The objective of this assignment is to compute each player’s expected tournament score using the Elo rating model and compare it with their actual score from the Project 1 tournament.
We will then identify:
The five players who most overperformed relative to expectation
The five players who most underperformed relative to expectation
Data Source
This analysis builds directly on the structured dataset created in Project 1.
The dataset contains exactly one row per player with the following variables:
Player Name
Player State
Total Points (Actual Score)
Pre-Tournament Rating
Average Pre-Tournament Rating of Opponents
This dataset was exported as a CSV file and uploaded to GitHub to ensure reproducibility. The file will be loaded into R using the GitHub raw link.
Elo Expected Score Formula
The Elo rating system models the probability that Player A defeats Player B using a logistic function based on rating differences.
The expected score formula used in this assignment is:
\[
E_A = \frac{1}{1 + 10^{(R_B - R_A)/400}}
\]
Where:
(R_A) = Player A’s pre-tournament rating
(R_B) = Opponent’s pre-tournament rating
400 = scaling constant in the Elo system
This formulation implies that a 400-point rating difference corresponds to approximately a 10-to-1 expected win ratio.
In the standard Elo system, expected score is computed separately for each game using each opponent’s rating, and then summed across games.
However, the Project 1 dataset retains only the average pre-tournament rating of opponents, rather than the full match level opponent list. Therefore, this analysis approximates expected tournament score by applying the Elo expected score formula using the player’s rating versus their average opponent rating, and then scaling by the number of rounds.
The expected per-game score is computed as:
\[
E = \frac{1}{1 + 10^{(\overline{R_{opp}} - R_{player})/400}}
\]
The expected tournament score is then:
\[
\text{Expected Tournament Score} = n \times E
\]
Where:
(n) = number of rounds (games) in the tournament
({R_{opp}}) = average opponent pre-rating
This provides a transparent approximation consistent with the Elo logistic framework given the available Project 1 output.
Implementation Steps
The analysis will proceed as follows:
Load Project 1 dataset
Load Project1_Chess_Summary.csv directly from the GitHub raw link using read.csv() to ensure reproducibility.
Compute expected per-game probability
For each player, compute expected per-game score using the Elo formula and the player’s pre-rating versus the average opponent pre-rating.
Multiply by number of rounds
Since the tournament consists of seven rounds, the expected per-game probability will be multiplied by the number of rounds played to estimate the player’s total expected tournament score.
Positive values indicate overperformance relative to rating.
Negative values indicate underperformance.
Rank players
Sort players by performance difference.
Report the top five overperformers and bottom five underperformers.
Final Output
The final output will include:
Player Name
Pre-Tournament Rating
Actual Score
Expected Score (approximate)
Performance Difference
Players will be ranked to identify:
Top 5 Overperformers
Top 5 Underperformers
This approach builds directly upon the structured output from Project 1 and applies the Elo rating model to evaluate tournament performance relative to expectation.
Codebase
library(tidyverse)
── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr 1.2.0 ✔ readr 2.2.0
✔ forcats 1.0.1 ✔ stringr 1.6.0
✔ ggplot2 4.0.2 ✔ tibble 3.3.1
✔ lubridate 1.9.5 ✔ tidyr 1.3.2
✔ purrr 1.2.1
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
library(knitr)
1. Load Project 1 dataset
This analysis uses the cleaned Project 1 summary dataset (one row per player) containing:
Because the Project 1 output retains only the average opponent pre-rating (not the full opponent list), I approximated the expected tournament score by computing expected per-game score versus the average opponent rating, then multiplying by the number of rounds played.
In this tournament there are 7 rounds, so we use n_rounds = 7.
ggplot(results, aes(x = performance_diff)) +geom_histogram(bins =15) +labs(title ="Distribution of Performance Difference (Actual - Expected)",x ="Performance Difference",y ="Number of Players" )
6. Brief Interpretation
Players with the largest positive performance differences significantly exceeded their expected scores based on pre-tournament ratings and average opponent strength. In several cases, lower-rated players substantially outperformed rating-based expectations.
Conversely, the largest negative differences indicate higher-rated players who scored fewer points than predicted by the Elo model.
Because expected scores were calculated using average opponent ratings (rather than game-by-game calculations), these results represent an approximation of true Elo expectations. However, the ranking clearly highlights players whose tournament performance diverged most from rating-based predictions.
