Assignment 5B – ELO Calculations Approach

Objective

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

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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.

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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.

Sources:
- Glickman, M. E. A Comprehensive Guide to Chess Ratings: https://www.glicko.net/research/acjpaper.pdf
- Elo rating system (Wikipedia): https://en.wikipedia.org/wiki/Elo_rating_system
- The Elo Rating System for Chess and Beyond (YouTube, 2019): https://www.youtube.com/watch?v=AsYfbmp0To0

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Approximation Strategy

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.

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Implementation Steps

The analysis will proceed as follows:

1. Load Project 1 dataset
   • Load Project1_Chess_Summary.csv directly from the GitHub raw link using read.csv() to ensure reproducibility.
2. 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.
3. 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.
4. Compute performance difference
   • Compute:

\[ \text{Performance Difference} = \text{Actual Score} - \text{Expected Score} \]

• Positive values indicate overperformance relative to rating.
• Negative values indicate underperformance.

5. Rank players
   • Sort players by performance difference.
   • Report the top five overperformers and bottom five underperformers.

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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.