Introduction

This assignment extends the chess tournament dataset from Project 1 by calculating each player’s expected score using the Elo rating system. The expected score represents the probability of a player winning or drawing against opponents, based on rating differences.

Since my Project 1 table only contains each player’s average opponent rating, I approximate their Elo expectation by treating all seven rounds as if they were played against an opponent of that average strength.

I then compare each player’s expected score to their actual score, identifying the five players who most overperformed and the five who most underperformed relative to expectations.

The Elo formula I use is taken from Wikipedia — Elo rating system (Mathematical details).

1. Load Data

The dataset from Project 1 is loaded below, showing the first few rows of player information.

chess_players <- read_csv("https://raw.githubusercontent.com/egabrielvice/assignments/main/chess_players.csv")
head(chess_players)
## # A tibble: 6 × 5
##   Name                State Points PreRating AvgOpponent
##   <chr>               <chr>  <dbl>     <dbl>       <dbl>
## 1 GARY HUA            ON       6        1794        1635
## 2 DAKSHESH DARURI     MI       6        1553        1503
## 3 ADITYA BAJAJ        MI       6        1384        1557
## 4 PATRICK H SCHILLING MI       5.5      1716        1604
## 5 HANSHI ZUO          MI       5.5      1655        1547
## 6 HANSEN SONG         OH       5        1686        1552

2. Elo Formula

The Elo expected score per game is defined as:

\[ E = \frac{1}{1 + 10^{(R_{opp} - R_{player})/400}} \]

where:
- \(R_{player}\) = the player’s rating,
- \(R_{opp}\) = the opponent’s rating.

Since the tournament had 7 rounds, I approximate each player’s total expected score as:

\[ E_{total} = 7 \times \frac{1}{1 + 10^{(\text{AvgOpponent} - \text{PreRating})/400}} \]

The R code below applies this formula to compute each player’s expected score and the difference between actual and expected results.

expected_prob <- function(Rp, Ro) {
  1 / (1 + 10 ^ ((Ro - Rp) / 400))
}

elo_results <- chess_players %>%
  mutate(
    ExpScore = 7 * expected_prob(PreRating, AvgOpponent),
    Diff     = Points - ExpScore
  )

3. Results

The table below shows the first ten players with their actual points, expected score, and difference.

elo_results %>%
  select(Name, State, PreRating, Points, AvgOpponent, ExpScore, Diff) %>%
  head(10) %>%
  kable(caption = "Preview of Elo Results (first 10 players)")
Preview of Elo Results (first 10 players)
Name State PreRating Points AvgOpponent ExpScore Diff
GARY HUA ON 1794 6.0 1635 4.998552 1.0014477
DAKSHESH DARURI MI 1553 6.0 1503 4.000242 1.9997582
ADITYA BAJAJ MI 1384 6.0 1557 1.888282 4.1117179
PATRICK H SCHILLING MI 1716 5.5 1604 4.590744 0.9092565
HANSHI ZUO MI 1655 5.5 1547 4.554232 0.9457676
HANSEN SONG OH 1686 5.0 1552 4.786714 0.2132858
GARY DEE SWATHELL MI 1649 5.0 1434 5.426070 -0.4260701
EZEKIEL HOUGHTON MI 1641 5.0 1508 4.777993 0.2220073
STEFANO LEE ON 1411 5.0 1525 2.391098 2.6089016
ANVIT RAO MI 1365 5.0 1544 1.841037 3.1589633

Next, I identify the five players who most overperformed relative to their expected score.

top_over <- elo_results %>%
  arrange(desc(Diff)) %>%
  select(Name, PreRating, Points, ExpScore, Diff) %>%
  mutate(ExpScore = round(ExpScore, 2), Diff = round(Diff, 2)) %>%
  head(5)

kable(top_over, caption = "Top 5 Overperformers (Actual − Expected)")
Top 5 Overperformers (Actual − Expected)
Name PreRating Points ExpScore Diff
ADITYA BAJAJ 1384 6.0 1.89 4.11
ZACHARY JAMES HOUGHTON 1220 4.5 1.18 3.32
ANVIT RAO 1365 5.0 1.84 3.16
JACOB ALEXANDER LAVALLEY 377 3.0 0.05 2.95
AMIYATOSH PWNANANDAM 980 3.5 0.68 2.82

Finally, I identify the five players who most underperformed relative to their expected score.

top_under <- elo_results %>%
  arrange(Diff) %>%
  select(Name, PreRating, Points, ExpScore, Diff) %>%
  mutate(ExpScore = round(ExpScore, 2), Diff = round(Diff, 2)) %>%
  head(5)

kable(top_under, caption = "Top 5 Underperformers (Actual − Expected)")
Top 5 Underperformers (Actual − Expected)
Name PreRating Points ExpScore Diff
ASHWIN BALAJI 1530 1.0 3.77 -2.77
LOREN SCHWIEBERT 1745 3.5 6.00 -2.50
GEORGE AVERY JONES 1522 3.5 5.75 -2.25
GAURAV GIDWANI 1552 3.5 5.51 -2.01
CHIEDOZIE OKORIE 1602 3.5 5.40 -1.90

4. Conclusion

Using the Elo expected-score formula, I estimated each player’s expected points based on their pre-rating compared to the average rating of their opponents. By comparing expected scores to actual tournament results, I identified the five players who most overperformed and the five who most underperformed relative to expectations.

This analysis illustrates how Elo ratings can be applied not only for ranking players but also for evaluating tournament performance. As described in Wikipedia’s Elo rating system, expected scores reflect the probability of winning or drawing against an opponent, and differences between actual and expected outcomes highlight players who performed above or below their rating-based predictions.

While minor differences exist across Elo implementations, the results consistently reveal which players exceeded expectations and which underperformed. Overall, this confirms the usefulness of the Elo system for benchmarking tournament performance.