This report uses the chess tournament cross-table from Project 1 (64 players, 7 rounds) to ask which players scored more, and which scored fewer, points than their ratings predicted. For every game we compute the expected score with the Elo formula from the pre-tournament ratings of the two players, add up each player’s expected points, and subtract them from the points they actually scored. We then list the five biggest overperformers and the five biggest underperformers.
Approach
I read the tournament text file and pulled out each player’s information and game results. I matched each game with the opponent’s record to check that the results agreed. Next, I used the players’ ratings to calculate the expected score for each game. I added those expected scores for each player, compared them with the points the player actually earned, and identified the players with the largest differences.
Formula and sources. The expected score of a player with rating \(R_A\) against an opponent with rating \(R_B\) is
\[E_A = \frac{1}{1 + 10^{(R_B - R_A)/400}}\]
Source: Wikipedia contributors, “Elo rating system”, https://en.wikipedia.org/wiki/Elo_rating_system (accessed September 24, 2026). The assignment also points to the video “The Elo Rating System for Chess and Beyond” (February 15, 2019).
Data.tournamentinfo.txt is the Project 1 text file, stored in this GitHub repository so the report is reproducible.
Step 1: Read the raw text and learn its layout
The file is not a table. Each player takes two lines (a player line and a detail line) followed by a separator line. We first label every line.
# A tibble: 6 × 5
pair_num name state total_pts pre_rating
<int> <chr> <chr> <dbl> <int>
1 1 GARY HUA ON 6 1794
2 2 DAKSHESH DARURI MI 6 1553
3 3 ADITYA BAJAJ MI 6 1384
4 4 PATRICK H SCHILLING MI 5.5 1716
5 5 HANSHI ZUO MI 5.5 1655
6 6 HANSEN SONG OH 5 1686
Ten players have a provisional rating in the file (for example R: 1220P13, a rating based on few games). We use the number before the P.
Step 3: One row per game
A result cell such as W 39 means “won against player 39”. Cells with only a letter (H, B, U, X) are rounds without an opponent: a half-point bye, a full-point bye, an unplayed round and a forfeit.
stopifnot(nrow(games) ==448)stopifnot(sum(is.na(games$opp_num)) ==40)stopifnot(all(is.na(games$opp_num) == games$result %in%c("H", "B", "U", "X")))# Mirror test: if A beat B in round r, B must show a loss to A in round rplayed <- games |>filter(!is.na(opp_num))mirror <- played |>rename(pair_num = opp_num, opp_num = pair_num) |># swap the two columnsmutate(mirror_result =case_when(result =="W"~"L", result =="L"~"W", TRUE~"D")) |>select(pair_num, opp_num, round, mirror_result)joined <-inner_join(played, mirror, by =c("pair_num", "opp_num", "round"))stopifnot(nrow(joined) ==nrow(played), all(joined$result == joined$mirror_result))# Points check: 204 points from played games + 16 from byes = 220 total pointsstopifnot(sum(games$result =="W") +0.5*sum(games$result =="D") ==204)stopifnot(0.5*sum(games$result =="H") +sum(games$result %in%c("B", "X")) ==16)
Step 4: Elo expected score for every game
For rounds without an opponent there is nothing to predict, so we treat them as neutral: the expected points equal the actual points. A bye therefore adds the same amount to both totals and cannot make a player look like an over- or underperformer.
elo_expected <-function(rating, opp_rating) {1/ (1+10^((opp_rating - rating) /400))}# Properties of the formulastopifnot(elo_expected(1500, 1500) ==0.5)stopifnot(near(elo_expected(1900, 1500), 0.9091, tol =1e-4))stopifnot(near(elo_expected(1900, 1500) +elo_expected(1500, 1900), 1))points <-c(W =1, D =0.5, L =0, H =0.5, B =1, X =1, U =0)games_elo <- games |>left_join(select(players, pair_num, rating = pre_rating), by ="pair_num") |>left_join(select(players, opp_num = pair_num, opp_rating = pre_rating), by ="opp_num") |>mutate(actual =unname(points[result]),expected =coalesce(elo_expected(rating, opp_rating), actual))stopifnot(nrow(games_elo) ==448)stopifnot(near(sum(games_elo$actual), 220), near(sum(games_elo$expected), 220))# Hand check: Gary Hua (pair 1) expected about 5.16 points, scored 6gary <- games_elo |>filter(pair_num ==1)stopifnot(near(sum(gary$expected), 5.1616, tol =1e-4))stopifnot(near(mean(gary$opp_rating), 1605.29, tol =1e-2)) # Project 1 example: 1605
Step 5: Expected and actual points per player
player_summary <- games_elo |>summarise(games_played =sum(!is.na(opp_num)),actual =sum(actual),expected =sum(expected),.by = pair_num) |>left_join(select(players, pair_num, name, state, pre_rating), by ="pair_num") |>mutate(diff = actual - expected) |>select(pair_num, name, state, pre_rating, games_played, actual, expected, diff)stopifnot(nrow(player_summary) ==64)# Actual points must reproduce the official totals in the filestopifnot(all(near(player_summary$actual, players$total_pts[match(player_summary$pair_num, players$pair_num)])))stopifnot(near(sum(player_summary$diff), 0)) # zero-sum: every game gives and takes the samestopifnot(sum(player_summary$games_played <7) ==23)
Step 6: The five biggest over- and underperformers
top5 <- player_summary |>slice_max(diff, n =5)bottom5 <- player_summary |>slice_min(diff, n =5)show <-function(d) { d |>transmute(Player = name, State = state, `Pre-rating`= pre_rating,Games = games_played, Actual = actual,Expected =round(expected, 2), Difference =round(diff, 2)) |> knitr::kable()}
Overperformers (actual points above the expectation):
show(top5)
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
Underperformers (actual points below the expectation):
show(bottom5)
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
2.0
4.40
-2.40
JARED GE
MI
1332
7
3.0
5.01
-2.01
RISHI SHETTY
MI
1494
7
3.5
5.09
-1.59
extremes <-bind_rows(over = top5, under = bottom5, .id ="group")ggplot(extremes, aes(reorder(name, diff), diff, fill = group)) +geom_col() +coord_flip() +scale_fill_manual(values =c(over ="#1F4E79", under ="#C0504D"),labels =c(over ="Overperformed", under ="Underperformed")) +labs(title ="Five biggest over- and underperformers vs. Elo expectation",x =NULL, y ="Actual minus expected points", fill =NULL) +theme_minimal() +theme(legend.position ="bottom")
The overperformers were mostly low-rated players (average rating about 1151) and the underperformers were mostly high-rated (about 1473). Some low ratings are unreliable: Jacob Alexander Lavalley’s rating of 377 is marked provisional (based on only 3 games), and so is Zachary James Houghton’s 1220. Aditya Bajaj’s 1384 is not provisional and his rating rose to 1640 after the tournament, so his result reflects real strength beyond his old rating.
Notes and limitations
Byes and forfeits (40 rounds) have no opponent, so they are treated as neutral (expected points equal actual points).
Provisional ratings (10 players) are noisy; large differences for those players partly reflect a bad starting rating.
Games played differ: 23 players played fewer than 7 games, so they cannot build up large positive or negative differences.
Formula variants: other implementations of Elo differ in small details; this report uses the standard logistic formula with a scale of 400.
What did the analysis show? TODO: name the biggest over- and underperformer and explain what “expected score” means.
Why might low-rated players overperform? TODO: think about provisional ratings, improving players, and regression to the mean.
What are the limits? TODO: byes, few games for some players, one tournament only, one formula.
How would you extend or verify this? TODO: e.g. compare with the post-tournament ratings in the file, use several tournaments, or use a different Elo implementation.
AI Use
Anthropic. (2026). Claude Sonnet 5 [Large language model]. https://claude.ai. Accessed September 24, 2026.
I used Claude to help me understand the Elo formula and regular expressions, work through parts of the R code, and proofread my explanations. I reviewed the code, checked the results against calculations, and revised the writing before submitting the report.
