1522	3.5	7	3.5	6.02	-2.52	0.0
25	LOREN SCHWIEBERT	MI	1745	3.5	7	3.5	6.28	-2.78	0.0

Check Special Rounds and Game Consistency

# Identify rounds without a standard rated opponent
special_rounds <- games %>%
  filter(!Result %in% c("W", "D", "L") |
           is.na(Opponent_Pair)) %>%
  select(
    Player_Name,
    Round,
    Round_Result,
    Result,
    Opponent_Pair
  )

knitr::kable(
  special_rounds,
  caption = "Rounds Without a Standard Rated Opponent"
)

Rounds Without a Standard Rated Opponent
Player_Name	Round	Round_Result	Result	Opponent_Pair
KENNETH J TACK	5	H	H	NA
MIKE NIKITIN	3	H	H	NA
MIKE NIKITIN	7	U	U	NA
EUGENE L MCCLURE	5	H	H	NA
GAURAV GIDWANI	7	U	U	NA
CHIEDOZIE OKORIE	7	U	U	NA
SIDDHARTH JHA	5	H	H	NA
AMIYATOSH PWNANANDAM	1	B	NA	NA
AMIYATOSH PWNANANDAM	5	H	H	NA
BRIAN LIU	5	H	H	NA
KYLE WILLIAM MURPHY	5	X	NA	NA
KYLE WILLIAM MURPHY	6	U	U	NA
KYLE WILLIAM MURPHY	7	U	U	NA
JUSTIN D SCHILLING	1	B	NA	NA
DANIEL KHAIN	3	H	H	NA
DANIEL KHAIN	5	H	H	NA
MICHAEL J MARTIN	6	H	H	NA
MICHAEL J MARTIN	7	U	U	NA
SHIVAM JHA	5	H	H	NA
JOSE C YBARRA	1	H	H	NA
JOSE C YBARRA	3	H	H	NA
JOSE C YBARRA	5	U	U	NA
JOSE C YBARRA	7	U	U	NA
LARRY HODGE	4	B	NA	NA
ALEX KONG	5	B	NA	NA
MARISA RICCI	1	H	H	NA
MARISA RICCI	5	H	H	NA
MICHAEL LU	7	B	NA	NA
VIRAJ MOHILE	6	B	NA	NA
SEAN M MC CORMICK	2	B	NA	NA
JULIA SHEN	6	H	H	NA
JULIA SHEN	7	U	U	NA
ASHWIN BALAJI	2	U	U	NA
ASHWIN BALAJI	3	U	U	NA
ASHWIN BALAJI	4	U	U	NA
ASHWIN BALAJI	5	U	U	NA
ASHWIN BALAJI	6	U	U	NA
ASHWIN BALAJI	7	U	U	NA
THOMAS JOSEPH HOSMER	6	H	H	NA
THOMAS JOSEPH HOSMER	7	U	U	NA

# Check that each rated game has a corresponding
# opponent entry in the same round
reciprocal_check <- rated_games %>%
  select(Pair_Number, Round, Opponent_Pair, Result) %>%
  left_join(
    rated_games %>%
      select(
        Opponent_Pair = Pair_Number,
        Pair_Number = Opponent_Pair,
        Round,
        Opponent_Result = Result
      ),
    by = c("Pair_Number", "Opponent_Pair", "Round")
  ) %>%
  mutate(
    Valid = case_when(
      Result == "W" & Opponent_Result == "L" ~ TRUE,
      Result == "L" & Opponent_Result == "W" ~ TRUE,
      Result == "D" & Opponent_Result == "D" ~ TRUE,
      TRUE ~ FALSE
    )
  )

# Identify inconsistent or unmatched games
invalid_games <- reciprocal_check %>%
  filter(!Valid)

# Display validation summary
validation_summary <- reciprocal_check %>%
  summarise(
    Total_Rated_Rows = n(),
    Valid_Rows = sum(Valid),
    Missing_Reciprocal_Rows =
      sum(is.na(Opponent_Result)),
    Invalid_Rows = sum(!Valid)
  )

knitr::kable(
  validation_summary,
  caption = "Reciprocal Game Validation Summary"
)

Reciprocal Game Validation Summary
Total_Rated_Rows	Valid_Rows	Missing_Reciprocal_Rows	Invalid_Rows
408	408	0	0

# Display any games requiring further review
if (nrow(invalid_games) > 0) {

  knitr::kable(
    invalid_games,
    caption = "Games Requiring Further Validation"
  )

  warning(
    "Some game records failed reciprocal validation. ",
    "Review these records before interpreting Elo results."
  )

} else {

  cat("All rated game records passed reciprocal validation.")

}

## All rated game records passed reciprocal validation.

Handling Special Rounds

The tournament dataset includes special round codes (H, U, B, and X) that represent administrative outcomes rather than standard games with a recorded opponent and result. These rounds were excluded from the Elo expected-score calculations because the formula requires both players’ pre-tournament ratings to estimate the probability of winning or drawing. Including rounds without a valid opponent rating could produce inaccurate expectations.

To maintain a fair comparison, I calculated each player’s actual score using only the rated games included in their expected-score calculation. This ensures that the difference between actual and expected performance reflects games played against identifiable opponents rather than administrative results.

Validate Expected-Score Ranges

Each single-game expected score must fall between 0 and 1. A player’s total expected score must also fall between 0 and the number of rated games. I check both conditions before ranking the players.

validation_checks <- tibble(
  Check = c(
    "All game expected scores are between 0 and 1",
    "All player expected totals are between 0 and rated games",
    "All 64 tournament players are included",
    "Actual rated-game points are between 0 and rated games"
  ),
  Passed = c(
    all(between(rated_games$Expected_Score_Game, 0, 1)),
    all(
      player_performance$Expected_Score >= 0 &
      player_performance$Expected_Score <= player_performance$Rated_Games
    ),
    nrow(player_performance) == 64,
    all(player_performance$Actual_Rated_Score >= 0 &
        player_performance$Actual_Rated_Score <= player_performance$Rated_Games)
  )
)

kable(
  validation_checks,
  caption = "Expected-Score Validation Checks"
)

Expected-Score Validation Checks
Check	Passed
All game expected scores are between 0 and 1	TRUE
All player expected totals are between 0 and rated games	TRUE
All 64 tournament players are included	TRUE
Actual rated-game points are between 0 and rated games	TRUE

stopifnot(all(validation_checks$Passed))

Five Players Who Most Overperformed

The five largest positive values of Performance_Difference identify the players who scored the most points above their Elo-based expectation in games with rated opponents.

top_overperformers <- player_performance %>%
  slice_max(
    order_by = Performance_Difference,
    n = 5,
    with_ties = FALSE
  ) %>%
  mutate(
    Expected_Score = round(Expected_Score, 2),
    Performance_Difference = round(Performance_Difference, 2)
  ) %>%
  select(
    Player_Name,
    Pre_Rating,
    Rated_Games,
    Actual_Rated_Score,
    Expected_Score,
    Performance_Difference
  )

kable(
  top_overperformers,
  caption = "Five Players Who Most Overperformed Their Expected Score"
)

Five Players Who Most Overperformed Their Expected Score
Player_Name	Pre_Rating	Rated_Games	Actual_Rated_Score	Expected_Score	Performance_Difference
ADITYA BAJAJ	1384	7	6.0	1.95	4.05
ZACHARY JAMES HOUGHTON	1220	7	4.5	1.37	3.13
ANVIT RAO	1365	7	5.0	1.94	3.06
JACOB ALEXANDER LAVALLEY	377	7	3.0	0.04	2.96
STEFANO LEE	1411	7	5.0	2.29	2.71

Five Players Who Most Underperformed

The five most negative values identify the players who scored the most points below their Elo-based expectation in games with rated opponents.

top_underperformers <- player_performance %>%
  slice_min(
    order_by = Performance_Difference,
    n = 5,
    with_ties = FALSE
  ) %>%
  mutate(
    Expected_Score = round(Expected_Score, 2),
    Performance_Difference = round(Performance_Difference, 2)
  ) %>%
  select(
    Player_Name,
    Pre_Rating,
    Rated_Games,
    Actual_Rated_Score,
    Expected_Score,
    Performance_Difference
  )

kable(
  top_underperformers,
  caption = "Five Players Who Most Underperformed Their Expected Score"
)

Five Players Who Most Underperformed Their Expected Score
Player_Name	Pre_Rating	Rated_Games	Actual_Rated_Score	Expected_Score	Performance_Difference
LOREN SCHWIEBERT	1745	7	3.5	6.28	-2.78
GEORGE AVERY JONES	1522	7	3.5	6.02	-2.52
LARRY HODGE	1270	6	1.0	3.40	-2.40
JARED GE	1332	7	3.0	5.01	-2.01
RISHI SHETTY	1494	7	3.5	5.09	-1.59

Visualize Performance Relative to Expectation

The chart provides an additional way to see how actual tournament performance differs from Elo expectation. Players above zero scored more points than expected, while players below zero scored fewer points than expected.

library(dplyr)
library(ggplot2)

# Combine the five biggest overperformers
# and five biggest underperformers
top_bottom_players <- player_performance %>%
  arrange(desc(Performance_Difference)) %>%
  slice_head(n = 5) %>%
  bind_rows(
    player_performance %>%
      arrange(Performance_Difference) %>%
      slice_head(n = 5)
  ) %>%
  mutate(
    Performance = if_else(
      Performance_Difference > 0,
      "Overperformed",
      "Underperformed"
    ),
    Player_Name = reorder(
      Player_Name,
      Performance_Difference
    )
  )

# Create horizontal bar chart
ggplot(
  top_bottom_players,
  aes(
    x = Player_Name,
    y = Performance_Difference,
    fill = Performance
  )
) +
  geom_col(width = 0.7) +
  geom_text(
    aes(
      label = sprintf("%+.2f", Performance_Difference)
    ),
    hjust = ifelse(
      top_bottom_players$Performance_Difference > 0,
      -0.15,
      1.15
    ),
    size = 3.5
  ) +
  coord_flip() +
  scale_fill_manual(
    values = c(
      "Overperformed" = "#2A9D8F",
      "Underperformed" = "#E76F51"
    )
  ) +
  labs(
    title = "Chess Tournament: Actual vs. Expected Performance",
    subtitle = "Five largest overperformers and underperformers",
    x = NULL,
    y = "Actual Score - Expected Score (Points)",
    fill = "Performance"
  ) +
  theme_minimal(base_size = 12) +
  theme(
    plot.title = element_text(face = "bold"),
    legend.position = "bottom",
    panel.grid.major.y = element_blank()
  ) +
  scale_y_continuous(
    expand = expansion(mult = c(0.15, 0.15))
  )

Conclusions

This analysis compares tournament results with the scores predicted by each player’s pre-tournament rating and the ratings of the opponents they actually faced. The final overperformance and underperformance tables identify the five players with the largest positive and negative differences between actual and expected score in rated games. The tables are generated from the source data at render time, so their numeric results are not manually entered.

The results should be interpreted as performance relative to rating-based expectation rather than as a ranking of the strongest players. Because administrative rounds are excluded from both sides of the comparison, the rankings describe performance in played rated games, not the difference from the official total including byes. A lower-rated player can overperform by scoring more points than expected against stronger opponents, while a highly rated player can underperform even with a relatively high tournament score if that score falls below expectation.

A useful extension would be to compare the logistic Elo probabilities used here with FIDE’s published scoring-probability table to see whether the choice of implementation changes the top-five rankings. Another extension would be to repeat the analysis across multiple tournaments to determine whether the largest overperformances persist or are specific to this event.

Reproducibility Information

The analysis reads the tournament text file directly from a public GitHub URL and does not depend on local file paths. The session information below documents the R environment and package versions used when the report is rendered.

sessionInfo()

## R version 4.5.2 (2025-10-31)
## Platform: aarch64-apple-darwin20
## Running under: macOS Sequoia 15.7.3
## 
## Matrix products: default
## BLAS:   /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib 
## LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## time zone: America/New_York
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] knitr_1.51      lubridate_1.9.4 forcats_1.0.1   stringr_1.6.0  
##  [5] dplyr_1.2.1     purrr_1.2.1     readr_2.2.0     tidyr_1.3.2    
##  [9] tibble_3.3.1    ggplot2_4.0.2   tidyverse_2.0.0
## 
## loaded via a namespace (and not attached):
##  [1] gtable_0.3.6       jsonlite_2.0.0     compiler_4.5.2     tidyselect_1.2.1  
##  [5] dichromat_2.0-0.1  jquerylib_0.1.4    scales_1.4.0       yaml_2.3.12       
##  [9] fastmap_1.2.0      R6_2.6.1           labeling_0.4.3     generics_0.1.4    
## [13] tzdb_0.5.0         bslib_0.10.0       pillar_1.11.1      RColorBrewer_1.1-3
## [17] rlang_1.1.7        stringi_1.8.7      cachem_1.1.0       xfun_0.56         
## [21] sass_0.4.10        S7_0.2.1           otel_0.2.0         timechange_0.4.0  
## [25] cli_3.6.5          withr_3.0.2        magrittr_2.0.4     digest_0.6.39     
## [29] grid_4.5.2         rstudioapi_0.18.0  hms_1.1.4          lifecycle_1.0.5   
## [33] vctrs_0.7.1        evaluate_1.0.5     glue_1.8.0         farver_2.1.2      
## [37] rmarkdown_2.30     tools_4.5.2        pkgconfig_2.0.3    htmltools_0.5.9

Citations

Elo rating system. (n.d.). Wikipedia. https://en.wikipedia.org/wiki/Elo_rating_system

FIDE. (2024). FIDE rating regulations. https://handbook.fide.com/chapter/B022024

OpenAI. (2026). ChatGPT [Large language model]. https://chatgpt.com.

LLM Transcript

DATA 607- Assignment 5B - Chess ELO Calculations

Chanice McKenzie