Chess ELO Calculations

Introduction

This analysis reuses the Project 1 chess tournament cross-table (tournamentinfo.txt) to calculate each player’s expected score for the tournament, based on the Elo rating system, and compares it to their actual score. The goal is to find the five players who most overperformed relative to what their pre-tournament rating predicted, and the five who most underperformed.

The Elo formula used to predict the outcome of a single game between two players is:

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

where \(E_A\) is player A’s expected score against player B (a number between 0 and 1), \(R_A\) is player A’s rating, and \(R_B\) is player B’s rating. A player’s expected score for the whole tournament is just the sum of this value across every real game they played.

Source for the formula: Elo Rating Algorithm. “The Elo Rating System for Chess and Beyond.” YouTube, 15 Feb. 2019, https://www.youtube.com/watch?v=AsYfbmp0To0.

Load and Parse the Tournament Data

This is the same parsing logic from Project 1: pull out each player’s pair number, name, state, total points, pre tournament rating, and the pair numbers of everyone they actually played (wins, losses, and draws only, byes, forfeits, and unplayed rounds are marked H, U, or X and don’t have a real opponent, so they’re excluded).

lines <- readLines("https://raw.githubusercontent.com/nowhyporque/data607DataAcquisitionAndManagement/refs/heads/main/Project1/tournamentinfo.txt")

Warning in
readLines("https://raw.githubusercontent.com/nowhyporque/data607DataAcquisitionAndManagement/refs/heads/main/Project1/tournamentinfo.txt"):
incomplete final line found on
'https://raw.githubusercontent.com/nowhyporque/data607DataAcquisitionAndManagement/refs/heads/main/Project1/tournamentinfo.txt'

data_lines <- lines[str_detect(lines, "\\|")]
data_lines <- data_lines[-(1:2)]

n_players <- length(data_lines) / 2
players <- vector("list", n_players)

for (i in seq_len(n_players)) {
  line1 <- data_lines[(i - 1) * 2 + 1]
  line2 <- data_lines[(i - 1) * 2 + 2]

  f1 <- str_trim(str_split(line1, "\\|")[[1]])
  f2 <- str_trim(str_split(line2, "\\|")[[1]])

  pair_num     <- as.integer(f1[1])
  name         <- f1[2]
  total_pts    <- as.numeric(f1[3])
  round_fields <- f1[4:10]

  state      <- f2[1]
  pre_rating <- as.integer(str_match(f2[2], "R:\\s*(\\d+)")[, 2])

  opp_matches   <- str_match(round_fields, "^[WLD]\\s*(\\d+)$")[, 2]
  opponent_nums <- as.integer(na.omit(opp_matches))

  players[[i]] <- tibble(
    pair_num = pair_num, name = name, state = state,
    total_pts = total_pts, pre_rating = pre_rating,
    opponents = list(opponent_nums)
  )
}

players_df <- bind_rows(players)
glimpse(players_df)

Rows: 64
Columns: 6
$ pair_num   <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, …
$ name       <chr> "GARY HUA", "DAKSHESH DARURI", "ADITYA BAJAJ", "PATRICK H S…
$ state      <chr> "ON", "MI", "MI", "MI", "MI", "OH", "MI", "MI", "ON", "MI",…
$ total_pts  <dbl> 6.0, 6.0, 6.0, 5.5, 5.5, 5.0, 5.0, 5.0, 5.0, 5.0, 4.5, 4.5,…
$ pre_rating <int> 1794, 1553, 1384, 1716, 1655, 1686, 1649, 1641, 1411, 1365,…
$ opponents  <list> <39, 21, 18, 14, 7, 12, 4>, <63, 58, 4, 17, 16, 20, 7>, <8…

Calculate Expected Score for Each Player

For every player, I look up the pre tournament rating of each real opponent they faced, run the Elo formula for each of those games, and add the results together. That sum is the player’s expected score for the tournament. Then I compare it to their actual score (the total_pts column already in the data) to get the difference.

rating_lookup <- setNames(players_df$pre_rating, as.character(players_df$pair_num))

expected_score_for <- function(own_rating, opponent_nums) {
  if (length(opponent_nums) == 0) return(0)
  opp_ratings <- rating_lookup[as.character(opponent_nums)]
  sum(1 / (1 + 10 ^ ((opp_ratings - own_rating) / 400)))
}

players_df <- players_df %>%
  rowwise() %>%
  mutate(
    expected_score = round(expected_score_for(pre_rating, opponents), 2),
    diff = round(total_pts - expected_score, 2)
  ) %>%
  ungroup()

results <- players_df %>%
  select(name, state, pre_rating, total_pts, expected_score, diff) %>%
  rename(
    `Player Name` = name,
    State = state,
    `Pre-Rating` = pre_rating,
    `Actual Score` = total_pts,
    `Expected Score` = expected_score,
    `Difference` = diff
  )

kable(head(results, 5))

Player Name	State	Pre-Rating	Actual Score	Expected Score	Difference
GARY HUA	ON	1794	6.0	5.16	0.84
DAKSHESH DARURI	MI	1553	6.0	3.78	2.22
ADITYA BAJAJ	MI	1384	6.0	1.95	4.05
PATRICK H SCHILLING	MI	1716	5.5	4.74	0.76
HANSHI ZUO	MI	1655	5.5	4.38	1.12

Five Players Who Most Overperformed

These are the players whose actual score beat their expected score by the widest margin, they did much better than their pre, tournament rating predicted.

overperformers <- results %>% arrange(desc(Difference)) %>% head(5)
kable(overperformers)

Player Name	State	Pre-Rating	Actual Score	Expected Score	Difference
ADITYA BAJAJ	MI	1384	6.0	1.95	4.05
ZACHARY JAMES HOUGHTON	MI	1220	4.5	1.37	3.13
ANVIT RAO	MI	1365	5.0	1.94	3.06
JACOB ALEXANDER LAVALLEY	MI	377	3.0	0.04	2.96
AMIYATOSH PWNANANDAM	MI	980	3.5	0.77	2.73

Five Players Who Most Underperformed

These are the players whose actual score fell short of their expected score by the widest margin — they did much worse than their pre-tournament rating predicted.

underperformers <- results %>% arrange(Difference) %>% head(5)
kable(underperformers)

Player Name	State	Pre-Rating	Actual Score	Expected Score	Difference
LOREN SCHWIEBERT	MI	1745	3.5	6.28	-2.78
GEORGE AVERY JONES	ON	1522	3.5	6.02	-2.52
JARED GE	MI	1332	3.0	5.01	-2.01
RISHI SHETTY	MI	1494	3.5	5.09	-1.59
JOSHUA DAVID LEE	MI	1438	3.5	4.96	-1.46

Conclusions

The Elo formula turns the gap between two players’ ratings into a probability of winning, and summing that probability across every game a player actually played gives their expected score for the tournament. Comparing that to their real score highlights who beat the odds and who didn’t.

The biggest overperformer was Aditya Bajaj, a 1384 rated player who was only expected to score about 1.95 points across the tournament but actually scored a full 6.0, four points better than predicted. On the other end, Loren Schwiebert, rated 1745, was expected to score around 6.28 points but only managed 3.5, nearly a three point shortfall. Since expected score is built entirely from pre tournament ratings, a large gap like this usually means the player’s rating didn’t reflect their actual playing strength at the time of this tournament, whether because they’d improved (or declined) since their rating was last updated, or the specific matchups they drew happened to favor (or not favor) them.