Assignment 5B: Chess ELO Calculations

DATA 607 · Week 5

Author

Dillon Leeper

Published

October 4, 2026

Overview

This assignment builds on the Project 1 tournament data: the same 64-player, seven-round USCF tournament report I parsed for Project 1. This time, instead of just averaging each player’s opponent ratings, I use the rating difference in every game to calculate each player’s expected score for the tournament, compare it to their actual score, and find the five players who most overperformed and the five who most underperformed.

Approach

I reused the Project 1 parsing logic, with one addition. Project 1 only needed the opponent’s pair number from each round (the 39 in W 39), so it threw away the result letter. Here I kept both the letter and the number, so each round tells me who the player faced and how the game ended.

From there I:

Built one row per player per round, with the result letter and opponent number.
Kept only real games (W, L, D) and scored them 1, 0, and 0.5. Rounds marked B, H, U, or X are byes, unplayed rounds, or forfeits, so they have no opponent rating to compare against.
Looked up each opponent’s pre-tournament rating and calculated the expected score for every game with the Elo formula.
Summed expected and actual scores per player and took the difference.
Ranked players by that difference to find the top five over- and underperformers.

One thing I changed from my Approach write-up: I compare expected score to the points a player earned in games actually played, not their listed total. The total in the report includes bye and forfeit points (a half-point bye counts as 0.5, a full bye or forfeit win as 1). Those rounds have no opponent, so they can’t have an expected score. Leaving them on the actual side only would make players with byes look like they overperformed when they really just got free points. For most players the two numbers are the same.

The Elo expected score formula

For a player rated \(R_A\) facing an opponent rated \(R_B\), the expected score for that game is:

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

This is a number between 0 and 1. Two equally rated players each have an expected score of 0.5. A player rated 400 points higher has an expected score of about 0.91, meaning they’d be expected to score roughly 9 out of 10 points over 10 games against that opponent. Summing \(E_A\) across all of a player’s games gives their expected score for the tournament.

Source: Elo, A. E. (1978). The Rating of Chess Players, Past and Present. Arco Publishing. This is the same logistic formula used by FIDE and the USCF for expected scores.

Load the data

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library(tidyverse)
library(knitr)

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data_url <- paste0(
  "https://raw.githubusercontent.com/dillonleeper/",
  "DATA-607/main/assignments/week04/tournament-results.txt"
)

tournament_lines <- readLines(data_url, warn = FALSE)

This is the same file from Project 1, read from its public GitHub URL. It has 196 lines: a short header, then two lines per player.

Parse the players and rounds

Each player takes up two lines in the report. The first line has the pair number, name, total points, and seven round fields like W 39. The second has the state and the pre-tournament rating, written like R: 1794 ->1817. Provisional ratings like 1641P17 are cut down to just the number.

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player_line_numbers <- grep("^\\s*[0-9]+\\s*\\|", tournament_lines)

split_fields <- function(line) {
  trimws(strsplit(line, "|", fixed = TRUE)[[1]])
}

players <- map_dfr(player_line_numbers, function(i) {
  player_fields <- split_fields(tournament_lines[i])
  detail_fields <- split_fields(tournament_lines[i + 1])

  tibble(
    player_number = as.integer(player_fields[1]),
    player_name   = player_fields[2],
    state         = detail_fields[1],
    total_points  = as.numeric(player_fields[3]),
    pre_rating    = as.integer(str_match(detail_fields[2], "R:\\s*([0-9]+)")[, 2]),
    rounds        = list(player_fields[4:10])
  )
})

players |>
  head() |>
  transmute(player_number, player_name = str_to_title(player_name), state, total_points, pre_rating) |>
  kable(col.names = c("#", "Player", "State", "Total points", "Pre-rating"),
        caption = "First six players")

First six players
#	Player	State	Total points	Pre-rating
1	Gary Hua	ON	6.0	1794
2	Dakshesh Daruri	MI	6.0	1553
3	Aditya Bajaj	MI	6.0	1384
4	Patrick H Schilling	MI	5.5	1716
5	Hanshi Zuo	MI	5.5	1655
6	Hansen Song	OH	5.0	1686

Now each player’s seven round fields get split into a result letter and an opponent number, giving one row per player per round:

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rounds <- players |>
  select(player_number, rounds) |>
  unnest_longer(rounds, values_to = "round_field", indices_to = "round") |>
  mutate(
    result          = str_extract(round_field, "^[A-Z]"),
    opponent_number = as.integer(str_extract(round_field, "[0-9]+"))
  )

rounds |>
  filter(player_number == 1) |>
  select(round, round_field, result, opponent_number) |>
  kable(col.names = c("Round", "Raw field", "Result", "Opponent #"),
        caption = "Gary Hua's seven rounds")

Gary Hua’s seven rounds
Round	Raw field	Result	Opponent #
1	W 39	W	39
2	W 21	W	21
3	W 18	W	18
4	W 14	W	14
5	W 7	W	7
6	D 12	D	12
7	D 4	D	4

That’s Gary Hua’s seven rounds: five wins and two draws, matching his 6.0 points.

Calculate expected scores

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rating_lookup <- setNames(players$pre_rating, players$player_number)

games <- rounds |>
  filter(result %in% c("W", "L", "D")) |>          # real games only
  mutate(
    player_rating   = rating_lookup[as.character(player_number)],
    opponent_rating = rating_lookup[as.character(opponent_number)],
    actual          = recode(result, W = 1, D = 0.5, L = 0),
    expected        = 1 / (1 + 10^((opponent_rating - player_rating) / 400))
  )

games |>
  filter(player_number == 1) |>
  select(round, result, opponent_number, player_rating, opponent_rating, actual, expected) |>
  mutate(expected = round(expected, 3)) |>
  kable(col.names = c("Round", "Result", "Opponent #", "Gary's rating",
                      "Opponent rating", "Actual", "Expected"),
        caption = "Gary Hua's games, scored")

Gary Hua’s games, scored
Round	Result	Opponent #	Gary’s rating	Opponent rating	Actual	Expected
1	W	39	1794	1436	1.0	0.887
2	W	21	1794	1563	1.0	0.791
3	W	18	1794	1600	1.0	0.753
4	W	14	1794	1610	1.0	0.743
5	W	7	1794	1649	1.0	0.697
6	D	12	1794	1663	0.5	0.680
7	D	4	1794	1716	0.5	0.610

For example, in round 1 Gary (1794) beat player 39, who was rated 1436. With a 358-point rating edge, his expected score for that game was about 0.89, so the win only beat expectation by about 0.11.

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elo_results <- games |>
  group_by(player_number) |>
  summarise(
    games_played   = n(),
    actual_score   = sum(actual),
    expected_score = sum(expected)
  ) |>
  right_join(
    players |> select(player_number, player_name, state, total_points, pre_rating),
    by = "player_number"
  ) |>
  mutate(
    across(c(games_played, actual_score, expected_score), \(x) replace_na(x, 0)),
    difference = actual_score - expected_score
  ) |>
  select(player_number, player_name, state, pre_rating, total_points,
         games_played, actual_score, expected_score, difference) |>
  arrange(player_number)

Checks

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gary <- elo_results |> filter(player_number == 1)

stopifnot(
  nrow(elo_results) == 64,
  !anyNA(games$opponent_rating),                  # every opponent found
  all(games$expected > 0 & games$expected < 1),
  gary$actual_score == 6,
  # games-only score never exceeds the listed total (byes only add points)
  all(elo_results$actual_score <= elo_results$total_points)
)

# Each game appears twice in `games`, once from each player's side. The two
# sides' expected scores add up to exactly 1, and so do their actual results,
# so both totals should equal the number of distinct games.
tibble(
  Check = c("Total expected points", "Total actual points", "Games played"),
  Value = c(round(sum(games$expected), 2), sum(games$actual), nrow(games) / 2)
) |>
  kable(caption = "All three should match")

All three should match
Check	Value
Total expected points	204
Total actual points	204
Games played	204

The totals line up: 204 games were played, and both the expected and actual scores across the whole tournament add up to 204. Every game shows up twice in games (once from each player’s side), and the two sides always split exactly 1 point between them, in expectation and in reality. That’s a good sign the rating lookups and the formula are working.

Results

Top 5 overperformers

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show_cols <- c("Player", "State", "Pre-rating", "Games", "Actual", "Expected", "Difference")

over <- elo_results |>
  slice_max(difference, n = 5)

over |>
  transmute(player_name = str_to_title(player_name), state, pre_rating, games_played, actual_score,
            expected_score = round(expected_score, 2), difference = round(difference, 2)) |>
  kable(col.names = show_cols, caption = "Biggest overperformers (actual minus expected)")

Biggest overperformers (actual minus expected)
Player	State	Pre-rating	Games	Actual	Expected	Difference
Aditya Bajaj	MI	1384	7	6.0	1.95	4.05
Zachary James Houghton	MI	1220	7	4.5	1.37	3.13
Anvit Rao	MI	1365	7	5.0	1.94	3.06
Jacob Alexander Lavalley	MI	377	7	3.0	0.04	2.96
Stefano Lee	ON	1411	7	5.0	2.29	2.71

Top 5 underperformers

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under <- elo_results |>
  slice_min(difference, n = 5)

under |>
  transmute(player_name = str_to_title(player_name), state, pre_rating, games_played, actual_score,
            expected_score = round(expected_score, 2), difference = round(difference, 2)) |>
  kable(col.names = show_cols, caption = "Biggest underperformers (actual minus expected)")

Biggest underperformers (actual minus expected)
Player	State	Pre-rating	Games	Actual	Expected	Difference
Loren Schwiebert	MI	1745	7	3.5	6.28	-2.78
George Avery Jones	ON	1522	7	3.5	6.02	-2.52
Larry Hodge	MI	1270	6	1.0	3.40	-2.40
Jared Ge	MI	1332	7	3.0	5.01	-2.01
Rishi Shetty	MI	1494	7	3.5	5.09	-1.59

Everyone, visualized

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highlight <- c(over$player_number, under$player_number)

elo_results |>
  mutate(
    group = case_when(
      player_number %in% over$player_number  ~ "Top 5 over",
      player_number %in% under$player_number ~ "Top 5 under",
      TRUE ~ "Everyone else"
    ),
    label = fct_reorder(str_to_title(player_name), difference)
  ) |>
  ggplot(aes(x = difference, y = label, fill = group)) +
  geom_col() +
  geom_vline(xintercept = 0, color = "grey40") +
  scale_fill_manual(values = c("Top 5 over" = "#2c5f8a",
                               "Top 5 under" = "#c0392b",
                               "Everyone else" = "grey75")) +
  labs(
    title = "Actual score minus expected score",
    x = "Points above (right) or below (left) expectation",
    y = NULL, fill = NULL
  ) +
  theme_minimal(base_size = 10) +
  theme(legend.position = "top")

Round by round

The chart above shows where everyone ended up. This one shows how they got there. Each line is one player’s running total of points above or below expectation: it starts at zero, rises when the player beats their expected score in a game, and falls when they come up short. Byes and unplayed rounds leave the line flat, since they have no expected score. The top five over- and underperformers are highlighted, and everyone else is in gray. (The report only lists ratings before and after the tournament, so this tracks results against expectation rather than a round-by-round rating.)

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round_progress <- rounds |>
  select(player_number, round) |>
  left_join(games |> select(player_number, round, actual, expected),
            by = c("player_number", "round")) |>
  mutate(round_diff = replace_na(actual - expected, 0)) |>   # byes and unplayed rounds add 0
  bind_rows(tibble(player_number = players$player_number, round = 0L, round_diff = 0)) |>
  arrange(player_number, round) |>
  group_by(player_number) |>
  mutate(running_diff = cumsum(round_diff)) |>
  ungroup() |>
  left_join(elo_results |> select(player_number, player_name), by = "player_number") |>
  mutate(group = case_when(
    player_number %in% over$player_number  ~ "Top 5 over",
    player_number %in% under$player_number ~ "Top 5 under",
    TRUE ~ "Everyone else"
  ))

# Name labels at the end of each highlighted line, spaced so they don't overlap
end_labels <- round_progress |>
  filter(round == 7, group != "Everyone else") |>
  arrange(running_diff) |>
  mutate(
    label_y = accumulate(running_diff, \(prev, y) max(y, prev + 0.32)),
    label_y = label_y - (mean(label_y) - mean(running_diff)),
    label = paste0(str_to_title(player_name), " (", sprintf("%+.2f", running_diff), ")"),
    .by = group
  )

highlight_colors <- c("Top 5 over" = "#2c5f8a", "Top 5 under" = "#c0392b")

ggplot(round_progress, aes(x = round, y = running_diff, group = player_number)) +
  geom_hline(yintercept = 0, color = "grey50", linewidth = 0.4) +
  geom_line(data = \(d) filter(d, group == "Everyone else"),
            color = "grey80", linewidth = 0.5) +
  geom_line(data = \(d) filter(d, group != "Everyone else"),
            aes(color = group), linewidth = 1.1) +
  geom_point(data = \(d) filter(d, group != "Everyone else"),
             aes(color = group), size = 1.4) +
  geom_segment(data = end_labels,
               aes(x = 7.05, xend = 7.35, y = running_diff, yend = label_y, color = group),
               linewidth = 0.3) +
  geom_text(data = end_labels,
            aes(x = 7.4, y = label_y, label = label, color = group),
            hjust = 0, size = 3.1) +
  scale_color_manual(values = highlight_colors) +
  scale_x_continuous(breaks = 0:7, limits = c(0, 10.2), expand = c(0.01, 0)) +
  labs(
    title = "Running points above or below expectation, round by round",
    x = "Round (0 = start)", y = "Points vs. expectation", color = NULL
  ) +
  theme_minimal(base_size = 11) +
  theme(legend.position = "none",
        panel.grid.minor = element_blank())

Most of the gray lines stay close to zero, meaning those players scored about what their ratings predicted. The top overperformers built their lead over several rounds rather than in one game. Aditya Bajaj, for example, lost his first game, beat a lower-rated player, and then won five straight against higher-rated opponents. Jacob Alexander Lavalley’s line has a different shape: with a 377 rating he was expected to lose almost every game, so his losses barely moved the line and his three wins account for nearly all of his gain. The underperformers mostly slide over several rounds too, which points to a whole tournament below their rating rather than one bad game.

Full table

Show code

elo_results |>
  arrange(desc(difference)) |>
  transmute(player_number, player_name = str_to_title(player_name), state, pre_rating,
            total_points, games_played, actual_score,
            expected_score = round(expected_score, 2), difference = round(difference, 2)) |>
  kable(col.names = c("#", "Player", "State", "Pre-rating", "Total points",
                      "Games", "Actual", "Expected", "Difference"),
        caption = "All 64 players, sorted by difference")

All 64 players, sorted by difference
#	Player	State	Pre-rating	Total points	Games	Actual	Expected	Difference
3	Aditya Bajaj	MI	1384	6.0	7	6.0	1.95	4.05
15	Zachary James Houghton	MI	1220	4.5	7	4.5	1.37	3.13
10	Anvit Rao	MI	1365	5.0	7	5.0	1.94	3.06
46	Jacob Alexander Lavalley	MI	377	3.0	7	3.0	0.04	2.96
9	Stefano Lee	ON	1411	5.0	7	5.0	2.29	2.71
2	Dakshesh Daruri	MI	1553	6.0	7	6.0	3.78	2.22
52	Ethan Guo	MI	935	2.5	7	2.5	0.30	2.20
51	Tejas Ayyagari	MI	1011	2.5	7	2.5	1.03	1.47
24	Michael R Aldrich	MI	1229	4.0	7	4.0	2.55	1.45
37	Amiyatosh Pwnanandam	MI	980	3.5	5	2.0	0.77	1.23
5	Hanshi Zuo	MI	1655	5.5	7	5.5	4.38	1.12
1	Gary Hua	ON	1794	6.0	7	6.0	5.16	0.84
4	Patrick H Schilling	MI	1716	5.5	7	5.5	4.74	0.76
59	Sean M Mc Cormick	MI	853	2.0	6	1.0	0.41	0.59
50	Shivam Jha	MI	1056	2.5	6	2.0	1.42	0.58
58	Viraj Mohile	MI	917	2.0	6	1.0	0.43	0.57
61	Jezzel Farkas	ON	955	1.5	7	1.5	0.97	0.53
7	Gary Dee Swathell	MI	1649	5.0	7	5.0	4.58	0.42
60	Julia Shen	MI	967	1.5	5	1.0	0.60	0.40
38	Brian Liu	MI	1423	3.0	6	2.5	2.13	0.37
14	Bradley Shaw	MI	1610	4.5	7	4.5	4.18	0.32
36	Siddharth Jha	MI	1355	3.5	6	3.0	2.70	0.30
28	Sofia Adina Stanescu-Bellu	MI	1507	3.5	7	3.5	3.31	0.19
62	Ashwin Balaji	MI	1530	1.0	1	1.0	0.88	0.12
34	Michael Jeffery Thomas	MI	1399	3.5	7	3.5	3.44	0.06
40	Forest Zhang	MI	1348	3.0	7	3.0	2.94	0.06
23	Alan Bui	ON	1363	4.0	7	4.0	3.94	0.06
6	Hansen Song	OH	1686	5.0	7	5.0	4.94	0.06
8	Ezekiel Houghton	MI	1641	5.0	7	5.0	5.03	-0.03
44	Justin D Schilling	MI	1199	3.0	6	2.0	2.07	-0.07
56	Marisa Ricci	MI	1153	2.0	5	1.0	1.08	-0.08
12	Kenneth J Tack	MI	1663	4.5	6	4.0	4.11	-0.11
32	Joshua Philip Mathews	ON	1441	3.5	7	3.5	3.72	-0.22
16	Mike Nikitin	MI	1604	4.0	5	3.5	3.80	-0.30
57	Michael Lu	MI	1092	2.0	6	1.0	1.30	-0.30
21	Dinh Dang Bui	ON	1563	4.0	7	4.0	4.32	-0.32
19	Dipankar Roy	MI	1564	4.0	7	4.0	4.33	-0.33
41	Kyle William Murphy	MI	1403	3.0	4	2.0	2.36	-0.36
55	Alex Kong	MI	1186	2.0	6	1.0	1.44	-0.44
13	Torrance Henry Jr	MI	1666	4.5	7	4.5	4.95	-0.45
27	Gaurav Gidwani	MI	1552	3.5	6	3.5	4.00	-0.50
49	Michael J Martin	MI	1291	2.5	5	2.0	2.54	-0.54
18	David Sundeen	MI	1600	4.0	7	4.0	4.59	-0.59
26	Max Zhu	ON	1579	3.5	7	3.5	4.10	-0.60
39	Joel R Hendon	MI	1436	3.0	7	3.0	3.62	-0.62
17	Ronald Grzegorczyk	MI	1629	4.0	7	4.0	4.66	-0.66
47	Eric Wright	MI	1362	2.5	7	2.5	3.19	-0.69
53	Jose C Ybarra	MI	1393	2.0	3	1.0	1.72	-0.72
11	Cameron William Mc Leman	MI	1712	4.5	7	4.5	5.34	-0.84
63	Thomas Joseph Hosmer	MI	1175	1.0	5	0.5	1.43	-0.93
22	Eugene L Mcclure	MI	1555	4.0	6	3.5	4.48	-0.98
48	Daniel Khain	MI	1382	2.5	5	1.5	2.53	-1.03
29	Chiedozie Okorie	MI	1602	3.5	6	3.5	4.60	-1.10
20	Jason Zheng	MI	1595	4.0	7	4.0	5.13	-1.13
33	Jade Ge	MI	1449	3.5	7	3.5	4.64	-1.14
64	Ben Li	MI	1163	1.0	7	1.0	2.27	-1.27
43	Robert Glen Vasey	MI	1283	3.0	7	3.0	4.33	-1.33
45	Derek Yan	MI	1242	3.0	7	3.0	4.37	-1.37
35	Joshua David Lee	MI	1438	3.5	7	3.5	4.96	-1.46
31	Rishi Shetty	MI	1494	3.5	7	3.5	5.09	-1.59
42	Jared Ge	MI	1332	3.0	7	3.0	5.01	-2.01
54	Larry Hodge	MI	1270	2.0	6	1.0	3.40	-2.40
30	George Avery Jones	ON	1522	3.5	7	3.5	6.02	-2.52
25	Loren Schwiebert	MI	1745	3.5	7	3.5	6.28	-2.78

Conclusions

The biggest overperformer was Aditya Bajaj, who started with a pre-tournament rating of 1384 and was expected to score about 1.9 points, but actually scored 6, beating expectation by 4.05 points. The biggest underperformer was Loren Schwiebert (rated 1745), who was expected to score about 6.3 points and scored 3.5.

Four of the five overperformers came in rated below the middle of the field, and the fifth (Stefano Lee, 1411) was right at it (the median pre-rating is 1407), while the underperformers are more of a mix. Some of the overperformance is probably real improvement, but some of it is how ratings work. A low or provisional rating is a rough estimate, so a player whose rating hasn’t caught up to their actual skill will look like a big overperformer. Jacob Alexander Lavalley is the clearest example: his pre-tournament rating of 377 was provisional, based on only 3 games, so the formula expected him to score almost nothing, and his rating jumped to 1076 after this tournament. Comparing each player’s difference to their post-tournament rating change would be a good next step, since the USCF uses a version of this same expected-score math to update ratings.

The full results are saved as elo-results.csv.

Data source

The tournament report was supplied by the course instructor and is stored in my repo as tournament-results.txt (from Project 1).

AI Disclosure and Citation

Claude (Opus 5.5, accessed through claude.ai) helped write the code and text in this document, including extending the Project 1 parser to keep each round’s result, the expected-score calculation and checks, the over/underperformer tables and charts (including the round-by-round chart), the decision to compare expected score against points from games actually played, and the document’s styling and table formatting.

Anthropic. (2026). Claude (Opus 5.5) [Large language model]. https://claude.ai/. Accessed October 3–4, 2026.